The Logistic Map

Chaotic Growth with the Logistic Model of P.-F. Verhulst
Hugo Pastijn
Department of Mathematics, Royal Military Academy B-1000 Brussels, Belgium
Hugo.Pastijn@rma.ac.be

Summary. Pierre-Fran¸ois Verhulst was born 200 years ago. After a short biograc phy of P.-F. Verhulst in which the link with the Royal Military Academy in Brussels is emphasized, the early history of the so-called “Logistic Model” is described. The relationship with older growth models is discussed, and the motivation of Verhulst to introduce diﬀerent kinds of limited growth models is presented. The (re-)discovery of the chaotic behaviour of the discrete version of this logistic model in the late previous century is reminded. We conclude by referring to some generalizations of the logistic model, which were used to describe growth and diﬀusion processes in the context of technological innovation, and for which the author studied the chaotic behaviour by means of a series of computer experiments, performed in the eighties of last century by means of the then emerging “micro-computer” technology.

1 P.-F. Verhulst and the Royal Military Academy in Brussels
In the year 1844, at the age of 40, when Pierre-Fran¸ois Verhulst on November c 30 presented his contribution to the “M´moires de l’Acad´mie” of the young e e
Belgian nation, a paper which was published the next year in “tome XVIII” with the title: “Recherches math´matiques sur la loi d’accroissement de la e population” (mathematical investigations of the law of population growth), he did certainly not know that his work would be the starting point for further research by Raymond Pearl and Lowell J. Reed [10, 11], by the famous
A.-J. Lotka [8] and independently by Volterra [16] and later by V.A. Kostitzin [5], in the ﬁelds of mathematical biology, biometry, and demography.
It was in these M´moires that he introduced a growth model for a closed e population (no immigration, no emigration) facing a living environment with limited resources for the subsistence of its members. The purpose was to predict the demographic evolution of the young Belgian society, and to answer the question about the maximum population size sustainable with these limited resources. He certainly did not expect that more than one century later, the study of a discrete version of this model would give rise to a new ﬁeld in science: chaos theory. By the time he presented his paper, he was already a member of the “Acad´mie” (Academy of Sciences) and “Professeur e d’Analyse a l’Ecole Militaire de Belgique” – Professor of Calculus at the Mil` itary Academy – which had been founded in 1834, and which became later

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our present Royal Military Academy (Ecole Royale Militaire - Koninklijke
Militaire School) in Brussels.
In 1844 he had already a remarkable career behind him. Pierre-Fran¸ois c Verhulst was born in Brussels, October 28, 1804. This year we commemorate the 200th anniversary of his birth. He was a member of a family that neglected nothing to facilitate study opportunities for the young boy. In the Ath´n´e e e
(secondary school – high school) of Brussels he received in August 1822 the ﬁrst prize of mathematics, shared with Plateau and Nerenburger, becoming later his colleagues in the class of sciences of the Acad´mie. Quetelet will e remind us later the excellent reputation of this Ath´n´e of Brussels, which e e previously as “Lyc´e” during the “Empire” period, had been the school of e several future polytechnicians. In September 1822, the young Verhulst wanted to register for the “exact sciences” at the university of Ghent, although he had not yet ﬁnished the complete curriculum at the Ath´n´e in Brussels. e e
In Ghent too he completed his studies successfully after three years, with a doctoral dissertation about the reduction of binomial equations. He obtained his doctoral degree on August 3, 1825 (see also “Pierre-Fran¸ois Verhulst’s c ﬁnal triumph” by J. Kint et al. in the present book).
After some teaching duties at the “Mus´e des Sciences et des Lettres” in e Brussels, he went to Italy to recover from fatigue and exhaustion, just before the Belgian Revolution of 1830 broke out. In the short period 1830-1831 he hardly thought about mathematics. He came back to Belgium and in spite of his illness, he enrolled in the army to participate in the battle against Holland. In 1832 he agreed to help Quetelet to establish the mortality tables for the young Belgian state. This collaboration with Quetelet, who was one of the ﬁrst professors of the newly founded Military Academy in Brussels, led in 1834 to join him to the Military Academy ﬁrst as a “R´p´titeur” of “Analyse” (cale e culus) without ﬁnancial remuneration. Very quickly he became professor at the Military Academy. Quetelet, in his “Notice sur Pierre-Fran¸ois Verhulst” c published in the “Annuaire de l’Acad´mie royale des Sciences, des Lettres et e des Beaux-Arts de Belgique” [12], is mentioning the care with which Verhulst prepared and permanently updated the lecture notes. Unfortunately to our knowledge no copies of these notes are still available in the present archives of the Royal Military Academy.
In 1841, after he bought at a public sale, an old issue of a book by Legendre about elliptic functions, he published a compilation of what was currently known about that subject in his book “Trait´ ´l´mentaire des fonctions ele ee liptiques”. This book is still available in the present collection of the Royal
Military Academy. After the publication of this book he is appointed as
“correspondant de la section des sciences” of the Acad´mie royale on May 7, e 1841. In December of the same year he is appointed member of the Acad´mie e royale, to replace Garnier, his former professor at the University of Ghent.
From that year on he developed an ever increasing interest for the application of mathematics in a political context. It was probably after the publication of

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Quetelet’s “Essai de physique sociale” that he got convinced about the idea that the sum of obstacles against an unlimited growth of a closed population is increasing proportionally to the square of the population level currently reached. It was the famous Fourier (“Th´orie de la Chaleur”) who made e an analogous proposal in the introduction of the Tome I of his “Recherches statistiques sur Paris” (1835), and Quetelet urged Verhulst to submit this hypothesis to the investigation of empirical data available for Belgium. The results of the early research of Verhulst on this subject have been published already in 1838, in Tome X of the journal “Correspondance Math´matique e et Physique”, with Quetelet as chief editor. The idea of what Verhulst called
“the logistic growth model” (“la courbe logistique”) was born.
In 1845 his communication to the Acad´mie with the title “Recherches e math´matiques sur la loi d’accroissement de la population” (Mathematical e investigations about the law of population growth) was published in Tome
XVIII of the report of the Acad´mie. A second version of a growth model is e presented by him on May 15, 1846, to the Acad´mie, in which he is actually e criticizing the logistic model presented by himself about one year earlier. The text was published in Tome XX of the M´moires de l’Acad´mie in 1847. The e e self-criticism about the logistic model in this publication, and the emphasis
Quetelet later puts in his “Notice sur Pierre-Fran¸ois Verhulst” [12] on Verc hulst’s hesitation and his own reluctance to accept the applicability of the logistic model, are probably the main reasons why after Verhulst’s death the logistic model was entirely forgotten for a long time.
In 1848 the King of the Belgians appointed him as President of the
Acad´mie royale des Sciences, des Lettres et des Beaux-Arts de Belgique. e Although his health condition became ever worse, he continued to deliver his lectures at the Military Academy and to take oﬃce as the President of the
Acad´mie royale de Belgique. On February 15, 1849, P.-F. Verhulst died at the e age of 44. One of his last publications of which there is still a copy available in the library of the Royal Military Academy is his very modest booklet of 1847
“Le¸on d’Arithm´tique d´di´e aux candidats aux ´coles sp´ciales” (Lesson in c e e e e e
Arithmetics to the candidates of the “special schools”) on 72 pages.

2 The Exponential Growth Process
Until the end of the 18th century human and raw material resources were seemingly so unlimited from a Western point of view, that really no obstacles were supposed to exist for the development of human activities and for the growth of human population. This mental state was at the basis of the industrial revolution. Engineers and other scientists considered almost everything to be known and almost everything could be achieved by man without the need for ecological considerations, related to constraints on natural resources.
With these ideas in mind it is quite natural to make very optimistic forecasting about the growth of an economic system and about human population.

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So, when dealing with human population growth, Malthus suggested that the growth speed of a population is proportional to the current population level.
When no other external constraints on the growth speed are considered, then the continuous time model corresponding to this hypothesis is: dx/dt = rx, with x denoting the population level at time t. The solution of this diﬀerential equation is an exponential function x(t) = x(0) exp(rt). The exponential growth model is clearly not appropriate to describe the evolution of a population over a long period of time, even if it approximates suﬃciently the growth phenomenon during a certain period (for instance during the start-up episode). This is essentially the consequence of the hypothesis of a constant growth rate r during the whole lifetime of the process, and of independence of this growth rate with regard to the current population level at time t.

3 Limited Growth Models
If we consider the coeﬃcient r as roughly the diﬀerence between the birth rate and the death rate (B − D), then this means that natality and mortality in a population are independent both on the age and on the level (density) of this population. This seems not to be true in the real world. The natality rate is mostly decreasing for higher population densities, whereas the mortality rate is generally increasing for higher densities. The most simple assumption about these relationships is (with positive coeﬃcients): B = b − bx and
D = d + dx.
Substitution into the Malthusian equation yields dx/dt = ex − f x2 with e = b − d and f = b + d.
This means that the obstacle against an unlimited growth rate is proportional to the square of the current population level at time t. Another way to obtain the same form for the diﬀerential equation is to consider the coeﬃcient r in the Malthusian equation as dependent (and decreasing) on x, instead of being constant. The most simple dependence would then be: r = g − hx. In this case we obtain: h dx
=g 1− x x, dt g with the same observation about the obstacle against unlimited growth. Of course it is possible to imagine more sophisticated slowdown functions relating the growth rate to the current population level at time t. For example such a slowdown function could be: x m−1 r = sg 1 −
,
k with s = −1 for m < 1 , s = 1 for m > 1 , and g, k, m real valued parameters.

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Notice that m = 2 is yielding the same quadratic right hand side of the equation as we mentioned previously. The particular case for m tending to 1 is known as the well-known Gompertz model. For all values of m diﬀerent from
1 and 2, the model is now known as the generalized logistic growth model.
For other old variants and generalizations we refer for instance to Lebreton and Millier [6], to De Palma et al. [1] and to Kinnunen and Pastijn [4]. For m = 2 we actually obtain the model introduced by P.-F. Verhulst in 1844 and which he called the logistic growth model (“la courbe logistique”).

4 The Logistic Growth Process
For continuous time, this process is described by a diﬀerential equation, which is a special case of the Riccati type. The solution is straightforward: for x dx = gx 1 − dt k

the closed form solution is x(t) =

k k − x(0) with C =
.
1 + C exp(−gt) x(0) The parameter k is the maximum size of the population, or the asymptotic value of x(t). This is the closed form of the continuous time logistic growth curve. Although there are two parameters, certain morphological aspects of this curve are rather rigid. So, for instance the only existing inﬂexion point, when x(0) is less than a certain value related to the equation parameters, has always the same ordinate. This was one of the main reasons for introducing variants and generalizations of this simple model in the late previous century, in order to have more ﬂexibility for ﬁtting the model to experimental data.
The reason why Verhulst called this curve “la courbe logistique” in his communication of November 30, 1844, is not clear. He does not give any explanation. One might guess that he refers to the term logistics, related to transportation and distribution in the supply chain of an army, analogous to the supply of subsistence means of a population which he considered to be limited. The term logistic was then already to a certain extent in use in the military environment. He could have been familiar with it, through his military contacts in the Military Academy in Brussels. Another possible root of the term logistic could have been the French word “logis” (place to live) which was of course related to the limited resources for subsistence of a population, Verhulst was dealing with in his model. It is however pure speculation, although the term was still in use in the Belgian army until the mid 20th century as a rank of a non-commissioned oﬃcer called “Mar´chal des e Logis”. Another explanation – probably the most likely one – is related to the
Greek word λoγσπκoζ, which means “the art of computation” (see also the

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“Dictionnaire Quillet de la langue fran¸aise” of 1961 for this meaning of the c French word “logistique”). With his high school education, where Greek and
Latin were key subjects at that time, he certainly must have known this term.
When we adopt this explanation, Verhulst simply called his curve “logistique” because it enabled him to predict the future population of Belgium – during the era without computers – by simple computations.
In the second degree right hand side of the equation, the slowdown term
−(g/k)x2 can be interpreted as the result of the interacting competition between the individuals of the population. This competition is proportional to the number of potential encounters per time unit, and is therefore proportional to x. This interpretation is of course a bit simplistic because it doesn’t take into account that the major reason for slowing down the growth speed is exogenously imposed by the limited capacity of the closed “adiabatic” world we are focusing on. A more chemico-physical interpretation of the right hand side is that the relative growth rate for this logistic model is proportional to the currently available non transformed resources (k − x). This idea stems from the dynamics of autocatalytic chemical reactions. Therefore in chemistry, the logistic curve is often referred to as the autocatalytic function. This last interpretation is perhaps a more fundamental one.
This theoretical justiﬁcation and the marvellous ﬁt of this model to real world data of some ﬁrst applications in economics and demography, let Kostitzin in 1937 [5] write: “Une population ferm´e tend vers une limite qui e ne d´pend que des coeﬃcients vitaux ; elle est ind´pendante de la valeur e e initiale x(0)”. This optimistic view on the self-regulating mechanism of human population growth is inspired by the conviction that the logistic model is of a universal validity and also by the bare mathematical fact that the asymptotic attractor of this model is always locally stable, when these “vital” coeﬃcients are positive – which is no restricting condition for real world growth processes. However, the simple outlook of this logistic equation makes us forget the complexity of the mechanisms in evolutionary processes.
With the model Verhulst introduced in 1844 he predicted that the maximum size of the Belgian population would be six million and six hundred thousand individuals. Presently Belgium has a population of roughly eleven million. In his communication of 1846, he adapts his logistic growth model.
The solution of the new diﬀerential equation is no logistic function any more.
Now his prediction of the maximum size of the Belgian population is about nine million and four hundred thousand people, which is remarkably closer already to the present population level of Belgium. The main diﬀerence between both models is the following in Verhulst’s own wordings. In the logistic model the sum of obstacles against unlimited growth is proportional to the excess population. This excess population is the diﬀerence between the current population at time t and a minimum level of the population which is sustainable by means of a given number of available resources, which are considered as constant. In the model of 1846 he considers the obstacles against unlimited

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growth to be proportional to the ratio of the excess population and the total population at time t. It is now obvious that the logistic model was not the most eﬀective to predict the long-term evolution of the Belgian population.
This continuous time model is ﬁnally not as universally valid as it was sometimes considered. In addition, it is now recognized that the continuous time model does not always reﬂect reality. When there are for instance jump-wise simultaneous behavioural changes of all the individuals of the population, the structural dynamics of the population may fundamentally change. This has been “re-discovered” in 1974 and published in 1976 by R. May [9]. When we construct a discrete version of the logistic diﬀerential equation, for instance by applying the common Euler method for numerical integration, then we obtain a discrete form of the logistic growth process: x(t + 1) = x(t) 1 + g

1 − x(t) k .

This equation is describing the evolution of a population which is progressing jump-wise with equally time spaced intervals. The memory of this system is only one time unit. This means that the future of this system only depends on the “now and here”, and that the role of the grandparents is instantaneously neglected. All individuals of the population have the same and synchronized reproductive behaviour. The importance of this model is due to its peculiar behaviour for some values of its parameters. This model has been extensively studied by May [9], and was one of the ﬁrst simple models to illustrate the phenomenon of chaos. With the advent of the so-called “micro-computers” of the eighties of last century, it became very easy to generate illustrations of this “chaotic” behaviour.

5 Attractors for the Discrete Logistic Model
If a dynamic system deﬁned by diﬀerence equations is allowed to evolve over a long time, starting from diﬀerent initial conditions, the information about these initial conditions may disappear as time is going on. From a set of diﬀerent initial conditions the system may tend to the same restricted region.
This restricted region is called an attractor, whereas the set of initial points that is “attracted” by this attractor is called the basin of attraction. We know that there are three types of attractors: static or ﬁxed point attractors, limit cycles or periodic attractors, and “chaotic” attractors. These three types of attractors have been illustrated for the equation x(t + 1) = rx(t)[1 − x(t)] which is a simpliﬁed form of the discrete version of the logistic model. These illustrations are widely present in the literature of the eighties of last century, and showed the existence of what we now call chaos, for values of the

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parameter r beyond 3. This logistic model is a member of a quadratic family
[the right hand side is of the second degree in x(t)]. It is also a member of a larger family of single peaked functions in the right hand side of the equation, for which general properties with respect to the chaotic behaviour have been studied. Last century many variants and generalizations of the logistic model have been introduced to describe the diﬀusion of new products and of technological innovations. These models have been summarized by De Palma et al. [1].
The most well-known are those of Gompertz (see supra), Blumberg with the diﬀerential equation dx/dt = rxa (1 − xb /k), Bertalanﬀy with the diﬀerential equation dx/dt = r1 xm − r2 xn , Bass with the diﬀerential equation dx/dt =
(a + bk/x)(K − x), all with positive parameters.
The Bass model was describing the evolution of several consumer goods markets in the USA (refrigerators, TV-sets, air-conditioners,. . . ) during the second half of last century. Several discrete versions of these models have been studied by the author, and their chaotic behaviour illustrated [4]. It was then also announced that variants and generalizations of these models, used for the description of the substitution process of old by new technologies (Blackman–
Fisher–Pry), and for the evolution of commercial naval transportation and railways in the USA (Sharif–Kabir), have chaotic attractors.
In the meantime, the study of chaos has achieved a certain degree of maturity, conditions for its generation having been discovered in a wide category of discrete models [7] and chaos itself having been considered in the general context of fractal geometry.

6 Conclusion
The maturity of the ﬁeld of chaos theory, and the fact that chaotic behaviour now pervades almost all the sciences, is an argument to include this topic in the future curricula of our engineering and science students. This inclusion is possible in a very early stage of the student’s curriculum. The minimal prerequisites are related to basic calculus. The logistic model of Verhulst still nowadays plays an important role in the ﬁrst introduction of chaos theory to undergraduate students (“Encounters with chaos”, Denny Gulick, 1992). We can be conﬁdent that through these undergraduate courses of chaos theory, the ideas of Verhulst will survive in another format however than for the purpose they were originally introduced. But that happens quite often in the history of science.

References
1. A. De Palma, F. Droesbeke, Cl. Lefevre, C. Rosinski: Mod`les math´matiques e e de base pour la diﬀusion des innovations, Jorbel, Vol 26, No 2, 1986, pp 37–69

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2. D. Gulick: Encounters with chaos (McGraw–Hill, New York 1992)
3. T. Kinnunen, H. Pastijn: Chaotic Growth – attractors for the logistic model of
P.-F. Verhulst. In: Revue X (RMA Brussels 1986), 4, pp 1–17, 1986
4. T. Kinnunen, H. Pastijn: The chaotic behaviour of growth processes, ICOTA proceedings, Singapore, 1987
5. V.A. Kostitzin: Biologie math´matique (Armand Colin, Paris 1937) e 6. J.D. Lebreton, C. Millier: Mod`les dynamiques d´terministes en biologie (Mase e son, Paris 1982)
7. T.-Y. Li, J. Yorke: Period three implies chaos, American mathematical monthly
(82), pp 985–992, 1975
8. A.J. Lotka: Elements of physical Biology (Williams & Wilkins, Baltimore 1925)
9. R. May: Nature 261, 459 (1976)
10. R. Pearl: Introduction to Medical Biometry and Statistics (W.B. Saunders,
Philadelphia London 1923)
11. R. Pearl, L.J. Reed: Metron 5, 6 (1923)
12. A. Quetelet: Notice sur Pierre-Fran¸ois Verhulst. In: Annuaire de l’Acad´mie c e royale des Sciences, des Lettres et des Beaux-Arts de Belgique (Impr. Hayez,
Brussels 1850) pp 97–124
13. P.-F. Verhulst: Trait´ ´l´mentaire des fonctions elliptiques (Impr. Hayez, Bruseee sels 1841)
14. P.-F. Verhulst: Recherches math´matiques sur la loi d’accroissement de la pope ulation. In: Mem. Acad. Royale Belg., vol 18 (1845) pp 1–38
15. P.-F. Verhulst: Deuxi`me m´moire sur la loi d’accroissement de la population. e e
In Mem. Acad. Royale Belg., vol 20 (1847) pp 1–32
16. V. Volterra: Le¸ons sur la th´orie math´matique de la lutte pour la vie c e e (Gauthier-Villars, Paris 1931)

Pierre-Fran¸ois Verhulst’s Final Triumph c Jos Kint1 , Denis Constales2 , and Andr´ Vanderbauwhede3 e 1

2

3

Ghent University, Faculty of Medicine and Health Sciences, Department of Pediatrics and Medical Genetics GE02 jos.kint@belgacom.net Ghent University, Faculty of Engineering, Department of Mathematical
Analysis TW16
Denis.Constales@UGent.be
Ghent University, Faculty of Sciences, Department of Pure Mathematics and
Computer Algebra WE01
Andre.Vanderbauwhede@UGent.be

The so-called Logistic function of Verhulst led a turbulent life: it was ﬁrst proposed in 1838, it was dismissed initially for being not scientiﬁcally sound, it became the foundation of social politics, it fell into oblivion twice and was rediscovered twice, it became the object of contempt, was subsequently applied to many ﬁelds for which it was not really intended and it sank to the bottom of scientiﬁc philosophy. Today it is cited many times a year. And last but not least, during the past three decades it has been claimed as the prototype of a chaotic oscillation and as a model of a fractal ﬁgure.
It is only now, 155 years after Verhulst’s death, that it becomes clear that his logistic function transcends the importance of pure mathematics and that it plays a fundamental role in many other disciplines. The logistic curve has lived through a long and diﬃcult history before it was ﬁnally and generally recognised as a universal milestone marking the road to unexpected ﬁelds of research. Only at the end of the 20th century did Verhulst’s idea enjoy its deﬁnitive triumph. But let us start at the beginning.
On August 3, 1825 the magniﬁcent auditorium of Ghent University was still under construction. It would only be completed early 1826. However, at 11 a.m. of that particular August 3, a small function was held in the provisional hall of the university. In the presence of the then rector of the university, Professor Louis Raoul, a mathematician of scarcely 21 years old defended his doctorate’s thesis. Even in those days, twenty-one was very young to take one’s PhD. It was clear that, from that moment on, PierreFran¸ois Verhulst would not go through life unnoticed. c 1 His Life
He was born in Brussels on October 28, 1804 as the child of wealthy parents.
As a pupil at the Brussels Atheneum, where Adolphe Quetelet was his mathematics teacher, he already excelled, and not only because of his knowledge of mathematics. He also had linguistic talents. Twice he won a prize for Latin

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poetry. However, he had a distinct preference for mathematics. His desire to study exact sciences was so strong that in September 1822, without even having completed his grammar high school, Verhulst enrolled as a student at the
University of Ghent. Evidently, his lack of formalism caused some problems when he tried to enrol, although, in those days such matters could easily be resolved with some negotiating and argumentation. It was here that he met
Quetelet again, this time as his algebra professor. Just like his studies at the
Brussels Atheneum, his academic performance at the University of Ghent was a success. In less than a year, between February 1824 and October 1824, he was honoured with two prizes, one at the University of Leiden for his comments on the theory of maxima, and a second time he won the gold medal of the University of Ghent for a study of variation analysis [1].
In 1825, after only three years of study, Verhulst took his PhD in mathematics with a thesis entitled De resolutione tum algebraica, tum lineari aequationum binominalium, in other words, with a thesis in Latin on reducing binomial equations (Fig. 1).

Fig. 1. Doctorate’s thesis of Pierre-Fran¸ois Verhulst from 1825 c Pierre-Fran¸ois Verhulst’s Final Triumph c 15

After his studies Verhulst returned to Brussels. He took a keen interest in the calculus of probability and in political economy, an interest which he shared with Quetelet. From then on Quetelet’s inﬂuence on Verhulst is marked. Indeed, on several occasions Verhulst did some computations to support research carried out by Quetelet.
Moreover, Quetelet’s inﬂuence was not limited to passing on ideas and stimulating research. It was through his agency that Verhulst was entrusted with a teaching assignment at the “Mus´e des Sciences et des Lettres” in e Brussels in April 1827. A job which he soon had to give up on account of his poor health. Verhulst would be in bad health all of his life as a result of a chronic illness, the nature of which could not be retrieved from the documents that are left from that period. A brief stay in Italy, shortly after his promotion, did not help much to improve his state of health. During his stay in Rome in
September 1830, the Belgian Revolution broke out in Brussels. In the mind of
Verhulst, who was 26 at that time, a rather peculiar idea began to take shape.
An idea only conceivable by young people who in their youthful exuberance and audacity let their imaginations run free. Verhulst always consistently acted upon the consequences of his principles with the self-conﬁdence of a profound conviction. He conceives the rather original idea that the papal state could use a constitution, just like Belgium, his own country which had just become independent. And of course he is not satisﬁed with the idea alone, but immediately prepares a draft constitution. It seems incredible, yet it is true: the draft constitution was given some consideration by a few cardinals of the papal Curia and was sent to various foreign ministries. However, the matter came to the attention of the Roman bourgeoisie who was not at all pleased with someone from Brussels lecturing the Italians on how to deal with their political matters. The Roman police ordered him to leave the country at once.
Verhulst retired to his residence for a couple of days and tried to barricade himself, expecting a siege by the police. But in the end, after having discussed the matter with some friends, he decided to obey the expulsion order and left
Italy. Queen Hortense of Holland – at that time living in Rome – made in her memoirs a lively account of the aﬀair. Translated from French: “. . . A young
Belgian savant, Mr Verhulst, had come to Rome for his health. He came very often to my house in the evening; we had frequent discussions together. He asked to speak to me one morning, and brought a plan for a constitution for the Papal States, which he wished to submit to my criticism before giving it to the cardinal-vicar to submit to the pope. I could not help laughing at the singularity of my position. I [the exiled Queen of Holland] to revise a constitution, and for the pope! That seemed to me like a real joke. But my young Belgian friend did not laugh. ‘I was talking yesterday evening,’ he said to me, ‘with several cardinals; their terror is great. I told them of the only way to save the church and the state. They agreed with all my observations.
And one of them wishes to submit them to the pope himself. Here is the constitution of which I have sketched the basis . . . ’” [2]

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Back in Brussels, in 1831, he writes a document on behalf of the recently established Congress – the present Belgian parliament – in which he deplores the situation at the university and formulates a way to resolve it [3].
He complained about the political favouritism in the appointment of university professors and the poor standard of the lectures. In spite of his rebellious attitude he is appointed professor at the Royal Military Academy in 1835, and in the same year he is also appointed professor of mathematics at the Universit´ Libre of Brussels, both newly established teaching institutes. e However, Verhulst had to give up his professorship at the Universit´ Libre e of Brussels in 1840, following a decision of the then Minister of War, which stipulated that professors at the Military School were not allowed to teach in other education institutes. It is not unlikely that Quetelet had a part in the appointments of Verhulst. In 1837 he married a miss Debiefve, who would bear him a daughter about a year later.
Verhulst and Quetelet were closely associated in their life and work [4].
They were both professors at the Military School, they were both members of the Acad´mie royale des Sciences et des Belles Lettres de Bruxelles e and they were both interested in mathematical statistics which could be the key to revealing the “natural laws” of human society. Although Verhulst hardly made any general statements regarding the purpose and methodology of these statistics, his practical routine was in line with the theories of
Quetelet. The application of mathematics was an essential feature. In both
Quetelet’s and Verhulst’s opinion scientiﬁc statistics should be based on a precise mathematical formula to make the accurate incorporation of statistical data possible. However, gradually a signiﬁcant diﬀerence arose in the approach of Verhulst and Quetelet. Verhulst was not in the least interested in what Quetelet called “applied statistics”. Verhulst was of the opinion that the calculations were only applicable if there was a direct relation between cause and eﬀect. Quetelet himself did not feel so strongly about such reservations. In contrast he always preferred to ﬁnd some analogy between physical laws and social phenomena. The debate on this problem, which must have been going on between Verhulst and Quetelet for several years, came to a sudden end with Verhulst’s untimely death [4]. It is diﬃcult to determine the precise nature of their relationship from the available documents of that period. Adolphe Quetelet (1796–1876) was eight years older than Verhulst.
It is true that Quetelet called Verhulst “successively my pupil, my fellowworker, my colleague at the Military School, my confrere at the university and the Academy and my friend”. However, according to several authors, the relationship between both men was not always as serene as it appeared at ﬁrst sight. There is one thing we know for sure: they were both interested in mathematical statistics capable of explaining the so-called natural laws of society. Quetelet spoke highly of Verhulst’s work, but he had more regard for his compilations than for his original ideas. On one particular occasion, at a public sale, Verhulst managed to get hold of a valuable edition of

Pierre-Fran¸ois Verhulst’s Final Triumph c 17

the complete works of the French mathematician Legendre (1752–1833). The satisfaction of having acquired these works inspired Verhulst to study the
“Trait´ des fonctions elliptiques” and to read the works of the German Abel e (1802–1829) and the Norwegian Jacobi (1804–1851), with the intention of making a compilation of all aspects related to elliptic functions. He read and summarized the works of these three famous mathematicians as well as every other document on this subject. Quetelet was full of praise about the result of this study entitled “Trait´ ´l´mentaire des fonctions elliptiques”, which, in eee fact, was nothing more than a critical r´sum´ of the works of others. Howe e ever, Quetelet did not approve of what was in fact Verhulst’s most original achievement, i.e., the logistic function. After the publication of his “Trait´ e ´l´mentaire des fonctions elliptiques” Verhulst was admitted as a member of ee the “Acad´mie royale” in 1841. In 1848 Verhulst is appointed director of the e scientiﬁc department and later, in spite of his deteriorating health, the king appointed him chairman of the Academy. He died a couple of months later on February 15, 1849, at the age of 44.
According to Quetelet, Verhulst was somewhat of an “enfant terrible” [1].
He was self-willed, a man with a social conscience and a man of principle, controversial and often an advocate of extreme ideas, but he also had a strong sense of justice and acted from a deep feeling for his duty. He was straightforward and consistent in his thinking, but on the other hand also conciliatory. As chairman of the Academy he shrank from anything that might have caused dissension. He was never oﬀensive, and the higher his position the more unassuming he became. Although he himself did not have the slightest inclination for losing his temper, he respected the short-temperedness of others. Although he loved taking part in debates, it was more out of a craving for knowledge than in a spirit of contradiction or with the intention of imposing his own views. He was noted for his unperturbed equanimity. It would have been diﬃcult to ﬁnd a man more conscientious. According to Quetelet’s testimony, this sense of duty was marked during the last years of his life, when he still went to work every day. It took him more than an hour to walk the short distance from his house to his oﬃce. People saw him trudge along the streets, resting with every step he took, to arrive ﬁnally at the academy, panting heavily and completely exhausted.

2 His Work in the Field of Population Growth
Verhulst’s ﬁrst research in the ﬁeld of population growth dates from shortly after the independence of Belgium. In order to grasp the full import of the research on population growth in the nineteenth century, one must recall the social climate of those days. During the ﬁrst half of the nineteenth century
Flanders went through the worst economic depression in its entire history.
Although under the “Ancien r´gime” in the 18th century it had been one of e the most prosperous regions of Europe, it became a backward and shattered

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region with an impoverished and destitute population in only a few decades’ time. In addition to sheer destitution, the pauperization of the population also resulted in demoralization, moral degeneration and social unrest. The same confusion was also seen in other European countries. The correlation between poverty and population was ﬁrst demonstrated by Thomas Robert
Malthus, in his famous Essay on the Principle of Population, which was published in 1798. Malthus stated that poverty is only the inevitable result of overpopulation. In turn, overpopulation was the natural result of the fundamental laws of human society. The ideas of Malthus were the subject of heated debates in the nineteenth century. The necessity of conducting a social policy to curb the pauperization of the population turned the study of the laws of population growth into a scientiﬁcally respectable subject. A new discipline, political economics, found enthusiastic adherents everywhere. A demographic study of the population was initially impeded by a lack of statistical material or, even worse, by the unreliability of the available material.
It was only in 1820 that progress was made in the methods of compiling and processing statistical data on which demographic conclusions could be based.
In Belgium it was again Adolphe Quetelet who organized the collection of data with regard to population ﬁgures. He was the initiator of the ﬁrst census carried out in 1829, the results of which were published in 1832. As chairman of the “Commission centrale de statistique” Quetelet was in charge of the general censuses of 1846, 1856, and 1866. Quetelet also laid the foundations of the international conferences of statistics, the ﬁrst of which took place in
Brussels in 1853.
It was against this background that Verhulst started his research on population growth. His research was based on the ideas of Malthus. In his opinion it could not be denied that the population grew according to a geometric sequence. On the other hand it was incontestable that a number of inhibiting factors also increase in strength as the population grows. Verhulst argued that, as a consequence, the growth of the population was bound by an absolute limit, if only because of the limited availability of habitable land and food supplies. This was an original interpretation, but also a deviation from the original concept of Malthus. Malthus’ hypothesis can be formulated by means of a diﬀerential equation (with p for the population ﬁgure) dp = mp . dt Integration of this equation produces the well known exponential growth curve, on which economic Malthusianism is founded. Verhulst did not accept this and considered an alternative. In order to implement the check on population growth, Verhulst had to subtract a still unknown factor from the right-hand side of the equation; a factor which, according to Verhulst, is dependent on the population ﬁgure itself. He started from the most obvious hypothesis, namely that the growth coeﬃcient m is not constant but in proportion to the distance of the population size from its saturation point.

Pierre-Fran¸ois Verhulst’s Final Triumph c 19

In other words Verhulst introduced an inhibitory term, proportional to the square of the population size. Consequently, Verhulst stated that dp = mp − np2 . dt The solution of this diﬀerential equation gave rise to a function which was to project the population growth p= np0

mp0 emt
,
+ m − np0

emt

where p0 represents the population ﬁgure at a given time t = 0. Verhulst veriﬁed this formula by comparing the real population ﬁgures of France, Belgium,
Essex and Russia with the result of his calculations. The correspondence was striking, although the available ﬁgures related to a period of only twenty years. Verhulst created a new term for his equation and called it the logistic function. Verhulst never explained why he chose the term “logistique”. Yet, in the nineteenth century this French term was used to designate the art of computation, as opposed to a branch of theoretical mathematics such as the theory of proportions and relations. The term was also frequently used in connection with logarithms in astronomic calculations.
As a matter of fact the military meaning of the word “logistic” also found its origin around that period. The third supplement to the sixth edition of the etymological dictionary of the Acad´mie Fran¸aise ﬁrst mentions the term in e c
1835. The military meaning of the word also comprises the calculation of the provisionment of an army or of a population. The “logistic problem” par excellence is the provisioning of the population. Through his contacts at the
Military School, Verhulst must have been familiar with military terminology. Verhulst probably used this term to launch the idea of an arithmetical strategy that could be used to calculate the saturation point of a population as well as the time at which that point would be reached within a given percentage. Verhulst’s results were published in 1838 [5] as a modest “Notice sur la loi que la population suit dans son accroissement” in the “Correspondance
Math´matique et Physique”, a journal of which Quetelet was editor-in-chief. e Verhulst regarded his work as a ﬁrst step towards a much more elaborate study which would be published in 1845 and 1847 in the form of a “M´moire e de l’Acad´mie royale des Sciences et Belles-Lettres de Bruxelles” [6, 7]. For e more details on the life of Verhulst, see [8] and its references.

3 The Logistic Function After 1849
From then on this logistic principle of Verhulst led a most peculiar life. It may be said that after Verhulst’s death his principle was completely forgotten. One

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can only guess why this was the case. But Quetelet’s rather ambiguous eulogy
[1] on Verhulst at the Academy a few months after his death had something to do with it. In a condescending, almost contemptuous tone Quetelet expresses his reservations with regard to Verhulst’s principle and even with regard to his former “friend” himself. Quetelet had previously considered another principle regarding population growth, founded on the analogy with a falling stone in a viscous medium which encounters more resistance as its speed of fall increases.
Verhulst considered this concept too dogmatic and had always rejected it strongly. For in Verhulst’s mind there was only one thing that mattered: to ﬁnd a correspondence between his calculations and the real population ﬁgures, whereas Quetelet attached greater importance to a formal analogy between the laws of physics and the behavioural pattern of a population: much more than Verhulst, Quetelet was obsessed with the notion – which was popular in the nineteenth century – to presuppose exact causal mechanisms without which the world would not be able to function. The title of his magnum opus “La Physique sociale” already outlines Quetelet’s tendency to compare human social behaviour to the laws of physics. However, to state that Quetelet’s attitude was the decisive factor in the scarce dissemination of
Verhulst’s ideas in the nineteenth century, would be a limited representation of the facts. At least as important was the fact that Verhulst’s work never developed into a practicable theory that could be tested by demographers.
John Miner of Johns Hopkins University translated Quetelet’s French eulogy on Verhulst into English and published it in 1933 [9].
Whatever the reason may be, it is a fact that Verhulst’s work was completely ignored during the whole nineteenth century. The logistic curve was rediscovered only in 1920. In that year two renowned American demographers, Raymond Pearl and Lowell Reed [10], who were not acquainted with
Verhulst’s publications, formulated the sigmoid growth curve a second time.
It was only when their manuscript was already at the printer’s that they were informed of Verhulst’s work which had been published 75 years earlier.
In later publications they recognise their omission and they adopt the term
“logistic” from Verhulst [11].
The data of the United States census available to Pearl and Reed only made up half of a logistic curve, and the population level was far from reaching its saturation point. Nevertheless, they endeavoured to make an extrapolation and stated that the American population – at that time only 80 million people – would grow to a saturation point of 198 million people and that this saturation point would only be reached by the end of the twentieth century.
Unlike Verhulst, Pearl and Reed did not deduce the curve’s equation from any preliminary thinking. On the contrary, reﬂexions on the inhibitive eﬀect of diminishing ambient factors as a result of the population growth only appear towards the end of the article, and only to support the application of the sigmoid curve. In other words, Pearl and Reed start from the idea that population growth follows a sigmoid curve. In addition they regard the sigmoid

Pierre-Fran¸ois Verhulst’s Final Triumph c 21

curve of population growth as a genuine principle of population growth. On the one hand this was based on the fact that the logistic curve supported the data fairly well, and on the other hand on the fact that, based on reasonable assumptions, it provided a fairly accurate picture of the future evolution of the population. In 1924, Pearl [12] compared his curve “in a modest way” with Kepler’s law of planetary motion and with Boyle’s law of gases. . . For many years, the emphasis which Pearl and Reed put on the systematic nature of the logistic curve led to many heated and bitter discussions which would only come to an end with Pearl’s death in 1940. In spite of, or maybe thanks to, these ﬁerce discussions, the logistic curve is sometimes also called the Verhulst–Pearl curve.
A ﬁrst sign of real recognition of Verhulst’s merits came in 1925 [13], when the English statistician Udny Yule recognised that Verhulst was far ahead of his time: “. . . Probably owing to the fact that Verhulst was greatly in advance of his time, and that the then existing data were quite inadequate to form any eﬀective test to his views, his memoirs fell into oblivion; but they are classics on their subject. . . ” But even that was not suﬃcient to make
Verhulst’s reputation and his name was lost again. Verhulst’s formula got its ﬁnal victory only after 1965. From then on scientists from various countries and domains start to refer to Verhulst’s publications (Fig. 2). There are at least ﬁve reasons for this.
First of all there is the major breakthrough of ecology as a new scientiﬁc discipline: on account of the scope of their research ecologists are particularly interested in the growth and the evolution of populations. Verhulst’s formula appeared to be an excellent basis for calculating ecological growth problems.
A second aspect of Verhulst’s formula was that it required a considerable

Fig. 2. Citations to the publications of Pierre-Fran¸ois Verhulst c 22

J. Kint et al.

degree of computation. It was only with the advent of the electronic calculator and later the computer that the laborious job of making endless calculations could be carried out with a minimum of eﬀort.
A third factor was the discovery that the S-shaped logistic function could also be applied to a wide variety of other ﬁelds, such as chemical autocatalysis, Michaelis–Menten kinetics, cancer chemotherapy, the Hill equation, the
Langmuir isotherm, velocity equations of the ﬁrst and second order of magnitude, oxidation-reduction potentials, erythrocyte haemolysis, the ﬂow of streaming gases, etc. Verhulst’s principle was even applied to economics and sociology. It seemed as if everything could be deﬁned using the same sigmoidal curve. Many scientists carried it beyond the limit and applied Verhulst’s formula, whether it was relevant or not. This led to a situation in which over the past thirty years Verhulst’s work was cited in just about every country of the world, from Brazil to the People’s Republic of China, from the Soviet Union to the United States of America. His publications are now cited about 15 times a year, which is quite remarkable considering that his work goes back more than one hundred and sixty years. It is quite amusing in this context to see that each year several authors mention 1938 and 1945 as the year of publication of his works, thinking that 1838 or 1845 must have been a printing error. The journal “Correspondance Math´matique et Physique” ended its e publications in 1841. It was in fact published and edited by Quetelet himself on behalf of the Belgian mathematicians. It would reappear only at the end of Quetelet’s life from 1874 to 1880 under the name of “Nouvelle Correspondance Math´matique et Physique” and from 1881 to 1961 as “Mathesis”. e 4 Verhulst’s Principle and Chaos Theory
But there is a fourth reason why the work by Verhulst received so much attention all of a sudden: its implication in chaos theory. Already in 1963
Edward Lorenz used a one-dimensional mapping equivalent to the Verhulst mapping to explain certain aspects of his by now famous simpliﬁed weather forecast model. In 1976 the biologist Robert May [14] stated explicitly that the logistic model should be studied as early as possible in one’s scientiﬁc education in order to start understanding nonlinear phenomena. Since the work of May, Feigenbaum [15], and others the Verhulst model has become the paradigm for the period-doubling route to chaos, as is for example nicely illustrated in “The Beauty of Fractals” by H.O. Peitgen and P.H. Richter [16]
(one of the ﬁrst mathematical “coﬀee table books”).
Meanwhile several authors have adopted this idea and it seems to be generally acknowledged now that Verhulst’s logistic function is the basis of modern chaos theory, although Verhulst himself had absolutely no idea that something like that lay hidden in his formula.
To obtain deterministic chaos from Verhulst’s formula one has to replace the continuous logistic diﬀerential equation by its discrete form

Pierre-Fran¸ois Verhulst’s Final Triumph c 23

pn+1 − pn = rpn (1 − pn ) or equivalently pn+1 = pn + rpn (1 − pn ).
In this diﬀerence equation pn denotes the population size at time n, and r > 0 is still the growth coeﬃcient; the carrying capacity has been normalized to 1. Using this prototype of a nonlinear iterative process one calculates the evolution of a population by starting with some initial population p0 (between 0 and 1) and by applying the formula again and again, thus obtaining successively p1 , p2 , p3 , and so on.
When carrying out this iteration scheme one ﬁnds that the resulting evolution of the population depends strongly on the value of the growth parameter r (Fig. 3):
1. For r < 2 the population sequence tends to the limit value 1. For r < 1 this happens in a monotone way, similar to the behaviour in the differential equation (Fig. 3(a)), but for 1 < r < 2 in an oscillatory way
(Fig. 3(b)). As r increases to 2 these oscillations also increase, both in amplitude and length: for r = 1.95 the limit is reached only after more than 2000 steps!
2. For values of r between 2 and 2.5699 . . . the sequence displays, after some initial steps, a periodic behaviour with a period which depends on
r. When r increases one ﬁrst observes an oscillation between a maximum and a minimum (period 2, Fig. 3(c)), then an oscillation between 4 different local extremes (period 4, Fig. 3(d)), and subsequently oscillations with period 8 (Fig. 3(e)), period 16, and so on. Such a period-doubling cascade has been identiﬁed as one of the typical ways in which a system can go from orderly to chaotic behaviour.
3. For most values of r larger than 2.5699 . . . (and less than 3) the sequence shows no regularity (periodicity) any more (Fig. 3(f)). For such values of r the system is “chaotic”, a regime which is mainly characterized by a few hallmarks as described in the next paragraph.
The main characteristic of a chaotic system is its extreme susceptibility to a change in the initial condition (illustrated for the Verhulst model in Fig. 4).
Two sequences with almost identical values for p0 will at ﬁrst behave in a virtually identical manner, but then suddenly diverge so that from then on there is no correlation between the two oscillations. A similar sensitivity is also observed with respect to a change in the growth parameter r. Another phenomenon is that a chaotic system sometimes seems to behave regularly for a number of steps in the iteration. For example, for r = 2.7 and p0 = 0.05 there is an apparent regularity (a ﬁxed point) between step 590 and step
670 (Fig. 5(a)); with r = 2.7001 and p0 = 0.05 there is an apparent periodtwo behaviour between step 298 and step 316 (Fig. 5(b)). Under further iteration these apparent regularities disappear again. Predictability and chaos alternate with each other, but in a basically unpredictable manner.

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Fig. 3. Deterministic chaos obtained from Verhulst’s formula

At the moment when the system becomes chaotic, the size of the population at each step in the iteration will be diﬀerent from its value at any of the previous steps. There is no stability or regularity any more. Moreover, the long-term evolution of the population will strongly depend on the chosen initial value p0 . Even the smallest deviation – say in the hundredth or thousandth decimal – from the initial value will have a signiﬁcant eﬀect and in the end, result in a totally diﬀerent evolution. It is important to notice that also our computers which work with a ﬁxed number of decimals, are subject to this type of unpredictability, however powerful they may be.

5 Logistic Fractal of Verhulst
And ﬁnally, a ﬁfth factor can be identiﬁed which contributes to the late triumph of Verhulst’s logistic function. Indeed, using the logistic formula, one can produce fractal ﬁgures comparable to the well-known Mandelbrot fractal. For that purpose we consider again the discrete Verhulst iteration, pn+1 = pn + rpn (1 − pn ),

Pierre-Fran¸ois Verhulst’s Final Triumph c 25

Fig. 4. Example of extreme susceptibility to the initial condition in Verhulst’s formula but this time we allow p and r to be complex, and therefore related to points in the plane. More precisely, p and r take values of the form a + bi, and are then identiﬁed with the point (a, b) in the plane. The iteration is started by ﬁxing a nonzero value for p0 , for instance 0.01 + 0.01i. For each value of r one can then calculate the resulting iteration sequence. One ﬁnds that there are two possible results: either the sequence stays bounded, or it diverges to inﬁnity. The r-values for which the sequence stays bounded form a set which we call a Verhulst fractal; observe that this Verhulst fractal depends on the choice of the initial value p0 . In a similar way as for the Mandelbrot set, such
Verhulst fractals are easily generated on a computer: points not belonging to the fractal evolve towards inﬁnity at diﬀerent speeds, and by assigning diﬀerent colors to diﬀerent speeds one obtains patterns such as in Fig. 6.
In this ﬁgure the black points form the Verhulst fractal; each picture in the sequence is an enlargement of part of the preceding picture. What we learn from these pictures is that the boundary of the Verhulst set has a fractal structure, in the sense that however much we enlarge this boundary, it will never become a simple line or curve. At each scale new details appear, and the ﬁgure never reaches a limit.

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Fig. 5. Predictability and chaos alternating in Verhulst’s formula

6 Conclusion
Hence, twice in the past three decades Verhulst’s logistic function obtained a new, additional meaning. The ﬁrst time as a model of a chaotic oscillation and the second time as an example of a fractal ﬁgure. The realization that complex phenomena can be represented by means of a simple algebraic equation has radically changed our way of thinking in the past years. Robert May was one of the ﬁrst people to understand its broader social signiﬁcance: “Not only in research, but also in the everyday world of politics and economics, we would all be better oﬀ if more people realized that simple non-linear systems do not necessarily possess simple dynamical properties.”
Verhulst’s function is but one of the many examples of a non-linear, chaotic system, although it clearly illustrates the essence of deterministic chaos. It also illustrates how a discovery can go through a real evolution of its own and how the underlying signiﬁcance of a discovery can change radically as a result of the evolution of its scientiﬁc context. Some scientiﬁc ideas have to wait for a long period before they come to their ﬁnal triumph.
Verhulst’s logistic function is certainly one among them.

Pierre-Fran¸ois Verhulst’s Final Triumph c 27

Fig. 6. The Logistic fractal of Verhulst for the value p0 = −10−7 ; each ﬁgure to the right and downwards is an enlargement of the preceding ﬁgure

References
1. A. Quetelet: Notice sur Pierre-Fran¸ois Verhulst, In: Annuaire de l’Acad´mie c e royale des Sciences, des Lettres et des Beaux-Arts de Belgique (Impr. Hayez,
Brussels 1850) pp 97–124
2. M´moires de la Reine Hortense, publi´s par le Prince Napol´on (Librairie Plon, e e e Paris 1927) pp 210–212
3. P.-F. Verhulst: M´moire sur les abus dans l’enseignement sup´rieur actuel e e
(Hayez, Brussels 1831)
4. G. Vanpaemel: Quetelet en Verhulst over de mathematische wetten der bevolkingsgroei, Academiae Analecta, Klasse der wetenschappen 49, 96 (1987)
5. P.-F. Verhulst: Correspondance Math´matique et Physique 10, 113 (1838) e 6. P.-F. Verhulst: Mem. Acad. Roy. Belg. 18, 1 (1845)
7. P.-F. Verhulst: Mem. Acad. Roy. Belg. 20, 1 (1847)
8. J. Mawhin: Les h´ritiers de Pierre-Fran¸ois Verhulst: une population dye c namique, Bull. Cl. Soc. Acad. R. Belgique, in press

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14.
15.
16.

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J.R. Miner: Human Biology 5, 673 (1933)
R. Pearl, L.J. Reed: Proc. Natl. Acad. Sci. 6 (1920) pp 275–288
R. Pearl, L.J. Reed: Metron 3, 6 (1920)
R. Pearl: Studies in Human biology (Williams and Wilkins, Baltimore 1924) p.
585
G. Udny Yule: J. Roy. Statist. Society 88, 1 (1925)
R. May: Nature 261, 459 (1976)
M. Feigenbaum: J. Statistical Physics 19, 25 (1978)
H.O. Peitgen, P.H. Richter: The Beauty of Fractals (Springer Verlag, Berlin
1986)

Limits to Success. The Iron Law of Verhulst
P.L. Kunsch
Vrije Universiteit Brussel, MOSI department, Pleinlaan 2 BE-1050 BRUSSELS pkunsch@vub.ac.be Summary. In this chapter we develop the point of view that Verhulst is a major initiator of systems thinking. His logistic equation is a system archetype, i.e. a simple system built with few feedback loops. In the Fifth Discipline [19] Peter
Senge calls this particular archetype “Limits to Success”. It can also be called the “Iron law of Verhulst”, expressing that trees can never grow to heaven. In a deeper analysis this equation illustrates the shifting loop dominance, one of the basic principles of system dynamics. The basic message is that the combination of some few archetypes, like the logistic growth, can aﬀord valuable insight into many complex systems such as the economy, environment, organisations, etc. This fruitful concept is illustrated by a simple model in behavioural ﬁnance describing the equity price evolution, and based on the interplay of three main growth archetypes: “Limits to Success”, “Tragedy of the Commons”, and “Balancing Loop with Delay”.

1 Introduction
Chaos theory is said to have been founded by the 1-D logistic equation. This is certainly true although, as it is well known, the merit of discovering chaos in the discrete formulation of this formula may be given to May [16] in 1976, more than one century later. In its original continuous format the logistic equation is unable to generate chaos. This is a consequence of the Poincar´– e Bendixon theorem, which says that there is no chaos on the line, or on the plane, thus at least 3-D is needed. In this chapter we develop the point of view that Verhulst, more directly, started “systems thinking” applicable to complex systems. There is clearly a straight line between Verhulst’s germane ideas and the feedback-centred thinking of System Dynamics (SD), developed by J.W. Forrester [6, 7] in the 1960’s, and used by the early Club of Rome in its famous book Limits to Growth [17]. What Verhulst’s equation simply says, is that there is shifting loop dominance between two feedback loops (FBL): a positive FBL initiates growth; it is brought into balance by a negative FBL with growing importance, incorporating the limits to growth in a ﬁnite world.
The association of FBL’s of diﬀerent polarities and the shifting dominance between them is indeed the central thought of SD to model complex reality in population dynamics, ecology, economy, organisations, etc. These ideas have been later translated into management recipes by Peter M. Senge in his famous book The Fifth Discipline [19]. Simple archetypes are presented

30

P.L. Kunsch

there as elementary building blocks, pervasive in all organizational problems.
All archetypes result in the association of one to three FBL’s with diﬀerent polarities. Senge argues that most dynamic patterns can be reproduced from the association of some of them.
In Sect. 2 we develop some basic concepts of systems thinking from this perspective. We use as a starting point the logistic equation as an important growth archetype in SD. In Sect. 3 we present two other growth archetypes,
“Tragedy of the Commons”, and “Balancing Loop with Delay”, developed along similar lines to Verhulst’s logistic equation. In Sect. 4 we present a simple behavioural model of stock-price evolution by combining the basic mechanisms imbedded in these archetypes. Three families of investors are interacting on the equity market: fundamentalists, opportunists and long-term traders. This model comprises at least three stocks, and, therefore, chaotic dynamics is possible, contrary to the case of the continuous 1-D logistic equation. A conclusion relative to systems thinking and its links to the Iron Law of Verhulst is given in Sect. 5.

2 The Logistic Equation, a Prototype of Systems Thinking
Figure 1 reproduces a possible inﬂuence diagram of the logistic equation of
Verhulst in the very framework in which it was originally published, i.e., population dynamics. It represents a one-stock, two-ﬂow System-Dynamics (SD) model of the evolution of a deer population; the latter is submitted to a food availability constraint. The only stock is represented by a rectangular reservoir, according to the tradition introduced by J.W. Forrester, the initiator of
SD, in the early sixties of the last century.

Fig. 1. The inﬂuence diagram of the logistic growth of a deer population

Limits to Success. The Iron Law of Verhulst

31

Calling P the population, its logistic growth is represented by the Verhulst equation in a modernized form, and slightly modiﬁed to explicitly include the deer death rate:
P
dP
= rP 1 −
− DP .
(1)
dt
K
According to the usage in ecology, r represents the fractional growth rate corresponding to the r-strategy in a biotope, and K the limiting population size at maturity, corresponding to the K-strategy; D is the fractional death rate per unit of time, such that D =1/Lifetime of deer.
Figure 2 shows the evolution of the population and of the two ﬂows,
“Births” and “Deaths”. At logistic equilibrium the two ﬂows become equal, so that the net ﬂow vanishes. Figure 3 is the representation in the phase plane
(deer population, net growth rate). The equation of the 1-D ﬂow on the r.h.s. of (1) is a parabola. All this is of course well known. The inﬂuence diagrams and the computations originate from the SD-code VENSIM R [23].
Let us spend some more time examining the two feedback loops (FBL) in Fig. 1. The positive FBL in the inﬂuence diagram represents the growth process. The induced growth pattern is exponential; it corresponds to the r-strategy. Except for the natural death rate, the only negative inﬂuence is between
“deer population” and “relative food availability”: both variables move in opposite directions. Assuming that less food means less non-lethal births

Fig. 2. The evolution of the stock and of the two ﬂows in the logistic-growth model of Fig. 1

32

P.L. Kunsch

Fig. 3. The phase plane (P, dP/dt) of the logistic equation showing the parabolic function on the r.h.s. of (1)

of fawns, a negative FBL is obtained. The induced growth pattern is goal seeking with a resulting equilibrium population size K; it corresponds to the
K-strategy.
The dynamic behaviour of this simple dynamic system is dictated by
“shifting loop dominance” between the two FBL’s in the left part of the diagram: – First the (+) FBL activates the r-strategy, i.e. nearly exponential growth, the (−) FBL remains weak because it is driven by the term rP (P/K) in
(1), which is still second-order, and nearly negligible;
– As P grows this latter term becomes larger, and progressive shifting loop dominance appears. This concept has been introduced by Forrester [6–8].
In this speciﬁc case this simply means that the weaker (−) FBL becomes increasingly active with respect to the (+) FBL. In the growth curve, an inﬂection point is visible when P = K/2;
– At equilibrium, both loops are equally active, and thus exactly in balance, and the nonlinear process of shifting loop dominance is then complete to realise the asymptotic equilibrium at P = K.
Shifting loop dominance is the central idea of FBL-thinking, and thus of SD [8]. The properties of nonlinear systems are changing in the phase space. Some loops are dominant, or simply active, while some other ones are dormant, or practically inactive. So that there are no universal properties any more, contrary to what happens in linear systems.

Limits to Success. The Iron Law of Verhulst

33

Even if all FBL’s are present from the beginning in the inﬂuence diagram of the model, much diﬀerent behaviour can be observed by numerical integration as the relative strengths of several FBL’s change along the way.
This explains why nonlinear systems often show counterintuitive behaviours as already stressed by Forrester in his Urban Dynamics [7]. This complexity can be observed with only few FBL’s, but it increases when there are many possible combinations of interacting FBL’s present in the model. Given n
FBL’s there are n(n + 1)/2 FBL pairs to be compared. A larger system can have hundreds, or thousands FBL’s!
This counterintuitive behaviour is a diﬀerent concept from deterministic chaos. It has to do with the co-existence of many possible attractors of different nature (strange attractors are just one family). Another complication arises because of the possible bifurcations when parameters in the system (like the birth fraction) change value. This further increases the unpredictability and in fact the complexity of the system behaviour.
The 1-D logistic equation is unable to generate chaos, when the integration is done properly. This is because of the Poincar´–Bendixon theorem, which e states that there can be no chaos on the line or the plane (see for example [9],
Chap. 5.8, on stability properties in nonlinear systems). Chaos is thus only potentially observable in nonlinear systems with three stocks and more.
In the 1-stock case, chaos will only be observed as the result of an improper choice of the integration time step, and in this case it is thus a mere mathematical artefact (see [15]). Equation (1) indeed needs ﬁrst to be numerically integrated, with introducing of a discrete time step. The Euler integration scheme in time t can be written as follows:
P (t + ∆t) = P (t) 1 + r 1 −

P (t)
K

∆t .

(2)

Assume that the initial condition is such that 0 < P (t = 0) < K. Because for all ﬁnite t, the exact solution of (1) is such that P (t) < K, if ∆t is small enough, P (t) will be increasing from P (0) without ever exceeding K, except when P (t) comes very close to K from underneath. One should then observe that for small enough ∆t’s:
P (t + ∆t) > P (t) > 0 when K − P (t) > > 0 ,

(3)

where < ∆t is a very small number. Numerically, for t suﬃciently large P (t) will slightly exceed K, so that the ﬂow of the r.h.s. becomes negative; P (t) will then gently oscillate with hardly observable amplitude around K. It can be intuitively understood that for larger ∆t steps, oscillations will become of larger amplitude; once situations arise wherein P (t) becomes signiﬁcantly larger than K, overshoots of larger amplitude then occur, making P (t) swinging hence and forth passing the K-value; the place where the population size
P crosses the horizontal line at the boundary value K then changes at each period. Chaos arises when the set of crossing points becomes inﬁnite. This

34

P.L. Kunsch

Fig. 4. When the time step is too large, the integration of the logistic equation with the Euler scheme generates similar pattern as in the logistic mapping, including chaos situation is shown in Fig. 4. It is observed that, contrary to expectations from the continuous (2), P swings widely above the K = 1 boundary value of the population. Similar evolutions appear to the logistic map when the growth parameter increases. Several authors have established a correspondence between this latter parameter, and the time step, obtaining herewith the bifurcation diagrams in function of ∆t. This discussion does not need to be reproduced here (see for example a review paper in [13]).

3 Archetypes
System Dynamics (SD) is a quantitative simulation technique; many authors in many diﬀerent ﬁelds, such as economics and ﬁnance, organisation, environment, macroeconomics, etc. use it. A recent handbook is Sterman [22].
Soft modelling with SD is also a possibility. This approach limits the elaboration of models to the ﬁrst qualitative step of establishing the inﬂuence diagram, and analysing the feedback-loop (FBL) structure to deduce some consequences for the system and to derive possible improvement strategies.
This approach has some merit, though it is sometimes of limited predictive value: as mentioned before, systems often behave in a counterintuitive way due to the complex feedback interactions, and numerical simulations are necessary to test the actual behaviour patterns. Peter M. Senge is the author

Limits to Success. The Iron Law of Verhulst

35

of The Fifth discipline [19]. His main message is that Systems Thinking is indispensable for understanding and curing organizational problems. Unfortunately the human mind has diﬃculty in abandoning linear thinking, which was well adapted to local conditions of human societies in the past, but becomes far less adapted to global societies today. Senge further argues that, in numerous cases, simple systems often consisting of two to three basic feedback loops (FBL) provide a suﬃcient insight on what is going wrong in the enterprise. These elementary systems, called archetypes, thus can be assembled as building blocks for modelling more complex situations. According to
Senge’s convictions most situations of crisis are reducible to a small number of archetypes. His book enumerates ten main archetypes. Additional ones have been developed in later books of Senge [20] on the basis of the work of
Kim [12]. The most important archetypes are centred on three main growth patterns: 1. Logistic growth of (1) is described as combining exponential growth embodied in a (+) FBL, and goal-seeking growth, embodied in a (−) FBL.
It represents the “Limits to Success” archetype in Senge’s book. The interplay between the two FBL’s leads to the described shifting loop dominance, as has been illustrated in the basic Verhulst model. The archetype is shown in a more general way in Fig. 5: the whole model rests on the assumption that some resource is limited and becomes inadequate at some point. A more business-oriented case is shown in Fig. 6, called “Doctor’s
Practice” [18]. It illustrates the interplay between on the one hand the growth process of the (+) FBL, around the mouth-to-mouth publicity of satisﬁed patients, and, on the other hand, the constraints of the time resource. The latter is impeding the further growth because of the (−)
FBL related to the diminishing acceptability of time spent in the waiting room. Fig. 5. The archetype “Limits to Success” as a generic model of Verhulst’s Iron
Law (according to [22])

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P.L. Kunsch

Fig. 6. The logistic growth in the doctor’s surgery, as a further illustration of
Verhulst’s Iron Law. The limited resource is here the time that the doctor can devote to his patients (according to [18])

2. Overshoot and collapse growth appears in a second archetype “Tragedy of the Commons”, according to the economist Garrett Hardin [10]. This type of growth is quite pervasive in complex systems (traﬃc congestion, exhaustion of depletable resources, collapsing of biotopes, etc.). It is obtained from Verhulst’s logistic growth by adding just one more (−) FBL, as shown in Fig. 7. In the ﬁrst archetype of logistic growth in Fig. 5, the resource is in some way an external parameter to the model, embodied in the constant K in (1). The second (−) FBL on the right of the drawing now includes the limiting resource in the model. It corresponds to an erosion mechanism. The growth goal K, instead of being constant, will now be suddenly and often unexpectedly be collapsing through the internal nonlinear forces in the system. In human systems, the erosion is caused by the inadequate use of a common good or resource (highway, oil, etc.) in the egoistic search for individual advantage.
The model in Fig. 8 has two stocks, i.e. two ordinary diﬀerential equations, for representing both state variables, in this example deer population P and vegetation level V . These equations look as follows:

Limits to Success. The Iron Law of Verhulst

37

Fig. 7. The extension of the logistic model to a two-stock model representing the erosion of the food resource in the archetype “Tragedy of the Commons”

Fig. 8. The extension of the deer model of Fig. 1, including the resource Vegetation into the model. The deer population collapses when the food resource is eroded away

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P.L. Kunsch

dP
= rP dt dV
= sV dt P
K
V
1−
L

1−

− D(V )P

(4)

− CP V .

(5)

The variables and parameters in (4), representing the deer-population dynamics, have the same meaning as in (1). D(V ), the death ﬂow, is a declining nonlinear function of its argument V to be represented by a lookup table. It is of course equal to the natural death rate when the food is abundant, and it grows to 100 % mortality when food is disappearing.
In (5), representing the vegetation dynamics, s and L are constants, and they correspond to r and K in (4). The parameter C represents the speciﬁc consumption of food per deer and time period.
3. The third archetype is called in Senge’s original work “Balancing Loop with Delay”. It is basically a goal-seeking loop. Delays may be present at several stages: when information is collected or processed to take action, or before action leads to a change in the state of the system. The generic archetype is shown in Fig. 9. All loops are negative, because each information delay corresponds to one or several one-stock systems with

Fig. 9. The archetype “Balancing Loop with Delay” in which an information signal within a negative goal seeking FBL is submitted to delays causing overshooting and oscillations Limits to Success. The Iron Law of Verhulst

39

Fig. 10. In this archetype the oscillations are caused by the information delay between the inventory state and the manpower hiring

an outgoing ﬂow (see [22], Chap. 11). This archetype is typical for the existence of business cycles. An example is shown in Fig. 10, representing a manpower-management problem; it consists in a goal seeking loop, which is itself a ﬁrst-order delay, embedded in a two-stock system: inventory and manpower, as follows:
Goal − S dS =
,
(6) dt Tadj where S is the stock due to achieve the goal, and Tadj is the time constant necessary for the goal-adjustment process; it also represents the time delay constant. Equation (6) is the equation of a linear proportional controller (e.g., a thermostat) used in engineered devices to bring the state variable (e.g., the room temperature) to a desired goal. More complex nonlinear controllers, used in engineering, can be developed for the same purpose.

40

P.L. Kunsch

4 Modelling a Bubble on the Stock Market
In this section we discuss the modelling of the development and crash of a speculative bubble on the equity market (EM). The recent history of the high-tech bubble mainly in the years 1997 to 2003 provides a good example.
The archetypes presented in the previous section are the starting basis for modelling, because of their characteristic growth patterns. When a ﬁnancial bubble ﬁrst builds up, exponential growth is observed; later on temporary plateaux appear reminding of logistic growth equilibrium; there are also pseudo-random oscillations reminding of the “Balancing Loop with Delay”; ﬁnally crashes resemble the patterns in the Tragedy of the Commons.
Financial crashes were qualitatively modelled as cusp catastrophes by Zeeman [24]. R.H. Day [4] was one of the ﬁrst authors who intensively worked in quantitative non-equilibrium models inspired from chaos theory in discrete nonlinear systems. Following ideas of Shiller [21], Day postulates two families of investors, smart and ordinary investors. The formers are called α-investors; the latter are called β-investors. α-investors use quantitative valuations from fundamental analysis, they are basically goal-seekers and they stabilise the market. Their investment proﬁle as a function of the price has a reverse shape, because they are contrarians. β-investors, by contrast, remain in phase with the price trend by using simple investment rules: they overreact to sudden price moves or to fads, creating volatility. Day combines both investors’ proﬁles to deﬁne iterative 1-D mappings of the stock price: p(t + 1) = f [p(t)]. The patterns he observes show phases of high volatility, betraying the existence of deterministic chaos like in the logistic mapping.
Unfortunately, with those models it is much more diﬃcult to generate more representative evolutions typical for EM, like bullish or bearish behaviours, bubble formation and crashes, etc.
The idea developed by Kunsch et al. [14] is to use a continuous model with at least three stocks, in order to have the possibility to observe chaos, and a number of FBL’s able to generate representative and more realistic
EM signatures. The objective of considering at least three stocks is easy to achieve by considering several investors’ families and information delays.
Each ﬁrst-order delay requires one stock. Additional budget stocks represent the ﬁnancial constraints of investors.
Of course an important literature exists dealing with nonlinear dynamic modelling of the EM (see for example [3], [2] from [11]), or with artiﬁcial stock markets [1]. The ambition is not to present an up-to-date review of these models generally placed in the ﬁeld of behavioural ﬁnance. Rather it is planned here to show how archetypes, inspired from Verhulst’s ancient contribution, are still a source of inspiration for complexity modelling.
The universe in the EM model presented here is very simple. There are only two assets: a risky asset quoted at a variable homogeneous price P (it could represent a common equity index like Eurostoxx 50), and a risk-free asset, e.g., a high-rating bond. This universe is frozen for a given simulation

Limits to Success. The Iron Law of Verhulst

41

run. This means ﬁrst that the total number of equity shares n is ﬁxed in all scenarios. Second, all economic parameters of the model are constant, including the growth rate of the fundamental value, and the risk-free rate.
Some ﬁxed constraints are imposed on the available budget of the investors and their borrowing capacity.
Three homogeneous groups of investors are considered, instead of two in
Day’s model. They are called α-, βS-, and βL-investors, where “S” stays for short-term, and “L” stays for long-term. Inﬂuence diagrams can be drawn to represent each investor’s behaviour. Several important feedback loops are identiﬁed. They assist the understanding of basic behavioural rules developed in the investors’ minds. Some characteristics are important to understand. Negative loops assist the goal-seeking approach of fundamentalists. α-investors therefore help stabilizing the stock prices. By contrast positive loops, activated by short-term traders (βS-investors) are responsible for amplifying perturbations or rumours. Sometimes such a loop can act as a virtuous circle, in case it triggers a desired growth eﬀect in prices thanks to long-term strategies of βL-investors. Sometimes the loop acts as a vicious circle, because it ampliﬁes the market volatility, or it triggers crashes. In the
EM model, the two roles will be played in turn.
The three families of investors are now described in more detail; it is shown in each case in which way they are representative of the previously introduced archetypes. Note beforehand that this model is very simpliﬁed because many variables are considered as exogenous parameters, to be held constant: the relative proportions of the diﬀerent investor types, and the riskfree interest rate among others. These assumptions could be removed at the cost of higher complexity (e.g. including these parameters as model variables into additional FBL’s), but that would be beyond the scope of the present work. The same presentation is adopted as in [14].
Note ﬁrst that the total equity price P is split up into the three components representing the contributions of investors from diﬀerent groups:
1
Mα + MβS + MβL n = Pα + PβS + PβL .

P (t) =

(7)

Mα , MβS , MβL represent the amounts of money invested by the three investor types in the EM; n is the constant number of shares; Pα , PβS , PβL represent the three components of the total price P , attributed to the three investor types. 4.1 Family of α-Investors α-investors are “smart investors” behaving in a similar way to the rational goal-seekers assumed in Day’s model. Their sole aim is to achieve convergence

42

P.L. Kunsch

Fig. 11. Negative Feedback loop of α-investors (goal-seeking behaviour)

towards a current goal price Gα for the stock price P . It is why the unique feedback loop visible in Fig. 11 is a negative goal-seeking FBL. The price component Pα obeys to a similar equation to (6): dPα Gα − P
=
dt
Tα

(8)

As said above, more complex goal-seeking formulations can be adopted.
Because there are possibly information delays in the price adjustment, damped oscillations caused by slight overshooting above the goal price may be observed.
In our model, the assumed current goal price Gα is the sum of two terms: fundamental value gα , and risk premium Aα,βL resulting from the investing behaviour of βL-investors, to be described later:
Gα = gα + Aα,βL

(9)

– The fundamental value gα results from fundamental analysis, e.g., Dividend Discount Model (DDM). Deterministic dividends are assumed here, because stochastic changes do not bring more understanding on causal mechanisms. Dividends are growing with the given constant industry growth rate: gα is growing at the same rate.
– The risk premium depends on the arbitraging behaviour of βL-investors between the stock return and the risk-free rate; this is explained below.
In case of a positive gap, they invest more money into the EM, creating herewith a price increase ∆PβL , i.e. a risk premium above the fundamental value. In this case α-investors also adjust their long-term expectations, and they follow the observed positive trend over the fundamental value.
In practice, the premium is incorporated into the goal price by α-investors only up to a certain point; this occurs with a time delay τ . In the model it

Limits to Success. The Iron Law of Verhulst

43

is assumed that, in a bullish market mainly driven by the premium term, max α-investors will cap their goal price by a maximum arbitrage value ∆gα .
The latter corresponds to an acceptable risk level. Thus the actual risk premium is given by the following equation: max Aα,βL = min ∆gα , delayτ [max(0, ∆PβL )] .

(10)

In the model α-investors do not experience any liquidity constraints. This is a reasonable assumption, as they stop anticipating further price growth, as the risk premium above the fundamental value becomes exceedingly large.
In conclusion, α-investors behave according to the goal-seeking part in the logistic equation (Verhulst’s Iron Law). Because the goal is changing under the eﬀect of βL-investor strategies, there is a need for information collecting: damped oscillations due to overshoots may be observed, as in the archetype
“Balancing Loop with Delay”.
4.2 Families of β-Investors
The β-investors are “ordinary investors” in Shiller’s sense [21]. They are not entirely rational with respect to the use of information coming from the market. They use diﬀerent approaches to process the information, from rules of thumb to advanced technical analysis. An important aspect is the time horizon of anticipation, covering a continuum between short-term to long-term.
The model only considers two extreme cases in a continuum: βS-investors have a short-time horizon (S); βL-investors have a long-term horizon (L). Also proportions of the two types are kept constant. More sophisticated models may consider intermediate investors’ proﬁles or varying proportions within the model.
Family of βS-Investors βS-investors are opportunistic traders who are following immediate price movements; they buy in case of a price increase, and they sell when the price is going down. Therefore they destabilise the goal-seeking eﬀorts of αinvestors, who are contrarians, and they cause permanent noise. The presence of a positive feedback loop, visible in the right part of Fig. 12, conﬁrms the existence of this destabilizing investment approach. The driver in this loop is the ﬁrst derivative of the price, initiating a vicious circle of growth or decay.
A negative loop is visible in the left part of the diagram. It becomes active as the available budget drops to zero, forcing βS-investors to limit their stock position or even to liquidate part of their portfolio. The dynamic equation of βS-investors has been assumed to be the following: dPβS = SβS f dt ¯ dP ,P dt − RβS (BβS ) ,

(11)

44

P.L. Kunsch

Fig. 12. Feedback structure of short-term traders, i.e. βS-investors. The budget acts as a control mechanism; the presence of information delays causes oscillations.
There is also a partial correspondence between this diagram and the archetype
“Tragedy of the Commons”

where SβS > 0 represents the strength of βS-investors on the market (assumed to be constant). The function f (., .) depends in a nonlinear way on
¯
the stock price P and the smoothed value of its ﬁrst derivative dP /dt. Its sign is the same as the sign of the latter, indicating that βS-investors are trend-followers modifying their positions according to increasing or declining prices. A smoothed signal is calculated as an information delay as in (6), so that this may be the cause of oscillating behaviour, as in the archetype “Balancing Loop with Delay”. Overshoots may also be observed, which bring the budget to negative values. In such situations the βS-investors have to liquidate part of their portfolios. This appears in the last term on the r.h.s. of (11):
RβS (BβS ) represents the reimbursement rate to bring the budget BβS back to balance, in case it becomes negative. In conclusion, the delay mechanism and the budget constraints make that βS-investors are a source of instability and create pseudo-random oscillations in the search for price equilibrium on the EM.
Family of βL-Investors βL-investors rather have a long-term perspective. They permanently compare the long-term stock return and the risk-free interest rate (irate , assumed to be constant in this simple model). In case of a positive spread, in favour of risky asset positions, they invest additional money, curbing on the growth of the stock price. Therefore a positive feedback loop is visible in the upper part of the diagram in Fig. 13. It is driven by the positive return spread between

Limits to Success. The Iron Law of Verhulst

45

Fig. 13. Feedback structure of long-term traders, i.e. βL-investors. The budget fuels growth up to a certain point just before collapse. There is a clear correspondence between this diagram and the archetype “Tragedy of the Commons”

risky and risk-free assets, creating a risk premium. As indicated in (8) and
(9), α-investors will adjust in part their goal price to follow the growing price trend caused by βL-investors. In contrast to βS-investors, βL-investors have a borrowing capacity. They invest the borrowed money reinforcing herewith the growing trend, and transform it progressively into a vicious circle. Of course at some point there is shifting-loop dominance in favour of negative
FBL’s like in the archetypes “Limits to Success”, and the “Tragedy of the
Commons”. Such a loop is visible in the lower part of the diagram in Fig. 1: it relates to the available money resources. βL-investors have an initial budget and borrowing capacity up to a given permissible debt level. In any case, their willingness to reimburse their loans will grow with the relative level of their debt expressed as a percentage of the value of their stock position. As long as some borrowing capacity remains, βL-investors further strengthen their stock positions. Above some debt threshold, they experience an incentive to liquidate at least part of their positions. Another negative loop is not directly visible in Fig. 13, however. It ﬁnds its origin in the cap imposed by α-investors on the permitted price growth above fundamentals, according to (10). The simpliﬁed equation representing the dynamics of βL-investors is as follows: dPβL = SβL (r − i) − RβL BβS , VβL , dt (12)

where SβL > 0 represents the strength of βL-investors on the market (assumed to be constant in this simple model). r represents the smoothed stock return, and i = irate , the risk-free interest rate. The growth pattern of the ﬁrst term on the r.h.s. of (12) is thus exponential. RβL represents the reimbursement rate of loans in case the current budget BβL is becoming negative

46

P.L. Kunsch

and exceeds the borrowing capacity, which depends on the current value VβL of the portfolio owned by βL-investors.
In conclusion βL-investors develop a growth mechanism that is similar to the archetype “Tragedy of the Commons”. The growth is fuelled on a basis of an artiﬁcial money-borrowing resource and the landing can be quite sudden and hard, as some limits in the borrowing capacities are exceeded. Note that α-investors contribute to deﬁne when the bubble crash will start by capping the risk premium, as shown in (10).
4.3 Some Results of Simulation
In short some results of the EM model are presented in the form of time diagrams. The latter represent the total stock price (upper curve represented by a thick line), and its three components, indicated as Palpha (heavy line),
PbetaST (medium-heavy line), and PbetaLT (thin line), according to (7). There are many possible choices for the parameters in the model, but we shall limit our discussion to a few typical cases for bubble growth and crash. We again adopt the presentation from [14].
A ﬁrst scenario represents a boundary situation, helpful to calibrate the model. The risk-free rate is assumed to be very high; it is then expected that the risk premium is vanishing, i.e. no βL-investors will be present on the market, and thus no bubble can appear. In this case the price will gently

Fig. 14. (From [14]). A calibration calculation in case there is no risk premium, because of high risk-free rates

Limits to Success. The Iron Law of Verhulst

47

follow the industry growth rate. The price components and the total price are shown in Fig. 14 conﬁrming our expectations. The total price rapidly comes to the fundamental value. The lower curve shows the contribution of βSinvestors following the growing trends. Sometimes budgets become negative, so that, according to (11), a broken line and oscillations back to positive budget values are observed.
The following ﬁgures show two situations, in which the risk-free interest rate is low, so that βL-investors can arbitrage. In Fig. 15 the market is initially bearish. The return spread is negative, i.e. in favour of risk-free investment.
For quite a long time, the market is in near-equilibrium at the goal price set by α-investors; the growth rate is equal to the industrial growth rate (7%
p.a.). However, the steady industrial growth brings about a fresh-born wave of βL-investors. As a result the goal price also shifts up. βL-investors soon ﬁnd their limits. The borrowing capacity is reached. At this point, βL-investors have to rapidly liquidate the largest part of their stock portfolio in order to bring down their loan debt to an acceptable level. This reimbursement constraint has the same eﬀect on the price as a reﬂecting barrier. The price bounces back creating chaotic ups and downs of the price in search for a new equilibrium value. βS-investors amplify the appeared volatility. The price volatility becomes so large that at some point the long-term return drops below the risk-free rate. βL-investors disappear from the scene after a crash of limited amplitude. The market moves to a new equilibrium following the natural trend of fundamental values.

Fig. 15. (from [14]). The case of low interest rates, and an initially bearish market

48

P.L. Kunsch

In Fig. 16 the same assumptions as in the previous ﬁgure are used for simulating an initially bullish market. A burst in price sets up immediately; it is accompanied by high volatility. The bouncing back of the price against the debt barrier induces still more volatility than in the previous case. This turbulent behaviour cannot maintain itself very long. The market moves to its fundamental equilibrium as in the previous ﬁgure. After a while, a new price upsurge is observed with still more volatility than previously in the growing phase. It lasts for quite some time, exhibiting swings of considerable amplitude. As before, a crash brings back the price to its natural equilibrium.
When pursuing the computation, regular replicas with similar shapes are periodically observed.

Fig. 16. (from [14]) The case of low interest rates, and an initially bullish market

The more detailed paper contains additional runs with other choices of parameters, and a comparison of simulations with real observations on the
EM. The readers are referred to this paper for more details. As a source of inspiration for more advanced models, Fig. 17 presents, without any further comments, the evolution of the Eurostoxx 50 index between October 1994 and September 2004 during the lifetime of the recent high-tech bubble.

Limits to Success. The Iron Law of Verhulst

49

Fig. 17. The evolution of the DJ Eurostoxx 50 from October 1994 to September
2004 (from the website wallstreet-online)

5 Conclusions
The author has attempted in this chapter to present Verhulst’s contribution in a somewhat diﬀerent light: discussions generally go about the links between the logistic equation and chaos theory. It is argued here that in some way
Verhulst was a pioneer of nonlinear system theory. Systems thinking is today becoming a necessity for survival: the world globalization forces us to think in terms of causality networks rather that in terms of isolated cause-eﬀect links [5]. Verhulst’s ancient contribution thus remains modern, and it is still needed. With the logistic growth, Verhulst introduced for the ﬁrst time in history an inﬂuence diagram, in which two feedback loops are competing for dominance. This simple system teaches us that exponential growth is impossible in the natural world, because constraints on resources must be taken into consideration. The Iron Law of Verhulst remains an important message, which is unfortunately not yet accepted by all. Without this insight it is impossible to start a reﬂection on how to remove the “Limits to Success”, which are today threatening the very long-term existence of mankind in the limited spaceship earth. 50

P.L. Kunsch

References
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W.B. Arthur, S.N. Durlauf, D.A. Lane (Santa Fe Institute Studies in the Sciences of Complexity, Addison Wesley, Reading MA 1997)
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51

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Recurrent Generation of Verhulst Chaos Maps at Any Order and Their Stabilization Diagram by Anticipative Control
Daniel M. Dubois asbl CHAOS, Centre for Hyperincursion and Anticipation in Ordered Systems,
Institute of Mathematics, B37, UNIVERSITY OF LIEGE,
12, Grande Traverse, B-4000 LIEGE 1, Belgium
Daniel.Dubois@ulg.ac.be
ulg.ac.be/mathgen/CHAOS

1 Introduction
In 1838, P.-F. Verhulst [1] proposed an extension of the Malthus equation of the continuous growth of a population. In the Malthus equation, the time derivative of the population is directly proportional to the population, and gives a solution as a temporal exponential curve without limitation and tending to inﬁnity. Verhulst introduced, to the Malthus equation, a negative term, proportional to the square of the population, with a view to obtaining a stable stationary ﬁnite state of the population. The Verhulst nonlinear equation is called the logistic equation, for which an analytical solution exists.
For a non-continuous growth of a population, the diﬀerential equation is traditionally transformed with the Euler algorithm to a discrete equation, called the Verhulst map in the framework of chaos theory. This chaos map gives solutions of several types depending on the value of the growth parameter of the population: stable ﬁxed points, bifurcations and then chaos.
Simple analytical solutions of this chaos map exist for particular values of the growth parameter. In the chaos zone, this solution depends directly on the initial population, so the future evolution of the population is not predictable for long times if the initial condition is not known with precision.
This phenomenon is called the sensitivity to initial conditions.
The purpose of this chapter is ﬁrst to demonstrate that the Verhulst map is not the correct discrete equivalent to the Verhulst logistic equation. Due to the discrete time interval, the square of the population must be transformed to the product of the population at time t and the same population at the following discrete time t + ∆t, giving a non-recursive equation called an incursive equation (for inclusive or implicit recursive equation) [4]. The solution of this incursive discrete equation is similar to the stable stationary solution of the diﬀerential Verhulst logistic equation. The same result can be obtained from an anticipative control to the Verhulst map.
Secondly, it will be shown that the Verhulst chaos map belongs to a class of chaos maps at any order n, where n is the degree of the map. In this

54

D.M. Dubois

class, the Verhulst map depends on the power two, n = 2, of the population.
The concept of canonical chaos map will be introduced. The set of canonical chaos maps is given by the Tchebychev polynomials that verify a very simple relation of recurrence. Closed form solutions exist for these canonical chaos maps. This set of canonical chaos maps will be transformed to a set of chaos maps depending on a growth parameter.
Thirdly, this chapter will deal with the control of these chaos maps at any order, with the method of the incursive predictive control that belongs to the class of model predictive controls for which the model is the equation of the map itself, and for which no setpoint is deﬁned [5]. Such a control transforms all the unstable states of the chaos maps to stable ﬁxed points. A new type of diagram will be introduced, which we shall call the stabilization diagram. The resulting stabilization diagrams show a number of stable ﬁxed points directly related to the order of the maps. Numerical simulations will be performed on these chaos maps for various orders: the ﬁrst return map, its bifurcation diagram and its stabilization diagram will be displayed as a function of the growth parameter.

2 Analytical Solution of Chaos Maps
This section deals with the solution of the Verhulst logistic equation and the deduction of the so-called Verhulst chaos map. The closed form solution of the chaos map is then demonstrated and its relation to the Shift map is established. 2.1 From the Verhulst Diﬀerential Logistic Equation to the Verhulst Chaos Map
The original Verhulst [1] diﬀerential growth equation is given by dN (t)
N (t)
= rN (t) 1 − dt K

(1)

where N (t) is the value of the population at the current time t, r is the growth rate and K is a limiting growth factor. The well-known analytical solution of this equation is given by
N (t) =

ert N (0)
1 + (ert − 1)N (0)/K

(2)

where N (0) is the initial condition of the population at time t = 0. This solution tends to N (t) = K, for t
1/r. This means that the system loses its initial value and that its future is completely deﬁned by the value of the parameter K, the value of which being ﬁxed. So, such an equation is

Recurrent Generation of Verhulst Chaos Maps

55

an anticipative system in the sense that the ﬁnal future value, K, of the population is already known and completely ﬁxed at the present time.
Let us now deduce the classical discrete version of this equation. In deﬁning P (t) = N (t)/K, (1) becomes dP (t)
= rP (t)[1 − P (t)] . dt (3)

With the forward derivative, the discrete equation is given by the well-known
Euler algorithm
P (t + ∆t) = P (t) + ∆t[rP (t) − rP 2 (t)]

(4)

where ∆t is the time step. The time step can be taken equal to one (in rescaling the growth rate r) without lack of generality. So, with ∆t = 1, this equation can be rewritten as
(1 + r)P (t) − rP 2 (t)
P (t + 1)
=
.
1 + 1/r
1 + 1/r

(5)

With the change of variables x(t) = P (t)/[1+1/r] and a = 1+r, the Verhulst chaos map is readily obtained: x(t + 1) = ax(t)[1 − x(t)]

(6)

with a ∈ [0, 4] and x(t) ∈ [0, 1].
Let us summarize some properties of this map [8].
The stationary states are given by x(t + 1) = x(t) = x0 , with x0 = 0 or x0 = 1 − 1/a. The stability criterion of the stationary solutions is given by: dx(t + 1)
= |a − 2ax0 | < 1 . dx(t) (7)

For x0 = 0, | dx(t+1)/dx(t) |=| a |< 1, so this stationary solution is stable for a < 1. This means that the population disappears due to a negative growth rate r = a−1 < 0. The case a = 1 corresponds to a null growth rate r = 0. For x0 = 1 − 1/a,| dx(t + 1)/dx(t) |=| −a + 2 |< 1, so this stationary solution is stable for 1 < a < 3, and unstable for a ≥ 3. For 3 ≤ a ≤ 4, bifurcations occur
(period doubling: 2, 4, 8, 16, . . .), and then chaos (period 3, 5, 7, . . . mixed with even periods 6, . . . ). Chaos begins to occur for the value of a related to the universal constant of Feigenbaum [7, 6]: ac = 3.569945672 . . . So the chaos zone is deﬁned for 3.569945672 . . . ≤ a ≤ 4
2.2 Analytical Solution of the Verhulst Chaos Map
The analytical solution of the Verhulst chaos map (6), for a = 4, x(t + 1) = 4x(t)[1 − x(t)]

(8)

56

D.M. Dubois

with x(t) ∈ [0, 1], is given by x(t) =

1 − cos(2t g)
2

(9)

with g = arccos[1 − 2x(0)] where x(0) is the initial condition at t = 0.
This solution of the Verhulst chaos map is an exact closed form solution, because it is not necessary to compute all the successive iterates of the equation to obtain any iterate x(t) for any t. From the ﬁxed value of the initial condition x(0) at time t = 0, the variable g is ﬁxed to a constant, and the successive values of x(t) as a function of time t, are given by the calculation of cos(2t g) for the values of time t = 1, 2, 3, . . ., that is cos(21 g), cos(22 g), cos(23 g), . . ., and, contrary to the solution of the original
Verhulst diﬀerential equation, the chaos map never loses its initial condition x(0), via the g function.
So a natural chaos system, if it is not perturbed by external eﬀects, never loses its initial condition, and its future is completely written in its initial condition. But man, who tries to predict the future of this system, is limited by the exact knowledge of the initial condition. Indeed, one can only measure the initial condition with a certain number of decimals. Therefore, the future of this system is written in the successive values of the decimals, in theory, until inﬁnity!
The next section explains the shift of the digits of the initial condition, written in binary.
2.3 The Verhulst Chaos Map Transformed to the Shift Map
In order to explain that the whole future of a chaotic system is written in the digits of its initial condition, let us consider the following change of variables x(t) =

1 − cos[2πy(t)]
= sin2 [πy(t)]
2

(10)

in the Verhulst map (8) which becomes successively:
1 − cos[2πy(t)]
1 − cos[2πy(t + 1)]
1 − cos[2πy(t)]
=4
1−
, or
2
2
2
1 − 2 cos[2πy(t + 1)] = 2{1 − cos[2πy(t)]}{1 + cos[2πy(t)]} , or
1 − 2 cos[2πy(t + 1)] = 2 − 2 cos2 [2πy(t)] , or cos[2πy(t + 1)] = −1 + {1 + cos[4πy(t)]} , or cos[2πy(t + 1)] = cos[4πy(t)]

(11)

and the following Shift map is obtained y(t + 1) = [2y(t)] mod 1

(12)

Recurrent Generation of Verhulst Chaos Maps

57

with y(t) ∈ [0, 1], where mod 1 is the modulo 1, which means that only the fractional part of 2y(t) is taken.
The exact closed form solution of (12) is given by y(t) = [2t y(0)] mod 1 .

(13)

For example, with y(0) = 0.3203125 . . ., for which x(0) = sin2 [πy(0)] = 0.7137775467151410471604834284444 . . . , the Table 1 gives the successive values of y(t) and x(t).
Table 1. Numerical example of the Shift map and the Verhulst map t y(t)
0 0.3203125
1 0.640625
2 0.28125
3 0.5625
4 0.125

y(t + 1) binary y(t) x(t) = sin2 (πy(t)) x(t)
0.640625 0.0101001
0.7137775
0.7137775
0.28125
0.101001
0.8171966
0.8171966
0.5625
0.01001
0.5975451
0.5975451
0.125
0.1001
0.9619397
0.9619397
...
0.001
0.1464466
0.1464466

x(t + 1)
0.8171966
0.5975451
0.9619397
0.1464466
...

In Table 1, the ﬁrst column gives the time steps t = 0, 1, 2, 3, 4, the second column shows the y(t) which are calculated from the shift map in the third column y(t + 1) = 2y(t) mod 1, the fourth column shows the binary value of the decimal y(t), the ﬁfth column shows the x(t) = sin2 (πy(t)) calculated from the shift map y(t), the sixth column gives the x(t) calculated from the
Verhulst map in column seven x(t + 1) = 4x(t)[1 − x(t)].
It is clearly shown in Table 1 that, on one hand, the iterations shift to the left the digits of the binary y(t), and on the other hand that the x(t) calculated from the shift map and from the Verhulst map are identical.
But it is possible to obtain any iterate y(t) at any time t, without computing all the preceding iterates, from the exact closed form solution.
Indeed, from the initial condition: y(0) = 0.3203125 . . ., (13) gives y(4) = [24y(0)] mod 1 = (5.125 . . .) mod 1 = 0.125 . . . and from (10), one obtains x(4) = sin2 [πy(4)] = 0.14644660940672623779957781894758 . . .
Let us now demonstrate that the chaos map does not represent a correct discrete Verhulst equation.

3 The Verhulst Incursive Map is the Correct Discrete Verhulst Equation
I proposed [4], several years ago, to transform the Verhulst chaos map (6) to the following Verhulst incursive map:

58

D.M. Dubois

x(t + 1) = ax(t)[1 − x(t + 1)]

(14)

where the limiting factor [1 − x(t)] is deﬁned in the next time step as [1 − x(t + 1)].
This incursive equation (for inclusive or implicit equation) can be transformed (see the demonstration in the following section) to the following recursive equation x(t + 1) = ax(t) 1 −

ax(t) ax(t) =
1 + ax(t)
1 + ax(t)

(15)

where the limiting factor is equal to the whole equation.
This Verhulst incursive map (15) gives a stable solution for any value of
a.
Indeed, the two stationary states of this incursive equation are the same as for the chaos map, x0 = 0 and x0 = 1 − 1/a. In applying the criterion of stability a dx(t + 1)
=
<1,
(16)
dx(t)
[1 + ax(t)]2 it is proved that, ﬁrstly, the stationary state x0 = 0, dx(t + 1)
= |a| < 1 , dx(t) (17)

is stable for a < 1, and, secondly, the stationary state x0 = (a − 1)/a, dx(t + 1)
= 1/a < 1 , dx(t) (18)

is always stable for a > 1.
The analytical solution of this nonlinear (15) is given by x(t) =

(1 − 1/a)Cat
1 + Cat

(19)

as a closed form solution, where C is a constant. The initial value x(0) deﬁnes the value of C by, x(0) = (1 − 1/a)C/(1 + C), so the solution x(t) =

1+

(at

at x(0)
− 1)x(0)/(1 − 1/a)

(20)

is similar to the solution of the original Verhulst diﬀerential logistic (2).
Let us demonstrate that the incursive (15) represents the correct discrete equivalent of the logistic (1). For that, let us write the solution (2) of the
Verhulst logistic diﬀerential equation as
N (t2 ) =

er(t2 −t1 ) N (t1 )
1 + (er(t2 −t1 ) − 1)N (t1 )/K

(21)

Recurrent Generation of Verhulst Chaos Maps

59

which gives the growth of the population from the time t1 to the time t2 .
Indeed, in taking t1 = 0 and t2 = t, the continuous solution (2) is obtained.
Deﬁning a time interval ∆t between t2 and t1 as t2 = t1 +∆t, (21) becomes
N (t1 + ∆t) =

1+

er∆t N (t1 )
− 1)N (t1 )/K

(er∆t

(22)

Choosing ∆t = 1, and t1 = t, one obtains
N (t + 1) =

er N (t)
1 + (er − 1)N (t)/K

(23)

and with the change in variables, er = a and N (t)/K = x(t)/(1 − 1/a), (23) is written as ax(t) (24) x(t + 1) =
1 + ax(t) which is the discrete (15), and this is the correct discrete algorithm for the
Verhulst logistic diﬀerential equation.
As result, we conclude that the chaos emerging from the so-called Verhulst chaos map is due to instabilities of the Euler algorithm and not from fundamental biological properties of the Verhulst logistic diﬀerential equation.
This (24) can also be obtained from an incursive control as shown in the next section.

4 Incursive Control for Stabilizing Chaos Maps
This section presents a survey of the role of incursive control with an example of control of the Verhulst chaos map, with simulations.
Let us begin by the deﬁnitions of recursive and incursive systems.
4.1 Recursive and Incursive Systems Applied to the Verhulst Chaos Map
A recursive system computes its vector current state x(t), at successive time steps t = 0, 1, 2, . . . , from a vector function R of its past and present states as x(t + 1) = R(. . . , x(t − 2), x(t − 1), x(t); p)
(25)
where the vector p is a set of parameters.
A weak incursive system [4] computes its current state at time t, as a function of its states at past times,..., t − 2, t − 1, present time, t, and even its predicted states at future times t + 1, t + 2, . . .. x(t + 1) = A(. . . , x(t − 2), x(t − 1), x(t), x∗ (t + 1), x∗ (t + 2), . . . ; p) (26)

60

D.M. Dubois

where the future states x∗ (t+1), x∗ (t+2), . . ., are computed with a predictive model of the system.
A strong incursive system [4] computes its current state at time t, as a function of its states at past times,..., t − 3, t − 2, t − 1, present time, t, and even its states at future times t + 1, t + 2, t + 3, .... x(t + 1) = A(. . . , x(t − 2), x(t − 1), x(t), x(t + 1), x(t + 2), . . . ; p)

(27)

where the future states x(t + 1), x(t + 2), . . ., are computed by the system itself. The Verhulst chaos map (6): x(t + 1) = ax(t)(1 − x(t))

(28)

is a recursive system, and the Verhulst incursive map (14): x(t + 1) = ax(t)[1 − x(t + 1)]

(29)

is a strong incursive system, because the future value in the saturation factor, [1 − x(t + 1)], is computed by the system itself. Indeed, in replacing successively x(t + 1) by ax(t)[1 − x(t + 1)] in (29), one obtains the following equation x(t + 1) = ax(t)(1 − ax(t)(1 − ax(t)(1 − ax(t)(1 − ax(t)(1 − ...)))))

(30)

which is an inﬁnite recursive equation that converges to x(t + 1) =

ax(t)
1 + ax(t)

(31)

which is the (15), given at the preceding section.
This incursive map (29) can be obtained from the following recursive map
(28) to be controlled x(t + 1) = ax(t)[1 − x(t)] + u(t)

(32)

by an incursive control u(t) u(t) = ax(t)[x(t) − x(t + 1)]

(33)

which can be transformed to a recursive control with (31) as follows u(t) = ax(t) x(t) −

ax(t)
1 + ax(t)

.

(34)

Indeed, in including (34) to (32), (31) is obtained. As it was shown in the preceding section, this incursive map is always stable. This incursive controller, which is a powerful tool for stabilizing chaos maps, belongs to a special class of controller, because no setpoint is deﬁned. Such a controller stabilizes by itself the unstable states to stable ﬁxed points.
In the next section, this incursive controller will be compared to model predictive controllers through numerical simulations in order to indicate how to stabilize the chaos of the Verhulst map.

Recurrent Generation of Verhulst Chaos Maps

61

4.2 Incursive and Model Predictive Controls of the Verhulst Chaos Map
Predictive models were developed over the past two or three decades in the ﬁeld of control, what is referred as the Model-based Predictive Control (MPC). These include Model Predictive Heuristic Control (MPHC) [9],
Dynamic Matrix Control (DMC) [3], Internal Model Control (IMC) [10] and
Generalized Predictive Control (GPC) [2].
The key diﬀerence between a conventional feedback control and a predictive control is that the control error e = x − r, which is the diﬀerence between the process output x and the setpoint r (the desired output), used by the predictive controller is based on future and/or predicted values of the setpoint x(t + τ ), and also on future and/or predicted values of the process output x(t + τ ), rather than their current values. See [1] for an overview of
MPC.
The basic principle of the model predictive control of a discrete system, with a time step ∆t, x(t + ∆t) = F [x(t)] + u(t)

(35)

where u(t) is a control action, consists in minimizing a cost function J, given for example by a weighted least squares criterion:
N

Nu

[x(t + i∆t) − r(t + i∆t)]2 +

J =E i=1 wi u(t − ∆t + i∆t)2

(36)

i=1

where {r(t + i∆t)} is the setpoint sequence (target tracking), {wi } is the weight sequence, and N and N u are ﬁxed integers representing the time horizons of the predicted outputs, x(t + i∆t), and control sequence, u(t −
∆t + i∆t). The anticipative outputs x(t + i∆t), i = 1 to N , are computed from the model of the process to be controlled. The Verhulst map (28), to be controlled by a control action u(t), in the chaos regime with a = 4, is written as x(t + 1) = 4x(t)[1 − x(t)] + u(t)
(37)
with the cost function (36) given by
J = [x(t + 1) − x0 ]2 + w1 u2 (t)

(38)

with ∆t = 1, N = N u = 1, with a constant setpoint r(t + 1) = x0 = 3/4, that is the unstable equilibrium of the map [5]. The objective of the control is to stabilize the map at its unstable equilibrium given by x0 = 3/4, obtained from x0 = 4x0 (1 − x0 ), corresponding to the equilibrium condition x(t + 1) = x(t) = x0 in (37) [without the control u(t)]. Indeed, in applying the classical criterion of stability given by dx(t + 1)
<1
dx(t)

(39)

62

D.M. Dubois

to the chaos map at the equilibrium x0 = 3/4, one obtains dx(t + 1)
=| 4 − 8x0 |= 2 dx(t) (40)

which is greater then 1, so this equilibrium state x0 is unstable.
Putting (37) into (38), one obtains the following cost function at the current time
J = [4x(t)(1 − x(t)) + u(t) − x0 ]2 + w1 u2 (t) .

(41)

The minimum of J is obtained with the condition dJ/du(t) = 0, so u(t) =

x0 − 4x(t)[1 − x(t)]
.
1 + w1

(42)

Now, let us put (42) into (37), to obtain x(t + 1) = 4x(t)[1 − x(t)] +

x0 − 4x(t)[1 − x(t)]
.
1 + w1

(43)

After elementary mathematical transformations, (43) becomes x(t + 1) =

x0 + 4w1 x(t)[1 − x(t)]
.
1 + w1

(44)

The equilibrium conditions x(t + 1) = x(t) = x0 of (44) are x0 = 0 and x0 = 3/4, as desired.
Applying the stability criterion (39) to (44), for x0 = 3/4, one obtains
4w1 − 8w1 x0
2w1
dx(t + 1)
=
= −
<1
dx(t)
1 + w1
1 + w1

(45)

and the chaotic map is stabilized to x = x0 = 3/4 for a weight in the range
0 < w1 < 1.
Let us show the diﬀerences between the incursive control and the model predictive control by considering the control of the chaos map (37), with the three following control functions [5]:
a) The incursive control [Equation (34), in the chaos regime with a = 4]: u(t) = 4x(t)

x(t) − 4x(t)[1 − x(t)]
1 + 4x(t)

(46)

b) The model predictive control [Equation (42)]: u(t) =

x0 − 4x(t)[1 − x(t)]
1 + w1

c) The incursive predictive control:

(47)

Recurrent Generation of Verhulst Chaos Maps

u(t) =

x(t) − 4x(t)[1 − x(t)]
1 + w1

63

(48)

This third control is based on the incursive control applied to the model predictive control by replacing the setpoint x0 by x(t) in the cost function
(38) as follows
(49)
J = [x(t + 1) − x(t)]2 + w1 u2 (t)
This new controller (48) is obtained by minimizing this cost function (49).
Let us point out that the control (46) is similar to the control (48) by taking a variable weight w1 = 1/4x(t).
Figures 1, 2 and 3 show the simulations of the chaos map (37), with the three controllers (46), (47) and (48).

Fig. 1. Incursive Control (46), starting at step 50, of the chaos map

Fig. 2. Model Predictive Control (47), starting at step 50, of the chaos map

64

D.M. Dubois

Fig. 3. Incursive Predictive Control (48), starting at step 50, of the chaos map

From the simulations, the best controller is the incursive control (46), in
Fig. 1, where it is well seen that the control is optimal.
In Fig. 2, the model predictive control (47), with w1 = 1/2, shows damped oscillations around the setpoint x0 = 3/4. Let us notice that the damping eﬀect depends on the value of the weight w1 .
In Fig. 3, the incursive predictive control (48) shows a better control than the model predictive control, with the same weight w1 . The incursive control and the incursive predictive control, ﬁnd by themselves the setpoint which is the unstable equilibrium of the chaos map. Indeed, these incursive controllers do not use an explicit setpoint, but an implicit setpoint given by the unstable equilibrium state which is stabilized. The incursive control minimizes the distance between x(t + 1) and x(t), instead of the distance between x(t + 1) and a setpoint x0 .
Let us now show how to generate Verhulst chaos maps at any order and how to control them.

5 Recurrent Generation of Chaos Maps at any Order
With a view to obtaining a simple rule to generate chaos maps at any order, let us make the following change of variables
X(t) = 1 − 2x(t)

(50)

in the Verhulst chaos map (8) which becomes
X(t + 1) = 2X 2 (t) − 1 with X(t) ∈ [1, −1], and the exact closed form solution (9) becomes

(51)

Recurrent Generation of Verhulst Chaos Maps

X(t) = cos(2t g)

65

(52)

with g = arccos[X(0)] or X(0) = cos g.
The purpose of the next section is to deﬁne a set of canonical chaos maps with a general exact closed form solution given by
X(t) = cos(nt g)

(53)

for n = 0, 1, 2, 3, 4, 5, . . . and g = arccos[X(0)] or X(0) = cos g.
5.1 Canonical Chaos Maps at any Order
The set of canonical chaos maps is deﬁned by the following general equation x(t + 1) = cos[n arccos x(t)]

(54)

with x(t) ∈ [1, −1] and n ∈ N . With the relation
2
4 cos[n arccos(x(t)] = xn (t) + Cn xn−2 (t)[x2 (t) − 1] + Cn xn−4 [x2 (t) − 1]2 + · · ·
(55)
m where Cn = n!/m!(n − m)!, (54) becomes
2
4 x(t + 1) = Tn [x(t)] = xn (t) + Cn xn−2 (t)[x2 (t) − 1] + Cn xn−4 [x2 (t) − 1]2 + · · ·
(56)
with x(t) ∈ [1, −1] and n ∈ N . This equation (56) is the set of canonical chaos maps, for which the exact closed form solution is given by

x(t) = cos(nt g)

(57)

for n = 0, 1, 2, 3, 4, 5, . . . and g = arccos[x(0)] or x(0) = cos g.
Let us remark that (55) corresponds to what is called the Tchebychev polynomials Tn given by
2
4
Tn (X) = X n + Cn X n−2 (X 2 − 1) + Cn X n−4 (X 2 − 1)2 + · · ·

(58)

which gives:
T0 = 1 ,
T1 = X ,
T2 = 2X 2 − 1 ,
T3 = 4X 3 − 3X ,
T4 = 8X 4 − 8x2 + 1 ,
T5 = 16X 5 − 20X 3 + 5X ,
T6 = 32X 6 − 48X 4 + 18X 2 − 1 ,
...

(59)

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where Tn is of degree n, and, for n > 0, the coeﬃcient of the term of the highest degree of Tn is 2n .
These Tchebychev polynomials verify a very simple relation of recurrence given by
(60)
Tn+2 = 2XTn+1 − Tn with the initial conditions T0 = 1 and T1 = X.
With the relation of recurrence (60), it is possible to generate the set of canonical chaos maps as follows: n=0 n=1

x(t + 1) = 1 x(t + 1) = x(t)

n=2 n=3 n=4

x(t + 1) = 2x2 (t) − 1 x(t + 1) = 4x3 (t) − 3x(t) x(t + 1) = 8x4 (t) − 8x2 (t) + 1

n=5 n=6 x(t + 1) = 16x5 (t) − 20x3 (t) + 5x(t) x(t + 1) = 32x6 (t) − 48x4 (t) + 18x2 (t) − 1

n=7

x(t + 1) = 64x7 (t) − 112x5 (t) + 56x3 (t) − 7x(t)
...

n = 10

x(t + 1) = 512x10 (t) − 128x8 (t) + 112x6 (t) − 400x4 (t) + 50x2 (t) − 1
...
(61)

Introducing a growth parameter a in (56), the set of chaos maps at any order is given by the following map x(t + 1) = aTn [x(t)]
2
4
= a{xn (t) + Cn xn−2 (t)[x2 (t) − 1] + Cn xn−4 [x2 (t) − 1]2 + · · · }

(62)

with a ∈ [−1, +1] and x(t) ∈ [−1, +1]. This set of chaos maps (62) can be controlled by an action control u(t) as follows x(t + 1) = aTn [x(t)] + u(t)

(63)

with the following general incursive predictive controller u(t) =

x(t) − aTn [x(t)]
1 + w1

(64)

where
2
4
Tn [x(t)] = xn (t) + Cn xn−2 (t)[x2 (t) − 1] + Cn xn−4 [x2 (t) − 1]2 + · · · , (65)

with a ∈ [−1, +1] and x(t) ∈ [−1, +1].
In Subsect. 5.2, some numerical simulations of these chaos maps and their stabilization will be given.

Recurrent Generation of Verhulst Chaos Maps

67

5.2 Simulation and Incursive Control of Chaos Maps
In this last subsection, we show the numerical simulation of the chaos maps
(62) for n = 2, 3, 5, 10, 25, and their stabilization diagram with the incursive predictive control (63), (64) for n = 2, 3, 5, 10, 25.
The chaos maps, which are numerically computed, are given by x(t + 1) = a[2x2 (t) − 1] x(t + 1) = a[4x3 (t) − 3x(t)]

for n = 2 , for n = 3 ,

(66)
(67)

x(t + 1) = a[16x5 (t) − 20x3 (t) + 5x(t)] for n = 5 ,

(68)

and by x(t + 1) = a[512x10 (t) − 128x8 (t) + 112x6 (t) − 400x4 (t) + 50x2 (t) − 1] (69) for n = 10.
Let us remark that it is also possible to obtain this chaos map n = 10 by inserting the canonical map n = 5 in the chaos map n = 2 as follows:
2
x(t + 1) = a{2T5 [x(t)] − 1} .

(70)

In the same way, the chaos map n = 25 is obtained in inserting the canonical chaos map n = 5 in the chaos map n = 5 as follows n = 25
5
3
(71)
x(t + 1) = a{16T5 [x(t)] − 20T5 [x(t)] + 5T5 [x(t)]} with T5 [x(t)] = 16x5 (t) − 20x3 (t) + 5x(t).
Figures 4 and 5 give the ﬁrst return diagram of the chaos maps n = 2 and n = 10 for a = 1.
Figures 6, 8, 10, 12 and 14 give the bifurcation diagrams of the chaos maps n = 2, 3, 5, 10, 25, with a ∈ [−1, +1].
Figures 7, 9, 11, 13 and 15 give the stabilization diagrams of the chaos maps n = 2, 3, 5, 10, 25, with a ∈ [−1, +1], by using the incursive predictive control. For example, the control for n = 3 is given by x(t + 1) = a[4x3 (t) − 3x(t)] + u(t) , x(t) − a[4x3 (t) − 3x(t)]
.
u(t) =
1 + w1

(72)
(73)

These stabilization diagrams of the chaos maps show that the number of ﬁxed points is equal to n = 2, 3, 5, 10, 25, respectively. The proposition to create such stabilization diagrams could be of great interest in practice.
Indeed, it is often impossible to detect, in real time, the unstable points of real chaotic systems. Instead of deﬁning setpoints arbitrarily, it would be more accurate to let the controller self-stabilizes an unstable point.

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Fig. 4. First return diagram for n = 2

Fig. 5. First return diagram for n = 10

Recurrent Generation of Verhulst Chaos Maps

Fig. 6. Bifurcation diagram for n = 2

Fig. 7. Stabilization diagram for n = 2

69

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D.M. Dubois

Fig. 8. Bifurcation diagram for n = 3

Fig. 9. Stabilization diagram for n = 3

Recurrent Generation of Verhulst Chaos Maps

Fig. 10. Bifurcation diagram for n = 5

Fig. 11. Stabilization diagram for n = 5

71

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D.M. Dubois

Fig. 12. Bifurcation diagram for n = 10

Fig. 13. Stabilization diagram for n = 10

Recurrent Generation of Verhulst Chaos Maps

Fig. 14. Bifurcation diagram for n = 25

Fig. 15. Stabilization diagram for n = 25

73

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6 Conclusions
This chapter ﬁrstly presented the Verhulst logistic diﬀerential equation and its analytical solution for various values of the growth parameter.
The classical discretization of this equation, with the Euler algorithm, gives the so-called Verhulst map, exhibiting ﬁxed points, bifurcations and, then chaos for successive values of the growth parameter.
In the chaos zone, the Verhulst map has a closed form solution depending on the initial condition. It was recalled that the Verhulst map can be transformed to the Shift map for which the successive iterates are given by the fractional part of the shift, to the left, of the binary digits of the initial condition. The ﬁrst part of this chapter showed that the bifurcations and chaos in the Verhulst map are due to the Euler algorithm and not to the fundamental properties of the original Verhulst equation. It was demonstrated that the correct algorithm is the incursive algorithm (for inclusive or implicit algorithm) which gives ﬁxed points similar to those of the original Verhulst equation, what we called the Verhulst incursive map.
Such a Verhulst incursive map can be obtained from the Verhulst chaos map with an incursive control, which is an anticipative controller.
The second part of this chapter demonstrated that the Verhulst chaos map belongs to a family of canonical chaos maps, given by the Tchebychev polynomials at any order (degree of the polynomials), for which closed form solutions exist.
These canonical chaos maps can be generated by a simple relation of recurrence. With introduction of a growth parameter into these canonical chaos maps, a set of Verhulst chaos maps at any order is proposed.
For some of these Verhulst chaos maps, their ﬁrst return diagrams and bifurcation diagrams are numerically computed.
Finally, the simulations of the anticipative control of these maps, with an incursive controller, give stable ﬁxed points which are exhibited in stabilization diagrams.
In conclusion, the incursive tool is a powerful anticipative method for stabilizing chaos, and open new avenues of research and development in the framework of the theory of chaos.

Acknowledgements
I would like to thank Professor Mitchell J. Feigenbaum for very fascinating discussions and for having pointed out that the canonical chaos maps are related to the Tchebychev polynomials.

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75

References
1. E.F. Camacho, C. Bordons: Model Predictive Control (Springer, Berlin 1999)
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Systems: CASYS’99 – Third International Conference, ed by D.M. Dubois (The
American Institute of Physics, AIP Conference Proceedings 517, 2000) pp 3–30
5. D.M. Dubois: Mathematical foundations of discrete and functional systems with strong and weak anticipations. In: Anticipatory Behavior in Adaptive Learning
Systems, State-of-the-Art Survey, ed by M. Butz et al., Lecture Notes in Artiﬁcial Intelligence, Springer, LNAI 2684, pp 110–132, 2003
6. M.J. Feigenbaum: J. Stat. Phys 19, 25 (1978)
7. M.J. Feigenbaum: Physica D 7, 16 (1983)
8. H.-O. Peitgen , J. Hartmut, D. Saupe: Chaos and Fractals, New Frontiers of
Science (Springer, Berlin 1992)
9. J. Richalet, A. Rault, J.L. Testud, J. Papon: Automatica, 14, 413 (1978)
10. D.E. Rivera, M. Morari: Ind. Eng. Chem. Process Des. Dev. 25, 252 (1986)
11. P.-F. Verhulst: Correspondance Math´matique et Physique 10, 113 (1838) e Coherence in Complex Networks of Oscillators
Pedro G. Lind1,2 , Jason A.C. Gallas1,3 , and Hans J. Herrmann1,4
1
2

3

4

ICP, Universit¨t Stuttgart, Pfaﬀenwaldring 27, D-70569 Stuttgart, Germany a Centro de F´ ısica Te´rica e Computacional, Av. Prof. Gama Pinto 2, 1649-003 o Lisbon, Portugal
Instituto de F´ ısica, Universidade Federal do Rio Grande do Sul, 91501-970
Porto Alegre, Brazil
Departamento de F´ ısica, Universidade Federal do Cear´, 60451-970 Fortaleza, a Brazil

Summary. We study fully synchronized (coherent) states in complex networks of chaotic oscillators, reviewing the analytical approach of determining the stability conditions for synchronizability and comparing them with numerical criteria. As an example, we present detailed results for networks of chaotic logistic maps having three diﬀerent scale-free topologies: random scale-free topology, deterministic pseudo-fractal scale-free network and Apollonian network. For random scale-free topology we ﬁnd that the lower boundary of the synchronizability region scales approximately as k−µ , where k is the outgoing connectivity and µ depends on the local nonlinearity. For deterministic scale-free networks coherence is observed only when the coupling is heterogeneous, namely when it is proportional to some power of the neighbour connectivity. In all cases, stability conditions are determined from the eigenvalue spectrum of the Laplacian matrix and agree well with numerical results based on histograms of coherent states in parameter space. Additionally, we show that almost everywhere in the synchronizability region the basin of attraction of the coherent states ﬁlls the entire phase space, and that the transition to coherence is of ﬁrst-order.

1 The Interplay Between Dynamics and Topology
The structure and dynamics underlying complex networks have been widely investigated, providing insight for many systems where they arise naturally
[1–3]. Complex networks appear in a wide variety of ﬁelds ranging from lasers
[4], granular media [5, 6], quantum transport [7], colloidal suspensions [8], electrical circuits [9], and time series analysis [10], to heart rhythms [11], epidemics [12, 13], protein folding [14], and locomotion [15] among others
[1–3].
From the mathematical point of view, a network is a graph, composed by nodes or vertices and by their connections or edges [2]. Sometimes, each node is characterized by some dynamical state (amplitude), which evolves according to some local contribution and to the interaction with the neighbourhood.
In other words, the complexity of the system underlying the network may be introduced either in the way nodes are interconnected (topology) or in the way nodes evolve in time (dynamics).

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When studying network dynamics one frequently assumes a regular topology (lattice) where each node evolves according to some more or less complicated dynamics, typically ﬁxed points [16], limit cycles [17] or chaotic attractors [18, 19]. One main goal of this approach is to study the so-called spatiotemporal chaos which appears in many diﬀerent spatially extended systems out of equilibrium, such as hydrodynamical ﬂows, chemical reactions and biological systems [20,21]. Two main topics in this context concern the study of mechanisms underlying pattern formation and pattern selection [20–24] and also the study of chaotic synchronization behaviour [16, 25].
Spatially extended systems are fundamentally modelled by (i) sets of coupled diﬀerential equations [20] with nonlinear terms, where both time and amplitude are continuous, (ii) cellular automata [22], where both time and amplitude are discrete or (iii) coupled map lattices [21], where time is discrete as in cellular automata, but where the space of states is continuous.
In all these models the underlying networks have connections whose range assumes all values from 1 (nearest neighbours) up to some maximum range, in particular the size of the system (global coupling regime). In other words, neglecting boundary conditions, these network systems assume translational symmetry and therefore the underlying network is called a regular network.
To study more complicated network structures, one usually neglects node dynamics and all complexity is introduced by the network topology, i.e. by the way nodes are connected to each other. This can be done in three diﬀerent ways [2]: by randomly connecting the nodes (random networks [26, 27]), by considering some random long-range connections in a regular network with some small range of couplings (small-world networks [28,29]), or by considering the introduction of new nodes which are connected to the previous ones following some rule of preferential attachment (scale-free networks [30]). For all these cases there is no translational symmetry and no typical range connection: connections do not have any ‘spatial’ interpretation. Therefore, one uses some general topological quantities to characterize each particular network, namely the average path length
, i.e. the average minimum number of connections linking two nodes, the clustering coeﬃcient C, deﬁned as the average fraction of neighbours which are connected to each other, and the distribution of connections P (k), representing the number of nodes having k connections. Table 1 shows the values of these three quantities for all three topologies. Random networks were introduced by Erd¨s and R´nyi in the late ﬁfties o e
[26] to study organizing principles underlying some real networks [27]. In random networks one deﬁnes some probability p(N ) for any two nodes to be connected in a total of N nodes. Consequently, the connections are typically long-range connections having a completely irregular structure. One main goal in studying random networks is to determine the critical probability pc (N ), beyond which some speciﬁc property is very likely to be observed,
e.g. the critical probability marking a transition to percolation [31]. One

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Table 1. Characterizing complex topologies with the topological quantities: average path length
, clustering coeﬃcient C and distributions P (k) of connections k.
Here N is the total number of nodes, p is the probability for two nodes to be
¯
connected, k is the average number of connections per node, C0 is the clustering coeﬃcient of the regular network from which the small-world network is constructed, and m is the number of initial connections of each new node in a scale-free network
Random

P (k)

Scale-free

ln N/ ln (pN )
C

Small-world
N for small p ln N for large p

ln N/ ln ln N

¯ k/N ¯
−k ¯ k e k /k!

C0 (1 − p)3
¯
¯ e−k kk /k!

∼ N −3/4
2m2 /k3

important feature of random networks, which also appears in real networks, is their small average path length
, i.e. the average distance between any two nodes increases slowly with the system size. However, unlike random networks, real networks also have large cluster coeﬃcients C.
Small-world networks were introduced recently by Watts and Strogatz [28] in order to implement both short and large C features. Small-world networks have short-range connections between neighbours, as in regular networks, but they also have long-range connections similar to random networks, without middle range ones. There are mainly two procedures to construct a small-world network: starting from the same regular network, where each site is coupled to some number of ﬁrst neighbours, one either rewires each regular connection with probability p (Watts–Strogatz model [28]) or adds a random connection to each node with probability p (Newman–Watts model [32]). The second procedure is more appropriate for most purposes, since it avoids the possibility of generating disconnected clusters [32].
Both random and small-world topologies produce typically networks where connections obey a Poisson distribution (see Table 1). However, there are real systems which are scale-free, i.e. where the connection distribution obeys a power law.
Scale-free networks were introduced by Barab´si and Albert [30] using a growing and preferential attachment: one starts with a small amount of fully interconnected nodes, and adds iteratively one node with m connections to the previous nodes, chosen from a probability function proportional to their number of connections. With this construction one obtains analytically [33–
36] a distribution of the connections P (k) ∝ k −γ , where γ → 3 as N →
∞, independently of the initial number of fully interconnected nodes and of
m. It is also possible to generate scale-free networks, either by imposing a priori a power-law distribution of all randomly distributed connections, or by following a deterministic iterative rule for new nodes. The ﬁrst procedure generates what is usually called a generalized random graph, while the latter was recently referred as deterministic scale-free network [37].

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Deterministic scale-free networks are hierarchical structures composed by some succession of generations of nodes, i.e. the set of new nodes appearing simultaneously at a given iteration during the ‘construction’ of the network, whose connections follow a particular power-law distribution [37–40], being more easier to handle. The main diﬀerence between random and deterministic scale-free networks arises from the local connectivity character of the latter: random constructions generate irregular long-range connections, while deterministic networks impose a succession of generations of new nodes which are, in some way, organized in ‘space’. Therefore, deterministic networks are applied for instance in spin systems [39], and geographical and social networks [39, 41].
After considering separately dynamical and topological complexity, the next logical step toward real network dynamics is to consider them together.
One important question addressed in this context is to know whether full synchronization between oscillators in such complex topologies would appear and under which conditions the full synchronized state would be stable. By full synchronization we mean the convergence of the amplitudes of all oscillators to the same value, evolving coherently from then on. Therefore we call henceforth these fully synchronized states coherent states, to distinguish them from partially synchronized conﬁgurations, when several diﬀerent clusters of nodes with the same amplitude are observed [16]. Synchronization and coherent behaviour of oscillator networks with complex topologies have been studied for the random topology [42–45] and the small-world topology [46–49], and also scale-free networks [45, 50–53]. In random networks, it is already known [42] that with high coupling strengths it is possible to fully synchronize oscillators and the corresponding stability condition may be computed [43] from the matrix of connections characterizing the network. In small-world networks, synchronizability is observed [47] only at the end of the small-world regime
(high values of p), and recently [46] it was found that heterogeneity in the coupling may destroy coherence. These ﬁndings for small-world networks are somehow contradictory to the ones of other studies [45, 48] and other topological quantiﬁers have been proposed [49]. In scale-free networks some recent studies indicate that synchronizability among oscillators depends on the average connectivity [52] and is robust to a delayed ﬂow of information [50] and to the removal of low-connected nodes [52].
In this chapter we describe the general approach to study coherent states in any general complex network of oscillators, and apply it to the particular case of a scale-free network of discrete-time oscillators, which is studied in great detail. We start in Sect. 2 by describing the stability analysis approach to the model introduced in (1) and deduce the corresponding conditions for synchronizability. In Sect. 3 we apply this stability analysis procedure to the particular case of scale-free networks, comparing our results with numerical simulations. The random scale-free case is treated in Sect. 3.1, where we show that the threshold value of such a transition as a function of coupling

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strength and outgoing connectivity obeys a power law with an exponent that depends on the nonlinearity, while deterministic scale-free networks are studied in Sect. 3.2, namely a pseudo-fractal network [38] and an Apollonian network [39, 40]. Discussion and conclusions are given in Sect. 4.

2 General Approach to Analyse Coherent States
For all the network topologies described above, if one considers discrete-time oscillators, namely maps of the interval, the equation of evolution for their amplitudes reads
(1)
xt+1 = f (xt ) − εGg(xt ) , where ε is the coupling parameter, t labels time, xt = (xt,1 , . . . , xt,N ) with xt,i representing the amplitude at time-step t of node i = 1, . . . , N , where N is the total number of nodes, f = (f (x1 ), . . . , f (xN )) and g = (g(x1 ), . . . , g(xN )) with f and g being real nonlinear functions, and G is the coupling (Laplacian) matrix, whose element Gij represents the relative strength with which node
N
i is coupled to node j, and satisﬁes the conditions j=1 Gij = 0 and Gii = 1 for all i = 1, . . . , N . In general G is a non symmetric matrix.
Usually, one chooses g(x) ≡ x when studying linear coupling, and g(x) ≡ f (x) when studying nonlinear coupling. Here we consider the nonlinear case.
Apart from this choice, all the information about the dynamics is introduced in the function f (x), while all the information about the coupling topology
(regular, random, small-world or scale-free) and the coupling regime (either homogeneous or heterogeneous) is included in the coupling matrix G.
From (1) one easily sees that the coherent state xt,1 = xt,2 = · · · = xt,N =
Xt evolves in time according to the local map Xt+1 = f (Xt ). There are two ways to study these coherent states: either by studying the stability of small perturbations of the coherent states or by making statistics over signiﬁcant large samples of initial conﬁgurations, counting how many converge to a coherent state. Some attention to the parameter ranges must be taken, since the basin of attraction of the coherent states may be bounded by regions of phase space where amplitudes diverge. In particular, for maps of the interval one has 0 ≤ ε ≤ 1 in order to guarantee convergence of any initial conﬁguration.
In this chapter we shall illustrate both analytical and numerical approaches for the particular case of scale-free networks. To this end, we deﬁne the coupling matrix as Gii = 1 and
Gij = −

α kj k∈Ki

α kk (2)

if node i is coupled to node j, with kj representing the number of neighbours of node j and Ki is the set of labels of all neighbours of node i. If nodes i and j are not coupled Gij = 0. The parameter α is a real number controlling

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the heterogeneity in the coupling: positive values of α enhance the coupling strength with sites having a large number of neighbours, while negative values favour sites having less neighbours. For α = 0 the coupling between each site and its neighbourhood is homogeneous.
For local dynamics we choose the well-known quadratic map f (x) = 1 − ax2 , where the free parameter a is restricted to the interval −0.25 ≤ a ≤ 2 and contains all possible dynamical regimes from a ﬁxed point (e.g. a = 0) to fully developed chaotic orbits (e.g. a = 2).
When determining the stability of coherent states, various criteria are possible. For instance, one could compute the maximum Lyapunov exponent and obtain the conditions where it is negative. However, such a Lyapunov exponent does not indicate the existence of a local instability in the synchronous state, which may pull the trajectories apart from the coherent manifold.
The correct approach is based on the variational version of (1) proposed by
Pecora and Carroll [54], which is valid for any network of identical oscillators as far as their local dynamics (quadratic map, Lorenz system, etc.) and their coupling regime (linear, nonlinear, etc.) are concerned. For the nonlinear coupling regime, the diagonal form of these variational equations reads [54–
56]
(3) ξt+1,i = exp [Λ(ελi )]ξt,i = [Df (X) − ελi Df (X)] ξt,i , for the coherent states xt,i = X, where Λ(ελi ) is the Lyapunov exponent,
Df (X) represents the identity matrix multiplied by the derivative of f (x) computed at x = X and λi are the eigenvalues of the coupling matrix G. If
N
G has zero-sum rows, i.e. j=1 Gij = 0 ∀i, and all its eigenvalues λ1 ≤ λ2 ≤
· · · ≤ λN are real and nonnegative, then λ1 = 0 corresponds to the mode parallel to the synchronization manifold and the largest Lyapunov exponent deﬁnes a master stability function [54]. The coherent state is stable whenever
Λ(ελi ) < 0 for i = 2, . . . , N [54–56].
In our case, it is easy to check from (2) that G has indeed zero-row sum, yielding λ1 = 0. Moreover, all the eigenvalues of the matrix G are real and
¯
¯ nonnegative, since det(G − λI) = det(G − λI) where G is a positive semidef¯ inite symmetric matrix, namely G = H1/2 K1/2 AK1/2 H1/2 with A being the adjacency matrix of the network [52], and matrices H and K being the diagα α onal matrices with elements Hii = 1/( k∈Ki kk ) and Kii = ki respectively.
From (3) and regarding the ordering of the eigenvalues λi one easily concludes that the stability condition reads εL ≡

¯
¯
1 − exp (−λ)
1 + exp (−λ)
<ε<
≡ εU , λ2 λN

(4)

¯ where λ is the Lyapunov exponent of the local single map. In particular there is a range of coupling strengths enabling synchronizability whenever
¯
¯ λN /λ2 < (1 + e−λ )/(1 − e−λ ) holds. Therefore, by computing the eigenvalues of the Laplacian matrix G one is able to ﬁnd the range of couplings for which

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coherent states are stable. This approach can be applied for any system ruled by (1).

3 Scale-Free Networks of Coupled Logistic Maps:
An Example
For the particular case of scale-free networks, recent results [45] show a transition to full synchronization for two particular values of the nonlinearity a in the homogeneous regime (α = 0), when either the coupling strength or the number of outgoing connections is varied. However, as far as we know there is no detailed study showing how these coherent states depend on all the model parameters. Therefore, we present in this section detailed numerical results concerning synchronization in oscillator networks with scale-free topologies.
Our purpose is to give a complete example of how to study coherent solutions in complex networks of oscillators, comparing both the stability analysis and the numerical approaches.
The stability analysis is carried out just by computing the boundary values εL and εU in (4) as functions of the model parameters. The ranges of values where εL < εU are the ones where coherent solutions appear. As stated above, for discrete oscillators, ruled by maps of the interval, the condition
0 ≤ ε ≤ 1 must be added.
Numerically, to detect coherent solutions from a given sample of initial conﬁgurations, we compute the standard mean square deviation [45]
N
2 σt =

(xt,i − xt )2 ,
¯

1
N

(5)

i=1

where xt is the average amplitude at a given time step t. Whenever σ 2 is zero
¯
within numerical precision, i.e. σ 2 ∼ 10−30 , all the nodes are synchronized at the same amplitude.
We divide our approach into two parts, one concerning random scale-free networks (Sect. 3.1) and the other one concerning deterministic scale-free networks (Sect. 3.2).
3.1 Random Scale-Free Networks
In this section, we use the algorithm of Barab´si and Albert [2, 30] to cona struct the random scale-free network (see Sect. 1), where at each node one places a chaotic logistic map. In a previous work [45] a transition to coherence between chaotic logistic maps was found for random scale-free networks, occurring for particularly high coupling strengths, typically of the order of εc ∼ 0.9. Our simulations have shown that these transitions occur after discarding transients of ∼ 104 time steps and they do not change signiﬁcantly

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(a)
(b)
Ν
1
0.8
0.6
0.4
0.2
0

-30

0

N
1

0.2 0.8
0.6
0.4
0.6
0.8
-25

-20

-15

log σ2

-10

-5

0

1

ε

14

0.4
0.2
0
0

12
10
-5

8
-10

-15

log σ2

6
-20

-25

k

4
-30

2

Fig. 1. Typical histograms of the standard mean square amplitude deviation σ 2 showing the sharp transition to coherence in function of (a) the coupling strength ε with k = m0 = 8, and (b) the outgoing connectivity k with ε = 0.95. Values of
N represent the fraction of conﬁgurations which converge to a coherent state from a total of 500 initial random conﬁgurations, after discarding transients of 104 time steps. Here a = 2, N = 1000, and α = 0

with the network size. Moreover, as shown in Fig. 1, this transition to coherence is robust with respect to initial conﬁgurations, by varying either the coupling strength ε (Fig. 1a) or the outgoing connectivity k (Fig. 1b). In particular, above the threshold εc ∼ 0.9, all initial conﬁgurations converge to a coherent state, indicating that in this parameter region the basin of attraction of coherent states ﬁlls almost the entire phase space.
From stability analysis, we ﬁnd that in the fully chaotic regime (a = 2) the transition to coherence occurs for gradually smaller coupling strength if the connectivity k is increased. Figure 2a shows the boundaries εL and εU as a function of k for a = 2 (solid lines) and a = 1.9 (dashed lines) with the same parameter values as in Fig. 1. As one sees, in both cases the lower boundary εL decreases when k increases, while the upper boundary εU increases beyond ε = 1. Therefore, one expects that the region of synchronizability increases for larger values of connectivity k. Figure 2a also shows clearly that for a = 2 the intersection between both boundaries, εL = εU , occurs just above k = 7, which explains why the transition to coherence in Fig. 1b occurs at this value.
For a = 1.9 this transition should occur near k = 5. Decreasing even more the nonlinearity coherent solutions are observed for even smaller connectivities and synchronizability regions increase, as shown in Fig. 2b. To see this feature more clearly we magnify in Fig. 2c the rectangle of Fig. 2b. As one sees, one exception occurs for a = 1.8, where the lower boundary is below the one for a = 1.7, due to the fact that for a = 1.8 the Lyapunov exponent of the logistic map is smaller than the one for a = 1.7, as illustrated below in Fig. 4. For all these values of a, the single uncoupled map shows chaotic orbits. Moreover, for any other network size N , the same curves are obtained.

Coherence in Complex Networks

85

1.4
1.4

(b)

1.2

1.2

ε

εU

1

εL

1

sync

0.8

sync

0.6

0.8

0.4

(a)
0.6

4

6

8

10

12

14

16

18

204

6

0.9

8

10

12

14

16

18

20

0.2

(c)

a=2 a=1.9 ε 0.7

a=1.7 a=1.6 0.5
0.3

a=1.5

6

7

a=1.8

8

9

10

11

12

13

14

k
Fig. 2. Boundary values εL and εU in (4) in function of the connectivity k (a) for a = 1.9 (dashed lines) and a = 2 (solid lines), and (b) for a = 1.5, 1.6, 1.7, 1.8, 1.9 and 2. The data in the rectangle in (b) is magniﬁed in (c). The regions labelled with ‘sync’ are the ones where coherent solutions are observed, i.e. εL < εU . Notice that in (c) the boundary for a = 1.8 is below the one for a = 1.7, contrary to other values (see text)

These analytical predictions extracted from the stability condition in (4) and shown in Fig. 2 are strongly corroborated with our numerical simulations as shown in Fig. 3. In Fig. 3a we plot the fraction F of initial conﬁgurations which converge to a coherent state for a = 2, while Fig. 3b shows the threshold values, εc and kc , at the transition curves where the entire sample of initial conﬁgurations converge to a coherent state, for the same values of a as in
Fig. 2c. Here, one clearly sees that there is a clear and sharp transition to coherence. Interestingly, the curves in Fig. 3b ﬁt very well the ones in Fig. 2c, which means that whenever the synchronizability condition εL < εU holds, coherent states ﬁll almost entirely the phase space.
Moreover, as illustrated in Fig. 3c, all curves obey a power law, within our numerical precision,
−µ
(6) εc ∝ kc .
For the six above values of a, the exponents are respectively µ = 0.2345,
0.2354, 0.2353, 0.2231, 0.2023 and 0.1804: the exponent is almost constant below a ∼ 1.7 and decreases above this value, as illustrated in the inset of
Fig. 3c.
To determine the nature of the transition to coherence seen in Figs. 3a and 3b, we plot in Figs. 3d and 3e the average standard deviation in the

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(a)

Coherent
F
1
0.8
0.6
0.4
0.2
0

40

Noncoherent

35
30
25

1
0.8

20
0.6

ε

15

0.4

k

10

0.2
0

5
0

1

(b)

(c)

a=2

log(εc)

εc

−0.4
0.8

a=2

−0.8

a=1.5

0.6

−1.2
0.25
0.23

0.4
0.21

1.4

0

10

−1.6

0.19

a=1.5
0.2

µ

20

30

40

50

1.6

2

kc

a

1.8

2.5

0.17

2

3

3.5

4

−2

log(kc)

0

log σ2

(d) ε=0.9 k=10

ε=0.8

−20

k=20
−30
0.667

0.668

(e)

0.669

0.877

ε

0.878 0

5

10

15

k

Fig. 3. Transition to coherence as a function of connectivity k and coupling strength
ε. (a) Fraction Nσ=0 of coherent states from 500 random initial conﬁgurations for a = 2. (b) Coherence transition curves in the (ε, k) plane for (from bottom to top) a = 1.5, 1.6, 1.8, 1.7, 1.9 and a = 2, and (c) the same transition in a log-log plot, showing power-law dependence between connectivity and coupling strength for the transition curves, with an exponent µ which depends on the value of a (see inset).
Here α = 0, L = 1000 and we used transients of 104 time steps. By increasing the transient size to ∼ 106 one clearly sees that the transition to coherence is of ﬁrst-order either (d) when varying the coupling strength ε or (e) when varying the outgoing connectivity k

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region where transition to coherence is observed, using much higher resolution. One clearly sees that the transition to coherence is of ﬁrst-order, when varying ε or k. That the transitions are indeed of ﬁrst order is easily recognised by the clear existence of hysteresis: when increasing either ε or k the conﬁguration eventually falls into a coherent state, no longer spontaneously desynchronizing, no matter how far the parameters are tuned back.
All results till now, concern the case of homogeneous coupling (α = 0).
Next, we study the case of heterogeneous coupling. Figure 4 shows the boundaries εL and εU in (4) as functions of the outgoing connectivity k, the nonlinearity a and the heterogeneity α, covering both the homogeneous and heterogeneous regimes. Figure 4a shows the two boundaries as a function of k for a = 2 and α = 0 (solid lines), α = 1 (dashed lines) and α = 2 (dotted lines).
As one sees for nonzero values of α the boundaries are no longer smooth curves, but instead they show ﬂuctuations as k is increased, enlarging and shrinking alternately the region of synchronizability, labelled as ‘sync’. When varying a (Fig. 4b) the boundaries are mainly controlled by the Lyapunov exponent of the local map (see (4)), where εL (resp. εU ) decreases (resp. increases) whenever a periodic window occurs [57]. The ﬂuctuations observed in Fig. 4a are clearly seen in Fig. 4c, where the stability boundaries are plotted in function of α ﬁxing k = 8 and a = 2 (solid lines) and a = 1.5 (dashed lines). The ﬂuctuations are much higher for α > 1 and for the fully chaotic regime both boundaries may even cross each other suppressing synchroniz2.5

1.4

ε

1.2
1.5

1.2

1

sync

sync
0.5

0.8
1

a=2 sync −0.5

0.6

0.8

a=1.5

−1.5

(b)

(a)
0.6

4

8

12

k

16

−2.5
1.5
20

1.6

0.4

(c)
1.7

1.8

a

1.9

20

0.5

1

1.5

2

2.5

3

0.2

α

Fig. 4. Boundary values εL and εU in (4) in function of (a) connectivity k with a = 2 and α = 0 (solid lines) α = 1 (dashed lines) and α = 2 (dotted lines), (b) nonlinearity a for k = 8 and α = 0, 1 and 2, and (c) heterogeneity α with k = 8 and a = 2 (solid lines) and a = 1.5 (dashed lines). The inset in (c) emphasises one small region where synchronizability is not observed, εL > εU (see text). Here N = 1000

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ability (see inset of Fig. 4c). Moreover, the lower boundary εL decreases till α ∼ 0.5, then increases till α ∼ 1 and decreases in average from there on.
All these analytical results computed from (4) and matrix G in (1) are corroborated by our numerical simulations. In particular, the boundaries εL and εU seen in Fig. 4c are obtained also when plotting the contour of Fig. 5a, where we plot the average standard deviation from a sample of 500 initial conﬁgurations and vary the coupling strength and heterogeneity for a = 2 and L = 1000. While Fig. 5a shows the numerical results for α > 0, i.e. in the case where nodes are more strongly coupled to the neighbours with higher connectivities, Fig. 5b shows the transition to coherence when α < 0. Here synchronizability is observed only for α −0.15 and for very high coupling strengths ε 0.95.

<logσ2>
0
-5
-10
-15
-20
-25
-30

<logσ2>

(a)
1

0

0.5

0
-5
-10
-15
-20
-25
-30

(b)
-0.2

0.95
1

-0.15

0.9
1.5

α

2

0.85
2.5

3

0.8
3.5

ε

-0.1
0.93

0.94

0.75
4 0.7

0.95

0.96

ε

0.97

α

-0.05
0.98

0.99

1

0

Fig. 5. Transition to coherence in function of coupling strength ε and heterogeneity α (a) when the most connected nodes dominate the dynamics (α > 0) and (b) when the coupling to nodes with the least neighbours is strengthened (α < 0). Here, we compute the average standard deviation from a sample of 500 initial conﬁgurations and ﬁx a = 2, k = m0 = 8 and N = 1000

We end our study of coherent solutions in random scale-free networks by investigating brieﬂy the role of hubs in the lattice. Instead of strengthening the coupling to the most connected nodes by increasing α > 0, we now ﬁx α = 0 and impose synchronization between all the nodes with more than a certain threshold kt of neighbours and observe which fraction of the initial conﬁgurations converges to a coherent state. In this case the transition to coherence converges asymptotically to a limit of the coupling strength, as shown in Fig. 6a. The same occurs when synchronization is imposed to all nodes with less than kt neighbours, as shown in Fig. 6b.
3.2 Deterministic Scale-Free Networks
In the previous section we focused on random scale-free networks, i.e. growing networks where new nodes are connected following probabilistic rules. In this section we study deterministic scale-free networks [37–39], using two diﬀerent deterministic topologies: the pseudo-fractal scale-free network introduced by

Coherence in Complex Networks

F

(a)

F

1

0.8

0.6

0.6

0.4

0.4

0.2

(b)

1

0.8

89

0.2
0

0
50
0.8

0.84

ε

0.92

150
0.96

1

200

200

1

100
0.88

160

0.8

ε

kt

0.6

120
0.4

80
0.2

40

kt

0

Fig. 6. Transition to coherence when synchronization is imposed to all nodes having a number of neighbours (a) larger than a threshold kt , and (b) smaller than kt
(see text). Here a = 2, k = 8, α = 0 and N = 1000

Dorogovtsev et al [38] and the Apollonian network introduced by Andrade et al. [39] and also studied in [40]. Both networks are illustrated in Fig. 7.
The pseudo-fractal network of Dorogovtsev is obtained, starting from three interconnected nodes, and at each iteration each edge generates a new node, attached to its two vertices. Figure 7a illustrates this network after three iterations, i.e. with three generations of nodes. The number of nodes
Nn and the number of connections Vn increase with the number of generations as [38]
Nn = 3 (3n + 1) ,
2
n+1

Vn = 3

(7a)

.

(7b)

From Fig. 7a one easily sees that this network has indeed a scale-free topology, since the number of nodes with degree k = 2, 22 , . . . , 2n−1 , 2n and 2n+1 is equal to 3n , 3n−1 , . . . , 32 , 3 and 3 respectively. In particular, the exponent of this power-law distribution is γ = 1 + ln 3/ ln 2. Moreover, the cluster

(b)
(a)

Fig. 7. Illustrations of two deterministic scale-free networks: (a) the pseudo-fractal network [41], and (b) the Apollonian network [39]. Identical symbols label nodes belonging to the same generation n (see text), namely for n = 0, for n = 1 and • for n = 2

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coeﬃcient of a node with degree k is C = 2/k, and the average path length is approximately
4 ln Nn /(9 ln 3).
The Apollonian network is constructed in a diﬀerent way: one starts with three interconnected nodes, deﬁning a triangle; at n = 0 one puts a new node at the center of the triangle and joins it to the three other nodes, thus deﬁning three new smaller triangles; at iteration n = 1 one adds at the center of each of these three triangles a new node, connected to the three vertices of the triangle, deﬁning nine new triangles and so on (see Fig. 7b). The number of nodes and the number of connections are respectively given by
Nn = 1 (3n+1 + 5) ,
2

(8a)

3 n+1
2 (3

(8b)

Vn =

+ 1) .

The distribution of connections obeys a power law, since the number of nodes with degree k = 3, 3 × 2, 3 × 22 , . . . , 3 × 2n−1 , 3 × 2n and 2n+1 is equal to
3n , 3n−1 , 3n−2 , . . . , 32 , 3, 1 and 3 respectively, and the exponent γ is the same as for the pseudo-fractal network. Moreover, a node with k neighbours has a cluster coeﬃcient of C
4/k as reported in [40], converging on average to
C∞ = 0.828, and the average path length grows weaker than ln Nn [39].
Although both networks have similar values for the topological quantities, they are quite diﬀerent from the geometrical point of view: the pseudo-fractal network has no metric, while the Apollonian network is embedded in Euclidean space and ﬁlls it densely as n → ∞, being particularly suitable to describe geographical situations [39].
For stability analysis purposes (see Sect. 2), the Laplacian matrix G of deterministic networks can be analytically determined from the adjacency matrix A = {aij }, since they are related by
G = I + AT ,

(9)

where I is the identity matrix and the values of matrix T = {Tij } are deﬁned by α
N
aji k=1 aik
(10)
Tij = − α .
N
N ajp apk p=1 k=1
A simple way to write the adjacency matrix of the pseudo-fractal network is
An =

An−1 Mn−1
MT
∅ n−1 ,

(11)

Nn ×Nn

where Nn is given by (7a), MT represents the transposed matrix of M and for each generation n = 1, 2, . . . the matrix Mn reads
Mn =

Mn−1 Mn−1 ∅
∅
∅ Bn−1

,
(2×3n−1 )×3n

(12)

Coherence in Complex Networks

⎡

with
Bn−1

A0
⎢ ∅
⎢
=⎢ .
⎣ .
.

∅ ...
A0 . . .
. ..
.
.
.
∅ ∅ ...

∅
∅
.
.
.
A0

91

⎤
⎥
⎥
⎥
⎦

(13)

3n−1 ×3n−1

and whose starting form is
⎡

⎤
011
M0 = A0 = ⎣ 1 0 1 ⎦
.
1 1 0 3×3

(14)

For the Apollonian network, the adjacency matrix is given by the same recurrence of (11), but this time with
⎡
⎤
0111
⎢1 0 1 1⎥
⎥
A0 = ⎢
(15)
⎣1 1 0 1⎦ ,
1110
and Mn being a matrix with (3n + 5)/2 rows and 3n columns and having in each column only three nonzero elements.
Figure 8 shows the eigenspectra of the Laplacian matrices for both the pseudo-fractal (Fig. 8a) and the Apollonian (Fig. 8b) networks, in function of the heterogeneity. As one sees for a = 2 (solid lines) synchronizability is observed only above α 1.5, and in particular there is no synchronizability for the homogeneous coupling regime (α = 0). Figure 9a shows the distribution of the average standard deviation over a sample of 500 initial conﬁgurations, from which one clearly sees that there are no coherent solutions. Here the standard mean deviation is characterized by some large value which is almost constant beyond the weak coupling regime (ε
0.2). In the weak coupling regime (ε
0.2) the standard mean square deviation is even larger, since the coupling is not strong enough to compensate the highly chaotic local dynamics (a = 2).
From Fig. 8 one also sees that, for the pseudo-fractal and α > 1.5, the upper threshold εU increases monotonically with the heterogeneity, while for the Apollonian network the upper threshold decreases. This particular diﬀerence between both networks should be due to their geometrical diﬀerences, in particular the fact that Apollonian networks are embedded in Euclidean space could explain in some way that stronger dominance in the coupling to the most connected nodes destroys coherence.
Choosing other values of a for which the local dynamics is chaotic, one ﬁnds that the form of de curves εL (α) and εU (α) does not change. These curves are only shifted: εL gets smaller, while εU increases. Figure 8 illustrates this for the particular case of a = 1.5. Decreasing even further the nonlinearity

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1.5

(a)

(b)

1

ε

Sync

Sync a=2 a=2

0.5

a=1.5

a=1.5

Pseudo−fractal
0

0

1

Apollonian
2

3

40

1

2

α

3

4

α

Fig. 8. Boundary εL and εU for synchronizability in function of the heterogeneity α for (a) the pseudo-fractal network and (b) the Apollonian network, with a = 2
(solid lines) and a = 1.5 (dashed lines). For each network we use 6 generations of nodes (see text). These eigenspectra are the same for any number of generations

below the accumulation point a = 1.411 . . . synchronizability is attained for any positive value of α, whenever the coupling strength is suﬃciently strong.
Moreover there is a complicated dependence of the average standard deviation on the coupling strength and nonlinearity. As shown in Fig. 9b for deterministic scale-free networks one ﬁnds two main regions in the (a, ε) plane:
(I) a region where the standard mean square deviation is large and varies
(b)

(a)

F

<log σ2>

1
0.8
0.6
0.4
0.2
0

0

-0.2
-0.4
-0.6
-0.8
-1

0.2
-0.9

0.4
-0.8

-0.7

-0.6

log

0.6
-0.5

σ2

1.5
1.6

ε

0

1.7

0.2
1.8

0.4

-0.4

-0.3

0.8
-0.2

1

ε

0.6

a

1.9

0.8
2

Fig. 9. (a) Typical histogram of the standard mean square deviation σ 2 for the pseudo-fractal network as a function of the coupling strength ε, with a = 2 and α = 0. A similar result is obtained for the Apollonian network. (b) Histogram of the standard mean square deviation σ 2 as a function of nonlinearity a and coupling strength ε, for deterministic scale-free networks with α = 0. The mean square deviation is averaged over a sample of 500 initial conﬁgurations and during 100 time steps, after discarding transients of 104 time steps

Coherence in Complex Networks

93

smoothly with the parameters and (II) a region where the mean square deviation is smaller but has larger ﬂuctuations. This second region, observed for a
1.7, is somehow surprising, since irregular variations of the standard mean square deviation occur for low nonlinearity and high coupling strengths, precisely where one would expect the most regular behaviour of the node dynamics.
As for the heterogeneous coupling regime (α = 0), Fig. 10 illustrates the transition to coherence by varying the heterogeneity α for the pseudo-fractal
(Fig. 10a) and the Apollonian network (Fig. 10b). For both networks, one sees that coherence sets in for α 1.5, and the contour of the histograms marking the transition to coherence ﬁts well the regions in Fig. 8 labelled as ‘sync’.
Moreover, from Figs. 10c and 10d one observes that all these transitions to coherence are of ﬁrst-order. log σ2

log σ2

(a)

0
-5
-10
-15
-20
-25
-30

(b)

0
-5
-10
-15
-20
-25
-30

2

α

3
4 1

0.9

0.8

ε

0.7

0.6

0.5

2

α

3
4 1

0.9

0.8

ε

0.7

0.6

0.5

log σ2

0

Pseudo fractal Apollonian

Pseudo fractal −20
Apollonian

(c)
−30
0.75

0.76

ε

0.77

0.955

ε

0.965

(d)
0.975

Fig. 10. Inducing transition to coherence by varying the heterogeneity α (see (1)) in scale-free networks. (a) Pseudo-fractal network and (b) Apollonian network. For strong heterogeneity coherence appears beyond a relatively high coupling strength, and disappears again for very large couplings (see text). For each network, we use
= 6 generations of nodes and ﬁx a = 2. (c) and (d) show high-resolution plots of σ 2 as a function of ε for α = 2, emphasizing the ﬁrst-order phase transition to coherence Finally, we study the role of hubs in deterministic scale-free networks, as we did in the previous section for random networks. To this end, we impose synchronization among g = 1, . . . , generations, with being the total number of generations, and observed in what conditions coherent states are observed. In the pseudo-fractal network the ﬁrst generation has N1 − N0 = 3 nodes, the second one has N2 − N1 = 9 nodes, and the nth generation has

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0

(a)

(b)

log σ2

g<l−2

g<l−2

g=l−2

g=l−2

−20

g=l−1
0.499

−30

0

0.5

0.501

0.2

0.499

0.4

0.6

ε

0.8

0

0.5

0.2

g=l−1

0.501

0.4

0.6

0.8

1

ε

Fig. 11. Transitions to coherence in deterministic scale-free networks, when synchronizing the ﬁrst g generations of nodes out of generations (see text). (a)
Pseudo-fractal network and (b) Apollonian network. The collective dynamical behaviour is quite insensitive to hubs (see text). Insets show that transitions to coherence are of ﬁrst-order. For each network, we use = 9 generations of nodes and a = 2 ﬁxed

Nn − Nn−1 = 3n nodes. In the Apollonian network the number of nodes appearing at each generation is precisely the same.
Figure 11 shows the standard mean square deviation as a function of coupling strength for pseudo-fractal (Fig. 11a) and Apollonian networks
(Fig. 11b). In each case we choose the fully chaotic map (a = 2) and impose synchronization among the nodes of the ﬁrst g generations by setting them to their mean amplitude at each time-step. In both cases, one sees that the standard mean square deviation remains large when synchronization is imposed to all g < − 2 generations. Coherent solutions are only observed for g = − 2 and g = − 1, beyond a coupling threshold which is smaller for the latter case. Surprisingly, for g = − 1 the transition to coherence occurs for the same coupling strength in both networks. This may be due to the fact that the fraction Ng /N = 3g− = 1/3 of nodes on which one imposes synchronization is the same for both networks and is high enough to suppress the inﬂuence of local connectivities.
For g = − 2 the pseudo-fractal network shows coherence only above very high coupling strengths, near ε ∼ 1, while for Apollonian networks the threshold is much lower. This diﬀerence in the coupling strength threshold is due to the fact that here the fraction of nodes Ng /N = 1/9 to which synchronization is imposed is small enough not to suppress the inﬂuence of local connectivities. Therefore, since the hubs in the pseudo-fractal network are less connected than the hubs in Apollonian networks, one needs higher coupling strength to observe coherence. For any higher value of generations the same results are obtained, since one has for the quotient of the number of nodes between two successive generations Nn /Nn−1 → 3 as n increases.

Coherence in Complex Networks

95

As a general remark, one observes from Fig. 11 that one needs to synchronize a rather high fraction of nodes ( 1/9) to induce coherence. Therefore, it seems that, dynamical collective behaviour on scale-free networks is quite insensitive to hubs. As shown in the insets of Figs. 11a and 11b, the transition to coherence is also of ﬁrst-order.

4 Discussion and Conclusions
In this chapter we studied fully synchronized solutions for three scale-free network topologies. The main conclusion is the following: in random scale-free networks synchronization of chaotic maps not only depends on the coupling strength but is mainly controlled by the outgoing connectivity k, which is a measure of the cohesion in the networks. Because of that, one ﬁnds coherent solutions in random scale-free networks of fully chaotic logistic maps
(a = 2) with outgoing connectivity k = 8 and homogeneous coupling, but not in deterministic scale-free networks, since they have rather small eﬀective outgoing connectivity, namely k = 2 for the pseudo-fractal network and k = 3 for the Apollonian network. Therefore, although the exponent γ of connection distributions in scale-free networks does not depend on the outgoing connectivity [2], we have shown that, in general, synchronization of chaotic maps in such coupling topologies is quite sensitive to it.
Our results were obtained both numerically, from histogram of signiﬁcantly large samples of initial conﬁgurations with a criterion for full synchronization based on the mean standard deviation of amplitudes, (5), and analytically from the eigenvalue spectra of the diagonalized variational equations computed at the coherent states, (3).
In particular, for random scale-free networks, the threshold values of the coupling strength obey a power law, (6), as function of the outgoing connectivity. The exponent of this power law depends on the nonlinearity a of the chaotic map, being almost constant below ac ∼ 1.7 and decreasing linearly above it. Interestingly this value of ac is in the vicinity of the bifurcation of the quadratic map where the period-3 window appears, and coincides with the appearance of other nontrivial behaviours in coupled map lattices with regular topologies, namely in the velocity distribution of travelling wave solutions [24].
For deterministic scale-free networks with homogeneous coupling, the same value ac indicates the threshold above which no coherent solutions are observed, independently of the coupling strength. Above ac , coherence is observed only for heterogeneous coupling, namely for α
1.5. However, for this range of values, we have also shown that coherence is also absent either for very small or for very large coupling strengths, due to spatial instabilities. Another particularly interesting result that still needs to be explained is that, for Apollonian networks, the coupling threshold beyond which coherence disappears gets smaller when the heterogeneity is further increased.

96

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This point is not observed for the pseudo-fractal network and may be due to the geometrical diﬀerences between both deterministic networks.
As a general property, we have shown that all transitions to coherence are of ﬁrst-order, indicating a similarity with other complex networks [29].
Furthermore, all results are robust not only against changes of the initial conﬁgurations of node amplitude but also, in random scale-free networks, against changes of the connection network. We also presented results indicating that in scale-free networks hubs play apparently no fundamental role in the dynamical collective behaviour.

Acknowledgments
The authors thank A.O. Sousa and C. Zhou for useful discussions. P.G.L. thanks Funda¸˜o para a Ciˆncia e a Tecnologia, Portugal, for ﬁnancial supca e port. J.A.C.G. thanks Conselho Nacional de Desenvolvimento Cient´ ıﬁco e
Tecnol´gico, Brazil and Sonderforschungsbereich 404 of DFG for ﬁnancial o support.

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Growth of Random Sequences
K. Austin and G. J. Rodgers
Department of Mathematical Sciences, Brunel University, Uxbridge, Middlesex
UB8 3PH, U.K.

Summary. We study the behaviour of two classes of random sequences. In these sequences each new element is generated by adding together two or more of the previous elements, or multiples of previous elements in the sequence. At least one of the elements is chosen randomly from a probability distribution of the previous elements. We ﬁnd that a wide range of diﬀerent types of behaviour emerge from linear to exponential growth and that the sequences exhibit a remarkably diverse phase space. Interestingly, new transitions in phase space are observed when the generating equations correspond to the backward diﬀerence equations.

Key words: Random sequences, Fibonacci sequence, Disordered systems,
Anderson model

1 Introduction
Integer sequences have attracted widespread interest in many ﬁelds of science since (in the 13th century) Fibonacci discovered a sequence whose numbers are found frequently both in nature and in man-made systems.
1.1 The Fibonacci Sequence
The numbers of the Fibonacci sequence,
Fn+1 = Fn + Fn−1 ,

(1)

are the same as the numbers of petals found on ﬂowers, the numbers of seeds on the spiral arms of a sunﬂower, the numbers that give the spacial patterns of leaves growing up the stems of plants and the spiral pattern of a snail’s shell. More recently, the Fibonacci numbers have also been found to have applications in man-made systems. For example, in computer science the Fibonacci numbers appear in database structures and random number generation. An important property of the Fibonacci sequence is the “golden ratio”. The Fibonacci sequence grows exponentially, with the nth element of
√
√ the sequence given by Fn ∼ [(1 + 5)/2]n . The ratio (1 + 5)/2 1.61803 . . . is found everywhere in nature and was even known about in the time of the ancient Greeks, who used it for its aesthetically pleasing proportions.

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Interest in integer sequences has more recently focussed on the introduction of random properties to incorporate stochasticity into the sequences.
In [1], Furstenberg and Kesten showed that for a general class of randomsequence generating processes, the absolute value of the nth member of the sequence will, with probability 1, get closer to the nth power of some ﬁxed number, as with the Fibonacci sequence. Furstenberg and Kesten’s work has been found to have many applications. It was used in the ﬁrst rigorous explanation of Anderson localization, which came long after the mathematics had been worked out.
1.2 Random Fibonacci Sequences
Viswanath [2] made use of Furstenberg and Kesten’s work in the study of a stochastic version of the Fibonacci sequence, xn+1 = ±xn ± xn−1 ,

(2)

where the signs are chosen independently with equal probability. Unlike the deterministic rules of the original Fibonacci sequence, Viswanath’s [2] rules are stochastic and generate not one but many diﬀerent sequences.
Viswanath [2] showed that when the absolute value of the numbers of any sequence generated by these rules is taken, the nth number in the sequence will, on average, get closer to the nth power of the number 1.1398... Therefore the sequences generated by these stochastic rules grow exponentially, with yn ≡ |xn | ∼ (1.1398...)n , just like the Fibonacci sequence, but with a new “stochastic golden ratio” of 1.1398...
1.3 Generalised Random Fibonacci Sequences
In [3], Embree and Trefethen generalised this result to show that for the sequence generating equation xn+1 = xn ± βxn−1 ,

(3)

β = β ∗ ∈ (0.702582, 0.702585) ,

(4)

there is a critical value of

above which the sequences grow exponentially, but below which they decay exponentially. At the critical value β = β ∗ , no exponential growth or decay occurs. When two constant parameters are used rather than one, xn+1 = αxn ± βxn−1 , the same results are obtained by simple rescaling. This is because the random signs ensure that the terms xn and xn−1 are uncorrelated.

Growth of Random Sequences

101

1.4 Alternative Random Sequences
An alternative stochastic version of the Fibonacci sequence is introduced in [4]. Rather than adding or subtracting the two preceding elements with equal probability as in [2] and [3], random elements are added according to
(i) xn = xn−1 + xq

and

(ii) xn = xp + xq ,

(5)

where p and q are chosen randomly between 0 and n − 1. The sequences produced by (i) grow more slowly than the sequences in [2] and [3], with stretched exponential growth. The value of the √ average nth element of the se−1/4 exp (2 n). The sequences produced quence is given by < xn >= An ∝ n n + 1, or have power-law growth, by (ii) grow linearly according to An with An ∼ n2c−1 , if a multiplicative factor is introduced into the generating equation so that xn = c(xp + xq ) [4].
1.5 Applications
Just as the Fibonacci sequence is linked to a variety of deterministic systems in physics, biology and computer science, so too are stochastic sequences linked with random or disordered systems. Stochastic sequences are present in the random walk that appears in models of many stochastic processes in a variety of ﬁelds. For example, Einstein’s explanation of Brownian motion was one of the earliest to incorporate the random walk [5]. Other examples include models of evolutionary biology to model adaptive walks on a ﬁtness landscape
[6] and to ﬁnancial markets [7] to model random elements of the markets such as investor speculation and consumer demand. Stochastic sequences also appear in a number of problems in one-dimensional disordered systems [8], such as the Anderson model describing the propagation of electrons in a metal with impurities. In this model the electrons interact with the inhomogeneous potential of the system. The rates of transition of the electrons from one site to another is a random function of position and therefore analogous to the random progression of the stochastic sequence.
1.6 More Random Sequences
In this chapter we investigate the eﬀect of generalising some of the random sequences introduced so far. We ﬁrstly consider sequences where the random elements are chosen randomly from a separable probability distribution of the previous elements in the sequence. We then investigate the eﬀect of introducing multiplicative constants to the individual terms in the generating equations of random sequences where the random elements are chosen from a uniform probability distribution of the previous elements in the sequence.
In Sect. 2, we consider the sequences generated by xn = xn−1 + γxq ,

(6)

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with γ > 0 and q chosen randomly from a probability distribution Pn (q), [9].
This is a generalisation of sequences considered in [4], where the case for
Pn (q) = 1/n and γ = 1 was considered. The mean and variance of the sequences generated by Pn (q) = 1/n and the sequences generated by powerlaw and exponential Pn (q) are obtained, as well as non-random Pn (q) = δq,β(n−1) .
In Sect. 3, we investigate the eﬀect of introducing multiplicative constants to the individual terms in the generating equations of the sequences studied in [4] and introduce a new sequence as an extension of these studies. In particular, we investigate the sequences generated by
⎧
(Section 3.1)
⎨ γxp + δxq ;
(Section 3.2) xn = αxn−1 + γxq ;
⎩
αxn−1 + βxn−2 + γxq . (Section 3.2)
Since the random signs in [2] and [3] are removed, the terms become correlated so that the growth of the resulting sequences depends on the values of the multiplicative constants.
We ﬁnd that for each type of generalisation, new types of behaviour emerge. In particular, the sequences are found to exhibit linear and exponential growth at the extremes of the free parameters, with intermediate types of growth in-between. We summarise and discuss these results in Sect. 4.

2 Sequences with Random Elements Chosen from a Probability Distribution
In this section we consider the sequences generated by xn = xn−1 + γxq ,

(7)

where q is chosen randomly from the separable probability distribution n−1 th
Pn (q) = P (q)/bn , with bn = element, q=0 P (q), [9]. The average n
An =< xn > is given by
An = An−1 +

γ bn n−1

P (q)Aq .

(8)

q=0

We consider uniform, power-law and exponential forms of P (q) and then go on to look at non-random Pn (q) = δq,β(n−1) .
2.1 Uniform P (q)
First we consider the case of P (q) = 1 and hence bn = n. Substituting this into (8), multiplying through by n and subtracting the equivalent expression for n + 1, we obtain

Growth of Random Sequences

An+1 − 2 +

γ−1 n+1 An +

n
An−1 = 0 . n+1 103

(9)

This is equal to the nth Laguerre polynomial, Ln (−γ), [10] and for large n yields the asymptotic solution
√
An ∼ kγ n−1/4 exp(2 γn) ,
(10)
[9]. This is the result obtained in [4] for γ = 1. We can calculate the average growth of the second moment, Vn = x2 for large n, by introducing two n averages n−1 Vn = x2 n and Mn =

< xn xr > .

(11)

r=0

Using (7), it is simple to show that Vn and Mn obey the coupled iterations
(n + 1)Vn+1 − [2n + (γ + 1)2 ]Vn + (n + 2γ)Vn−1 = 2γ(Mn − Mn−1 )

(12)

and
(n + 1)Mn+1 − (2n + 2γ + 1)Mn + nMn−1 = (n + γ + 1)Vn − nVn−1 . (13)
We can then take the continuum limit for continuous n to give the secondorder diﬀerential equations
[nV (n)] − (2γ + 1)V (n) − γ 2 V (n) = 2γM (n)

(14)

[nM (n)] − M (n) − 2γM (n) = [nV (n)] + V (n) + γV (n) .

(15)

and
In the limit n →√ we can equate leading order terms to give the solutions
∞,
√
V (n) ∼ nφ exp(δ n) and M (n) ∼ nφ+1/2 exp(δ n), where δ= 2γ 4 + γ +

16 + γ 2

for γ > 0, [9]. This is in good agreement with [4], where δ numerically for γ = 1.

(16)
4.3 was obtained

2.2 Power-Law P (q)
If P (q) takes the form of a power-law, such that
P (q) = (q + 1)α

(17)

and γ = 1, ﬁve diﬀerent classes of behaviour emerge for the cases i) α > −1, ii) α = −1, iii) −2 < α < −1, iv) α = 2 and v) α < −2. We consider each of these in turn.

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i) α > −1
For α > −1, bn /bn+1 = 1 − (α + 1)/n + O(1/n2 ) for large n. From (8) we obtain α+1
An−1 .
(18)
An+1 − 2An + An−1 = n This equation can be solved exactly in terms of generalised Laguerre polyno(−α) mials [10]. In this case, An ∼ Ln (−(α + 1)), and we obtain the asymptotic solution 1
An ∼ cα (2α+1)/4 exp[2 (α + 1)n]
(19)
n as n → ∞, with
(α + 1)(2α−1)/4 e−(α+1)/2
√
cα =
,
(20)
2 π
[9].
ii) α = −1
When α = −1, bn /bn+1 = 1 − 1/(n + 1) log n + O(1/n(log n)2 ) and (8) gives
An+1 − 2An + An−1 =

1
An−1
(n + 1) log n

(21)

as n → ∞. Using the WKB method [11], An is found to obey
An ∼ √

1 exp 2 n log n

n log n

(22)

for large n. The growth of the sequences for α = −1 is slower than for α > −1 as Pn (q) is no longer as strongly dominated by q ∼ n. iii) −2 < α < −1
For −2 < α < −1, bn /bn+1 = 1 − (n + 1)α /ζ(−α) + O(nα−1 ) where ζ is the
Riemann zeta function [12]. For large n (8) gives
An+1 − 2An + An−1 =

(n + 1)α
An−1
ζ(−α)

(23)

and the WKB approximation [11] yields
An ∼

1 nα/4 exp

n1+α/2
2
. α + 2 ζ(−α)

(24)

Again, the growth of the sequences is found to be slower than for α ≥ −1 as
Pn (q) is not as strongly dominated by q ∼ n as for α > −1 or for α = −1.

Growth of Random Sequences

105

iv) α = −2
For α = −2 the large n behaviour of An is purely power-law with
1

An ∼ n 2 where the exponent 1/2 + 1/

+√1

ζ(2)

(25)

ζ(2) ≈ 1.108 is greater than 1.

v) α < −2
Finally, for α < −2, the right hand side of (23) can be neglected so that (23) yields An ∼ n
(26)
as n → ∞. The sequences are now dominated by Pn (q) ∼ 1, resulting in linear growth.
2.3 Exponential P (q)
Next we consider an exponential form of P (q), P (q) = aq . Again there are found to be diﬀerent solutions, in this case for diﬀerent values of a. For i) a = 1 the solution An = Ln (−1) is recovered. When ii) a < 1, bn → 1/(1 − a) for large n and An = n. When iii) a > 1, bn ∼ an /(a − 1) as n → ∞ and hence n
1
An ∼ 1 + 1 −
.
(27) a We ﬁnd that the average sequence growth is linear for a < 1 when Pn (q) is dominated by q ∼ 1, exponential for a > 1, when Pn (q) is dominated by q ∼ n and intermediate for a = 1.
2.4 Non-Random P (q)
Next we look at the non-random sequence xn = xn−1 + xβ(n−1)

(28)

where 0 ≤ β ≤ 1 is ﬁxed and the index in the last term on the right hand side is rounded to the nearest integer. If β = 0, the sequence grows linearly, with xn = n and for β = 1, the growth of the sequence is exponential, with xn = 2n . For 0 < β < 1 we can solve (28) for large n by taking the continuum limit of continuous n to give dx = x(βn) . dn (29)

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By substituting a power series solution for x(n) into (29) and solving for the coeﬃcients, we obtain
∞

x(n) = r=0 1

nr β 2 r(r−1)
.
r!

(30)

For large n this summation is dominated by the term r ∼ log(n)/ log(1/β) and evaluating the summation for large n yields xn ∼ exp

(log n)2
1
2 log( β )

.

(31)

Hence for 0 < β < 1 an intermediate type of growth occurs.

3 Random Sequences with Multiplicative Constants
In this section we investigate the eﬀect of introducing multiplicative constants to the generating equations of the random sequences studied in [4]
Table 1. Summary of results for Sect. 2. Here, ζ is the Riemann zeta function [12] and cα and kγ are coeﬃcients for the Laguerre polynomials Ln (−α) (−(α + 1)) and
Ln (−γ) respectively, [13]
Pn (q)

An α < −2

n

α = −2

(q + 1)α

n2

−2 < α < −1

1+√1 ζ(2) 1

α

n4

α = −1

√ 1 n log n

α > −1

cα

1

n

aq

n1+α/2
2 √ α+2 ζ(−α)

exp

exp

2α+1
4

2

exp[2

n log n

(α + 1)n]

α<1

n

α=1

√ kγ n−1/4 exp(2 γn)

α>1 δβ(n−1) 1+

β=0

exp

β=1

2n

1 a n

0<β<1

1−

(log n)2
1
2 log( β )

n

Growth of Random Sequences

107

and also introduce a new random sequence as an extension of these studies.
We consider the sequences generated by xn = γxp + δxq , xn = αxn−1 + γxq and xn = αxn−1 + βxn−2 + γxq in turn.
3.1 Sequences Generated by Randomly Chosen Previous Elements
Introducing multiplicative constants to the generating equation xn = xp + xq studied in [4] gives
(32)
xn = γxp + δxq where γ and δ are constants and p and q are chosen randomly between 0 and n − 1. The average value of xn , An , can be shown to obey γ+δ An = n n−1

Ar

(33)

r=0

and can be simpliﬁed to give the recursion relation
An =

n+γ+δ−1
An−1 . n (34)

For large n, the solution to this equation is An ∼ nγ+δ−1 . This is the result obtained in [14] for δ = 0. It is a simple matter to show that adding extra random terms with multiplicative constants to (32) simply multiplies this result by extra constant powers of n. Following the method of [9], in Sect. 2, we next introduce the averages n−1 Vn = x2 n and

Mn =

< xn xr >

(35)

r=0

to give the two coupled iterations
(n + 1)2 Vn+1 − [(δ 2 + γ 2 + 2n)n + (δ + γ)2 ]Vn

(36)

+[(δ + γ )n − (δ − γ) + (n − 1) ]Vn−1 = 4αγ(Mn − Mn−1 )
2

2

2

2

and
(n + 1)Mn+1 − (2(α + γ) + n)Mn = (α + γ)Vn .

(37)

Since we are only interested in the large n behaviour of the sequences, we take the continuum limit of the iterations for continuous n to give the second order diﬀerential equations
(n + 1)2 V (n) + [(4 − δ 2 − γ 2 )n+
(δ − γ)2 ]V (n) + 2(1 − δ 2 − γ 2 )V (n) = 4δγM (n) and (38)

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(n + 1)M (n) + (n + 1)M (n) + (1 − 2α − 2γ)M (n) = (α + γ)V (n) . (39)
2

2

In the limit n → ∞, the solutions are V (n) ∼ nα +γ −1 and M (n) ∼
2
2 nα +γ −1 for (α − 1)2 + (γ − 1)2 = 1 in the region [ α ≥ 0, γ ≥ 0 and α + γ ≥ 1 ]. This was the result obtained in [14] for α = 2 and γ = 0.
For 1 < (α − 1)2 + (γ − 1)2 < 2 in the region [ α ≥ 0, γ ≥ 0 and
2
2 α + γ ≥ 1 ], the solutions are V (n) ∼ nα +γ −1 and M (n) ∼ n2α+2γ−1 , and for (α − 1)2 + (γ − 1)2 = 2 in the region [ α ≥ 0, γ ≥ 0 and α + γ ≥ 1
], the solutions are V (n) ∼ n2(α+γ−1) and M (n) ∼ n2α+2γ−1 . Thus for
1 ≤ (α − 1)2 + (γ − 1)2 ≤ 2 in the region [α ≥ 0, γ ≥ 0 and α + γ ≥ 1] we were able to ﬁnd the average growth of the second moment, Vn = xn 2 , and show that in this region it always diverges faster than the mean, An =< xn >.
3.2 Sequences Generated by One Randomly Chosen and One Non-Random Previous Element
Adding multiplicative constants to xn = xn−1 + xq [4] gives the more general equation (40) xn = αxn−1 + γxq .
An is then given by
An = αAn−1 +

γ n n−1

Aq .

(41)

q=0

Following the method that was used in Sect. 2 to obtain V (n) and M (n), we obtain the second order diﬀerential equation
(n + 1)A (n) + [(1 − α)n + 1]A (n) + (1 − α − γ)A(n) = 0

(42)

for large n in the limit of continuous n. We ﬁnd that the form of the solution to this equation is dependent upon the multiplicative constant α, with α = 1 a special case. In particular, there are diﬀerent solutions for i) α > 1, ii) α = 1 and iii) α > 1. We consider these in turn.
i) α > 1
For α > 1, the solution is exponential, with
A(n) ∼ exp [(α − 1)n] .

(43)

In this region the growth of the average sequence is dominated by the term xn−1 in the sequence generating equation so that the sequence growth is exponentially fast.

Growth of Random Sequences

109

ii) α = 1
For α = 1, there is intermediate growth of the average nth element, given by
√
An ∼ exp [2 γn]

(44)

as found in [4] and [9]. This ‘special’ case corresponds to the ﬁrst backward diﬀerence equation with an added random term. Here the sequences are not so strongly dominated by the xn−1 term in the generating equation and the growth is slower than exponential. iii) α < 1
When α < 1, the average sequence growth is purely power-law, with
An ∼ n

α+γ−1
1−α

.

(45)

In this region the sequences are no longer dominated by the xn−1 term and the solution has the same form as the solution to (32).
To summarise, on average the sequences are found to grow exponentially fast when α > 1, as in this region the growth is dominated by the xn−1 term in the generating equation. For α < 1, the average sequence grows according to a power law as the xn−1 term no longer dominates. For α = 1, an intermediate type of growth occurs when the generating equation corresponds to the ﬁrst backward diﬀerence equation. Results for Sect. 3 are summarised in Table 2.
3.3 Sequences Generated by One Randomly Chosen and Two Non-Random Previous Elements
The next sequences we study are those generated by the equation xn = αxn−1 + βxn−2 + γxq .

(46)

This equation looks like the original Fibonacci sequence but with an extra random term and multiplicative constants. Following the method of Sect. 3.2, we ﬁnd that the average of the nth element obeys the third order diﬀerential equation −βnA (n) + [(β + 1)n + 1]A (n)

(47)

+[(1 − α − β)n + β + 1]A (n) − (1 − α − β − γ)A(n) = 0 in the limit of continuous n as n → ∞. The resulting sequences are dependent on both of the constants α and β, with four cases; i) [α + β > 1, β = 0] and
[α + β = 1, α > 2], ii) α = 2 and β = −1, iii) [α + β = 1, α < 2] and iv) α + β < 1.

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K. Austin and G.J. Rodgers

i) [α + β > 1, β = 0] and [α + β = 1, α > 2]
We ﬁnd that there are two regions in phase space, [α + β > 1, β = 0] and
[α + β = 1, α > 2] where the average sequences grow exponentially according to β + 1 + (β + 1)2 + 4β(1 − α − β) n .
(48)
An ∼ exp
2β
In these regions the sequence growth is dominated by the xn−1 and xn−2 terms. For β = 0 and α > 1, the average sequence still grows exponentially, but according to (43). iii) α = 2 and β = −1
For α = 2 and β = −1, there is the ‘special’ solution
An ∼ exp

3
(γn2 )1/3 ,
2

(49)

when the generating equation corresponds to the second backward diﬀerence equation with an added random term. This is a new intermediate type of sequence growth resulting from the extra term, xn−2 , in the sequence generating equation. iv) [α + β = 1, α < 2]
For [α + β = 1, α < 2] there is another intermediate type of growth given by
An ∼ exp 2

γn β+1 ,

(50)

corresponding to the solution for (40) when α = 1.
v) α + β < 1
Finally for α + β < 1, the solution is again purely power-law, with
An ∼ n

α+β+γ−1
1−α−β

.

(51)

The sequences are no longer dominated by the xn−1 and xn−2 terms in the generating equation and the solution takes the same form as that for (40) when α < 1.
In summary, we ﬁnd that there are two regions of phase space, [α + β > 1] and [α + β = 1, α > 2], where the average sequence grows exponentially fast. In these regions the sequence growth is dominated by the xn−1 and xn−2 terms in the sequence generating equation. For α + β < 1, the sequence

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111

growth is no longer dominated by the xn−1 and xn−2 terms and the sequences grow according to a power law. For [α + β = 1, α < 2], an intermediate type of growth occurs, and interestingly a new intermediate type of growth occurs for [α + β = 1, α = 2], when the generating equation corresponds to the second backward diﬀerence equation. Results for Sect. 3 are summarised in
Table 2.
Table 2. Summary of results for Sect. 3 xn An

γxp + δxq

nγ+δ−1

αxn−1 + γxq

α+γ−1
1−α

α<1

n

α=1

√ exp [2 γn]

α>1

exp[(α − 1)n]

αxn−1 + βxn−2 + γxq α + β < 1

n

α+β+γ−1
1−α−β

[α + β = 1, α < 2]

exp 2

[α = 2 and β = −1]

exp

[α + β > 1, β = 0]

exp

γn β+1 3
(γn2 )1/3
2
β+1+

√

(β+1)2 +4β(1−α−β) n 2β

and [α + β = 1, α > 2]

4 Discussion
We have generalised previously studied random sequences in two diﬀerent ways to study the eﬀect on their behaviour. In Sect. 2 we considered sequences of the form xn = xn−1 + γxq , where q = 0, 1, 2, ..., n − 1 is chosen randomly from the probability distribution Pn (q). We found the exact solution for the mean of the sequence < xn >= An when Pn (q) = 1/n and showed that the average of the second moment x2 = Vn diverges faster n than the mean. We also considered power-law, exponential and non-random forms of Pn (q) and found that the sequences can grow exponentially, linearly, or with intermediate growth. The sequences were found to exhibit exponential growth when Pn (q) is dominated by q ∼ n and linear growth when q ∼ 1.
Between these regimes the sequences exhibit an intermediate type of growth.
The boundaries of the diﬀerent types of behaviour are summarised in Table
1.

112

K. Austin and G.J. Rodgers

In Sect. 3, we investigated the eﬀect of introducing multiplicative constants to the individual terms in the generating equations of both new and previously studied random sequences. We obtained analytical results for the mean of the random sequences generated by (i) xn = γxp + δxq , (ii) xn = αxn−1 + γxq and (iii) xn = αxn−1 + βxn−2 + γxq , where p and q are chosen randomly between 0 and n − 1, and α, β, γ and δ are constants, with γ > 0 and δ > 0. We found that the solutions depend on all the multiplicative constants, while the form of the solution depends on α and β for γ > 0 and δ > 0. For (i), each element in the sequence is made up from randomly chosen previous elements and the average sequence grows according to a power law. We were also able to ﬁnd the average second moment,
Vn = x2 , for 1 ≤ (α − 1)2 + (γ − 1)2 ≤ 2 in the region [ α ≥ 0, γ ≥ 0 and n α + γ ≥ 1 ] for these sequences and to show that in this region, Vn = x2 n always diverges faster than An =< xn >. For (ii) and (iii), we found that the type of growth that the average sequences exhibit is highly sensitive to the constant multiplicative parameters α and β. In (ii), when α > 1, the next element in the sequence is dominated by the most previous elements and the average sequence grows exponentially. For α < 1, the elements in the sequences are no longer dominated by the most previous elements and the average sequence grows according to a power law, as with (i). There is a transition at α = 1, when the generating equation corresponds to the ﬁrst backward diﬀerence equation with an added random term. In this case an intermediate type of growth occurs and we ﬁnd that the average sequence grows as a stretched exponential. For (iii), a similar behaviour occurs, where in this case for α + β > 1, the average sequence grows exponentially and for α + β < 1, the average sequence grows according to a power law. However, for (iii), the intermediate behaviour becomes more complex, with three types of behaviour emerging for α + β = 1. For α + β = 1, α > 2, the average sequence continues to grow exponentially. For α + β = 1, α < 2 the average sequence grows as a stretched exponential, but for α = 2, β = −1, another intermediate type of behaviour is observed when the generating equation corresponds to the second backward diﬀerence equation with an added random term. This suggests that if we add more non-random previous elements with multiplicative constants to the generating equations, we obtain extra dimensions in phase space for the solutions to the average sequence growth. Also, new transitions are observed when the generating equations correspond to the backward diﬀerence equations.
Although these sequences are fairly straightforward generalisations of those previously studied, they are found to display a wide range of new and interesting behaviour with a remarkably diverse phase space.

Acknowledgments
KA would like to thank the EPSRC for their ﬁnancial support.

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113

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A. Einstein, R. Furth, A.D. Cowper: Investigations on the Theory of the Brownian Movement, (Dover Publications Inc., New York 1956)
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J.M. Luck: Syst`mes D´sordonn´s Unidimensionnels, (Alea-Saclay, 1992) e e e I. Krasikov, G.J. Rodgers, C.E. Tripp: J. Phys. A 37, 2365 (2004)
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Logistic Population Growth and Beyond:
The Inﬂuence of Advection and Nonlocal Eﬀects
Emilio Hern´ndez-Garc´ and Crist´bal L´pez a ıa o o
Instituto Mediterr´neo de Estudios Avanzados IMEDEA (CSIC-UIB), Campus a de la Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain emilio@imedea.uib.es clopez@imedea.uib.es

1 Introduction
By introducing the logistic equation in the context of demographic modelling [1], J.-F. Verhulst made seminal contributions to at least two important ﬁelds of research: On the one hand, to the quantitative approach to Population Dynamics, a subject in which substantial development beyond Verhulst’s ideas only occurred 100 years after him. On the other, to the basics of Nonlinear Science through a detailed study of the predictions of a nonlinear model and its comparison with experimental data.
The dynamics of biological populations in aquatic environments [2–4] is an excellent framework to see recent developments in which these disciplines work together. Growth, limitation, competition, predation and all the other kinds of biological interactions appear in this context, and nonlinear processes are relevant both in the biological dynamics and in the motion of the turbulent ﬂuid in which the population lives.
In this contribution we present two examples in the above ﬁeld. In both cases a prominent role is played by the logistic growth process (i.e. population growth limited by ﬁnite resources), but other ingredients are also included that strongly change the phenomenology. In Sect. 1, a phytoplankton population experiencing logistic growth is studied, but in interaction with zooplankton predators that maintain it in a state below the carrying capacity of the supporting medium. In the appropriate parameter regime the system behaves in an excitable way, with perturbations inducing large excitationdeexcitation cycles of the phytoplankton population. The excitation cycles become strongly aﬀected by the presence of chaotic motion of the ﬂuid containing the populations.
In Sect. 2, an individual based model of interacting organisms is presented, for which logistic growth is again the main ingredient. Reproduction of a given individual is limited by the presence of others in a neighbourhood of ﬁnite size.
This nonlocal character of the interaction is enough to produce an instability of the basic state of particles homogenously distributed, and clustering of http://www.imedea.uib.es/PhysDept 118

E. Hern´ndez-Garc´ and C. L´pez a ıa o the individuals occurs, which form groups arranged in an hexagonal lattice
(when the population lives in a two-dimensional space).

2 Plankton Dynamics Driven by an Environmental Open Flow
Plankton is the generic name given to a large number of aquatic organisms, most of them microscopic, living in the oceans, lakes or rivers, and characterized by the fact that they are transported by water currents in an approximately passive manner [2]. A major distinction in this group should be made between phytoplankton, the organisms with photosynthetic capabilities, that can grow from nutrients in the water and light, and zooplankton, the small predators that consume them. This distinction leads to the simplest models of aquatic population dynamics, which take into account just these two trophic levels.
Here we consider the phytoplankton-zooplankton competition model proposed in [5]. Phytoplankton growths logistically, and in addition it is limited by the zooplankton grazing, which itself dies grazed by upper trophic levels not explicitly modelled. The fundamental feature of the model is its excitable behaviour. In just a few words, this means that activator and inhibitor variables can be identiﬁed. The activator (phytoplankton in our case) displays some kind of autocatalytic growth behaviour, but the presence of the inhibitor (zooplankton) controls it so that the dynamical system has a stable ﬁxed point of low phytoplankton population as the unique global attractor.
The essence of the excitability phenomenon is the presence of a threshold, such that if the system is perturbed above it, variables reach the stable ﬁxed point only after a large excursion in phase space in which phytoplankton population grows to values close to the carrying capacity of the medium.
This behaviour usually appears when the activator has a temporal response much faster than the inhibitor, which then takes some time before stopping the growth of the activator. Observed plankton bloom phenomena have been interpreted in this dynamical context [4, 5].
Explicitly, the dynamics of the space- and time-dependent phytoplankton,
P = P (x, t), and zooplankton, Z = Z(x, t), concentrations is ruled by
P
∂
P + v · ∇P − D∇2 P = r βP 1 −
− f (P ) Z
∂t
K
∂
Z + v · ∇Z − D∇2 Z = r [f (P ) Z − ωZ] .
∂t

(1)

The left-hand-side terms represent the transport processes: both species are advected by the same ﬂuid ﬂow characterized by the velocity ﬁeld v = v(x, t), that we assume to be incompressible and not altered by the back-inﬂuence of the biological concentration ﬁelds. We choose v to be two-dimensional

Logistic Population Growth and Beyond

119

to emphasize the role of horizontal transport [3], and deﬁned on a twodimensional square box domain Ω = [0, 2L] × [−L, L] with Cartesian coordinates x = (x, y). The diﬀusion operator D∇2 , assumed to be the same for both species, models small-scale complex turbulent motions not explicitly included in v. The right-hand-side contains the biological interactions terms: r controls the ratio of the transport time scales to the biological activity time scales, sets the ratio of phytoplankton- to the much larger zooplankton-growth time scale, K is the phytoplankton carrying capacity, β the phytoplankton growth rate, ω a linear zooplankton mortality, and f (P ) =

2
P0

P2
+ P2

(2)

is a Hollings type III response function, describing the zooplankton predation on phytoplankton. As demonstrated in [5], is the functional form in (2) the responsible for the excitable character.
Following [5] we take non-dimensional units such that β = 0.43, K = 1,
= 0.01, P0 = 0.053, and ω = 0.34. This corresponds to phytoplankton doubling times of the order of days, and zooplankton time scales in the range of months. Biological concentrations have been scaled so that the phytoplankton carrying capacity (of the order of 100 µg of Nitrogen equivalent per liter) is the unity. For these parameter values the biological dynamical system is in the excitable regime. Excitable behaviour generally appears for small enough, which biologically means that, as already stated, time scales for phytoplankton growth are much shorter than for zooplankton. This is the biologically relevant case. We study the inﬂuence of transport by varying its relative strength via the parameter r. D is ﬁxed to 10−5 and L = 9, which means that the diﬀusive spatial scale corresponding to the phytoplankton doubling time is between three and four orders of magnitude smaller than system size.
Since the velocity ﬁeld v = (vx , vy ) is two-dimensional and incompressible it can be written in terms of a stream function Ψ (x, y, t):
∂Ψ
∂y
∂Ψ
vy = −
.
∂x

vx =

(3)

We consider the following stream function [6]:
Ψ = Ψ0 tanh

y
(x − L)2 + y 2
+ µ exp − d 2σ 2

cos (k(y − vt)) .

(4)

It represents an oceanic jet perturbed by a localized wave-like feature, trapped by topography or some geographical accident. The ﬁrst term is the main jet, of width d, ﬂowing towards the positive x direction with maximum velocity

120

E. Hern´ndez-Garc´ and C. L´pez a ıa o Ψ0 /d at its centre. The wave-like perturbation, of strength µ, is represented by the second term. It is localized in a region of size σ around the point
(x, y) = (L, 0), and the wavenumber and phase velocity (directed towards the positive y direction) are k and v, respectively. The complete velocity ﬁeld is time-periodic with period 2π/kv.
It is very important to note that the region Ω is open with respect to this ﬂow, so that we have the situation of chaotic scattering [7]: particles enter Ω from the left, following essentially straight trajectories, experience the irregular motion called transient chaos [7] when reaching the wave region
(which in consequence becomes a mixing region), and ﬁnally they leave the system. For µ large enough, recirculation in the mixing region gives birth to a chaotic saddle in Ω. The chaotic saddle is the (fractal) subset of the mixing region where particles are trapped forever. It is formed by an inﬁnite number of bounded hyperbolic orbits in the mixing region. The stable manifold of the saddle contains orbits coming from the inﬂow region but never escaping from the mixing zone. Concerning the unstable manifold, if a droplet of dye
(or any other passive tracer) is injected into the mixing region, most of it will be advected downstream in a short time. But part of the dye will remain close to the chaotic saddle for very long times, and continuously ejected along its unstable manifold. In this way passive tracers such as dye traces out the unstable manifold of the chaotic saddle, giving rise to the fractal patterns characteristic to open ﬂows. We next study how these ﬂow structures aﬀect the plankton dynamics given by the right-hand-side of (1). Pictures of the stream function and of the chaotic saddle can be seen in [6].
Equations (1) are solved by a semilagrangian method. The ﬁxed point representing stable phytoplankton-zooplankton coexistence in the absence of ﬂow and diﬀusion is given by P = Pe and Z = Ze , with Pe = P0 σ/(1 − σ) =
2
2
0.03827 and Ze = β(1 − Pe /K)(P0 + Pe )/Pe = 0.04603. We choose these values to be imposed as Dirichlet boundary conditions on the boundary of
Ω. In this way ﬂuid particles enter in the system with a plankton content corresponding to the equilibrium concentrations, which is a rather natural condition from the biological point of view. During an excitation phase, the values of phytoplankton concentration rise to P ≈ 0.8 − 0.9
Since (Pe , Ze ) is a stable equilibrium point, dynamics will be trivial without an initial seed to trigger the excitation dynamics. Our initial condition is a localized patch of high phytoplankton concentration close to the left part of Ω: P (x, t = 0) = Pe + Q exp[−((x − x0 )2 + y 2 )/l2 ], Z(x, t = 0) = Ze .
We take Q = 0.5, x0 = 0.3L, and l = 0.11L. The jet transports the patch towards the scattering or mixing region, where interesting dynamics occurs.
The ﬂow parameters are d = 1, Ψ0 = 2, σ = 2, k = 1, and v = 1, giving a ﬂow period T = 2π/kv = 2π.
We now consider µ = 3 (a chaotic saddle is present in the system above µ ≈ 2). For small r, the biological dynamics is slow compared to the time scales of stirring by the ﬂow. The phytoplankton patch is strongly deformed

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121

when reaching the scattering region. Plankton is stretched into long and thin ﬁlaments that become rapidly diluted into the surrounding unexcited ﬂuid by the eﬀect of diﬀusion. Thus excitation is destroyed by the fast stirring and by diﬀusion. Increasing r, i.e. by making the biological dynamics faster or the ﬂow slower, a dramatic change occurs. The transition to the new regime occurs around r ≈ 1. Plankton is again stretched in ﬁlaments but the width stabilizes and the excitation becomes distributed in the system, without leaving it (it remains forever) and oscillating in shape following the period of the ﬂow. Some features of the distributions of both phytoplankton and zooplankton (see Fig. 1) seem to mimic the shape of the unstable manifold of the chaotic saddle, so that in these zones we can say that plankton is basically covering it with a ﬁnite width. Summing up, the relevant result we have shown is the following: transient chaos, characteristic of open ﬂows, plus transient excitation, characteristic of excitable systems, give rise to a permanent pattern of high biological activity (excitation).

Fig. 1. Distribution of phytoplankton (left) and zooplankton (right) at time t = 100 and parameters r = 10 and µ = 3. Dark grey corresponds to low concentration and lighter grey to higher concentration. A state of permanent excitation is sustained in the region close to the chaotic saddle and its unstable manifold (being the domain far from this region in the unexcited equilibrium state). The shape of the distributions changes in time with the period of the ﬂow

The explanation for the observed behaviour can be elaborated along the lines of previous works [8, 9] as follows: The tendency of the chaotic ﬂow to stretch ﬂuid elements into long and thin ﬁlaments competes with the eﬀect of diﬀusion and biological growth, which tends to expand excited regions, so that a compensation can be achieved in some parameter range. Above a given biological growth rate, steady ﬁlament solutions appear via a saddlenode bifurcation. This can be explicitly shown in simpliﬁed models capturing some of the features of the full system (1). When this steady solution does not exist, initial perturbations decay as in the usual excitation-deexcitation cycle,

122

E. Hern´ndez-Garc´ and C. L´pez a ıa o so that excitation disappears at long times. At suﬃciently large biological growth rate, however, the steady ﬁlament solution exists, is stable, and the initial perturbation can be locally attracted by it. Chaotic ﬂow deforms the simple ﬁlament solution obtained under simpliﬁed assumptions [8,9], but the results in [8] indicate that it still provides a useful description of the process.
Chaotic stretching and folding of the excited ﬁlament in a closed system ends up when it ﬁlls the whole domain, after which an homogeneous deexcitation ﬁnishes the excitation cycle. In an open system, however, the continuous outﬂow of excited material inhibits the ﬁlling of the full domain, so that distributions related to relatively simple ﬁlament steady solutions can persist permanently. 3 Nonlocal Logistic Growth
In this section we discuss the interacting particle model with non-local interactions introduced in [10]. It is just a simple modiﬁcation of the Brownian
Bug model of [11] where there is an ensemble of diﬀusing particles (the bugs), each of them dying or duplicating with given probabilities per unit of time.
As we shall see below the modiﬁcation consists in the introduction of an interaction among the particles so that the birth rates for any of them diminish in regions of high particle density. This is precisely Verhulst logistic mechanism, here implemented in a particle model instead that in the original Verhulst equation for the global population.
The microscopic rules are enumerated in the following. Let N (t) be the number of bugs in the system (a two-dimensional periodic box Ω of size L×L; in all our computer simulations we shall take L = 1):
1. There is an initial population of N (t = 0) = N0 bugs or particles, randomly located.
2. One particle, j, is selected at random and it reproduces (i.e., it is transformed into two particles) with a rate (probability per unit time) λ(j) or dies with a rate β(j). Both rates are not constant but depend on the number of particles surrounding the particle j. Explicitly we take: λ(j) = max 0, λ0 −

1 j
N
Ns R

,

(5)

β(j) = max 0, β0 −

α j
N
Ns R

,

(6)

and

j where λ0 and β0 are constants. NR denotes the total number of particles which are at a distance smaller than R from particle j (excluding the particle j itself). R is thus a range of interaction, Ns is a saturation parameter, and α controls the asymmetry between its inﬂuence on death

Logistic Population Growth and Beyond

123

and on reproduction (in the rest of this paper we take α = 0 so that only birth rates are modiﬁed by the local density of particles). In the case of reproduction, the newborn particle is located at the same place as the parent particle. The process is repeated a number of times equal to N (t), so that each particle is checked for birth or reproduction once in average.
3. Each particle moves in random direction a distance drawn from a Gaussian distribution of standard deviation σ (this models Brownian motion).
4. When advection is considered, the particles are transported by an external ﬂow to be described later.
5. Time is incremented an amount τ = 1, and the algorithm repeats.
Figure 2 shows typical spatial conﬁgurations observed at large times under this algorithm. When the maximum growth rate µ ≡ λ0 −β0 is small (µ < µc ), population dies at long times. Above this value of µ, an active phase with a persistent average number of individuals is attained. The nature of the spatial
1
0.8

y

0.6
0.4
0.2
0
1
0.8

y

0.6
0.4
0.2
0

0

0.2

0.4

0.6

x

0.8

1

0

0.2

0.4

0.6

0.8

1

x

Fig. 2. Long-time spatial structures for the interacting particle model. Left column corresponds to two patterns with the same value of D = 10−4 , and two diﬀerent values of µ = 0.5 (up) and µ = 0.9 (bottom). Right column corresponds to ﬁxed µ = 0.7, and D = 10−4 (upper), and D = 10−5 (bottom). In all the plots, Ns = 50 and R = 0.1

124

E. Hern´ndez-Garc´ and C. L´pez a ıa o distribution in the active phase depends on the values of the parameters. For large enough value of the eﬀective diﬀusion coeﬃcient D ≡ σ 2 /τ , the spatial distribution of particles is homogeneous on average, whereas clear clustering occurs for small D. As in the Brownian Bug model [11], diﬀerent clusters are coming from diﬀerent families (i.e. each cluster is made of descendants of a diﬀerent initial reproducing individual). But the most striking feature is that they organize in a periodic pattern. The periodicity of the pattern is of the order of R, the interaction range. In addition to decreasing D, this transition to a periodic organization occurs by increasing R and, for small enough D, by increasing µ.
Particle clustering seems to be a rather natural way to make compatible the high local growth at relatively large value of µ, with the reduction of this growth that a too crowded neighbourhood would imply: the empty space between the clusters acts as a buﬀer zone keeping the competition for resources less limiting than in a homogeneous distribution. We believe that this dispersion of the total population in small groups over a large spatial area is a general consequence of the logistic mechanism when applied to particle systems. We try now to understand this pattern forming process. To this end we write down a mean-ﬁeld-like description of the model, which completely neglects ﬂuctuations, and check if the clustering instability appears there. The mean-ﬁeld equation is written in terms of an expected density φ(x, t) as follows
∂t φ(x, t) = D∇2 φ(x, t) +
(λ0 − β0 )φ(x, t) −

1
Ns φ(x, t) |x−r|<R

dr φ(r, t) .

(7)

This expression can be understood as a nonlocal version of the Verhulst logistic equation, complemented also with the diﬀusion term arising from the Brownian motion of the bugs. With more generality, the nonlocal term may be written as φ(x, t) A drG(x − r) φ(r, t), where A ⊂ Ω. Our model corresponds then to a kernel G(x) given by
G(x) =

1 if |x| ≤ R
0 if |x| ≥ R .

(8)

Stationary homogenous solutions of equation (7) are the empty phase φ(x, t) = 0, and the active or survival phase φ(x, t) = φs = µNs /πR2 (remember that µ = λ0 −β0 ). For µ < 0 the only stable solution is the absorbing one; the transition to the survival state is approached at µ = 0, and this state is stable for a range of positive values of µ. At the deterministic level the transition is transcritical. We note that the critical value µ = 0 is smaller than the one observed numerically in the particle model (µc > 0), this deﬁciency of the mean-ﬁeld description being a well known consequence of neglecting ﬂuctuations. Logistic Population Growth and Beyond

125

We perform now a stability analysis of the φs solution by considering small harmonic perturbations around it, φ(x, t) = φs + δφ(x, t), with δφ(x, t) ∝ exp(λt + ik · x). After simple calculations (see details in [10]) one arrives to the following dispersion relation λ(K) = −DK 2 −

2µ
J1 (KR) ,
KR

(9)

where K = |k|, and J1 is the ﬁrst-order Bessel function. It is clear that the relevant parameters in the problem are µ and D/R2 (in fact the precise dimensionless combinations are µτ and Dτ /R2 , see [12]). The eigenvalue λ(K) (which is in fact a function of KR, µ, and D/R2 ) is real and can be positive for some values of the parameters. This is shown in Fig. 3 where we plot λ against K for ﬁxed D/R2 and diﬀerent values of µ around the critical value µP = 185.192D/R2 , which is the value of µ at which λ(K) becomes positive [10]. Positive values of λ(K) in a range of values of K imply instability of the homogeneous distribution against perturbations containing this range of wavenumbers (K around Kc = 4.779/R) and pattern formation with the corresponding wavelengths (≈ 1.31R).
0.01
µ=0.10 µ=0.15 µ=0.185 µ=0.2 µ=0.25

0
-0.01
λ

-0.02
-0.03
-0.04
-0.05
30

40

50
K

60

70

Fig. 3. Linear growth rate λ vs wavenumber K from (9) for diﬀerent values of µ close to the onset of pattern formation. We take R = 0.1 and D = 10−5 so that µP = 0.185

The behaviour of the deterministic (7) is thus clear: for µ < 0 the only stable solution is φ = 0. Then there is an interval, 0 < µ < µP , where one has the homogeneous density φ = φs , and for µ > µP spatial patterns emerge. This last transition can also be crossed by decreasing D/R2 at ﬁxed µ > 0. The details of this sequence of transitions and the critical parameter values do not coincide with the behaviour of the particle model, implying that ﬂuctuations

126

E. Hern´ndez-Garc´ and C. L´pez a ıa o Fig. 4. Steady spatial pattern from the deterministic (7). µ = 0.70, R = 0.1,
D = 10−5 , and Ns = 50. Note the strong similarity with the pattern in the bottomright plot in Fig. 2

are rather important. However, the presence of a pattern forming instability is well explained and even the selected pattern wavenumber (Kc ≈ 4.779/R) is quantitatively reproduced by the mean-ﬁeld approach [10]. Figure 4 shows a steady pattern of density which is the solution of (7) reached at long times.
It is analogous to the one shown for the discrete model in the bottom-right panel of Fig. 2, conﬁrming for the full nonlinear model (7) the behaviour identiﬁed from the linear stability analysis of the homogeneous solutions.
So far the only motion considered for the bugs has been Brownian motion.
If they live in a turbulent aquatic medium, they will be also subjected to straining ﬁelds that will deform the clusters and, as in the previous Section, alter the population dynamics. We have implemented in the fourth step of the algorithm deﬁning the particle model a simpliﬁed ﬂow consisting in shear motions alternating in direction: if we denote by (xi (t), yi (t)) the coordinates of the particle i at time t, after one iteration of the map they become xi (t ) = xi (t) + A cos(yi (t)) , yi (t ) = yi (t) + A cos(xi (t )) ,

(10)
(11)

where t = t+τ . The parameter A gives the strength of the ﬂow and, depending of its value, particles can follow regular or chaotic trajectories. Next we

Logistic Population Growth and Beyond

127

1
0.8

y

0.6
0.4
0.2
0
1
0.8

y

0.6
0.4
0.2
0

0

0.2

0.4

0.6

x

0.8

1

0

0.2

0.4

0.6

0.8

1

x

Fig. 5. Snapshots of the long-time spatial structure for the distribution of particles with an external ﬂow. From top to bottom and left to right, A = 0, A = 0.1, A = 0.5, and A = 1. The other parameters: µ = 0.9, D = 5 × 10−6 , Ns = 50, and R = 0.1

analyze the behaviour of the model when the ﬂow changes. In Fig. 5 we show instantaneous conﬁgurations of the particle distributions as they are being stirred by the ﬂow at diﬀerent values of A. Increasing A leads to increasingly chaotic trajectories and mixing behaviour in the ﬂow. It is seen that the periodic array of clusters in the absence of ﬂow becomes more ﬁlamentallike as the ﬂow strength increases. The shape of the ﬁlamental structures reﬂects the known unstable and stable foliation of phase space for the map
(11). Inhomogeneity persists for rather strong ﬂow, but ﬁnally the distribution becomes homogenized. At this point, the particle distribution should be very close to Poissonian, with density given by the homogeneous solution of (7). This is indeed what is observed in Fig. 6 where we plot time evolution of the total number of particles. For large values of A the total number of particles, N (t), ﬂuctuates around the homogeneous deterministic solution value φs = µNs /πR2 . For smaller ﬂow strength A the spatial structure in the neighbourhood of each particle becomes relevant, and the number of particles approaches the value in the absence of ﬂow, corresponding to the pattern state of clustered particles. Thus, we see that the nonlocal interactions lead to

128

E. Hern´ndez-Garc´ and C. L´pez a ıa o 2200
2000

N(t)

1800
1600
1400
1200
0

2000

4000

t

6000

8000

10000

Fig. 6. N (t) vs time for diﬀerent values of the external ﬂow strength, A. From top to bottom: A = 0, A = 0.01, A = 0.05, and, ﬂuctuating around the value φs = µNs /πR2 (horizontal white line), A = 1 and A = 3. The other parameters are µ = 0.8, Ns = 50, D = 10−5 , and R = 0.1

a coupling between ﬂow and population dynamics, mediated by the changes in local distribution geometry that the ﬂow induces.

4 Summary
In this contribution we have presented results on two model systems coming from the context of population dynamics in aquatic ﬂowing media. In both cases the logistic mechanism of P.-F. Verhulst is an essential ingredient, although the presence of nontrivial predation dynamics in one case, and of a ﬁnite range of interaction in the other lead to interesting new phenomena, namely excitability and pattern formation, respectively. In both model systems the consideration of ﬂuid ﬂow leading to chaotic trajectories (the process known as chaotic advection or Lagrangian turbulence [13]) has additional impact on the dynamics.

Acknowledgements
Financial support from MEC (Spain) and FEDER from grants CONOCE2
(FIS2004-00953) and IMAGEN (REN2001-0802-C02-01/MAR) is greatly acknowledged. C.L. is a Ram´n y Cajal fellow. o Logistic Population Growth and Beyond

129

References
1. P.-F. Verhulst: Corresp. Math. Phys. 10, 113 (1838)
2. K.H. Mann, J.R.N. Lazier: Dynamics of Marine Ecosystems. BiologicalPhysical Interactions in the Oceans (Blackwell Scientiﬁc Publications, Boston
1991)
3. A.P. Martin: Progr. Oceanogr. 57, 125 (2003)
4. E. Hern´ndez-Garc´ C. L´pez, Z. Neufeld: Spatial Patterns in Chema ıa, o ically and Biologically Reacting Flows. In: Chaos in Geophysical Flows, ed by G. Bofetta, G. Lacorata, G. Visconti, A. Vulpiani (OTTO editore, Torino 2003) pp 35–61. Available from the publisher website: http://www.otto.to.it/otto/index2 e.php
5. J.E. Truscott, J Brindley: Bull. Math. Biol. 56, 981 (1991)
6. E. Hern´ndez-Garc´ C. L´pez: Ecological Complexity 1, 253 (2004) a ıa, o 7. E. Ott, T. T´l: Chaos 3, 417 (1993) e 8. Z. Neufeld, C. L´pez, E. Hern´ndez-Garc´ O. Piro: Phys. Rev. E 66, 066208 o a ıa, (2002)
9. E. Hern´ndez-Garc´ C. L´pez, Z. Neufeld: Physica A 327, 59 (2003) a ıa, o 10. E. Hern´ndez-Garc´ C. L´pez: Phys. Rev. E 70, 016216 (2004) a ıa, o 11. W.R. Young, A.J. Roberts, G. Stuhne: Nature 412, 328 (2001)
12. C. L´pez, E. Hern´ndez-Garc´ Physica D, in press (2004) o a ıa: 13. H. Aref: Phys. Fluids 14, 1315 (2002)

Predator-Prey Encounters Studied as Relative Particle Diﬀusion
J. Mann1 , S. Ott2 , H. L. P´cseli3 , and J. Trulsen4 e 1

2

3

4

Risø National Laboratory, DK-4000 Roskilde, Denmark jakob.mann@risoe.dk Risø National Laboratory, DK-4000 Roskilde, Denmark soeren.ott@risoe.dk University of Oslo, Institute of Physics, Box 1048 Blindern, N-0316 Oslo,
Norway
hans.pecseli@fys.uio.no
University of Oslo, Institute of Theoretical Astrophysics, Box 1029 Blindern,
N-0315 Oslo, Norway jan.trulsen@astro.uio.no Summary. The feasibility of an experimental method for investigations of the particle ﬂux to an absorbing surface in turbulent ﬂows is demonstrated in a Lagrangian as well as an Eulerian representation. A laboratory experiment is carried out, where an approximately homogeneous and isotropic turbulent ﬂow is generated by two moving grids. The simultaneous trajectories of many small approximately neutrally buoyant polystyrene particles are followed in time. In a Lagrangian analysis, we select one of these as the centre of a “sphere of interception”, and obtain estimates for the time variation of the statistical average of the inward particle ﬂux through the surface of this moving sphere. The variation of the ﬂux with the radius in the sphere of interception, as well as the variation with basic ﬂow parameters is well described by a simple model, in particular for radii smaller than a characteristic length scale for the turbulence. Applications of the problem to, for instance, the question of the feeding rate of micro-organisms in turbulent marine environments are pointed out.

1 Introduction
Often the problem of turbulent diﬀusion in neutral turbulent ﬂows is analysed in terms of an initial value problem [1, 2]. However, for many applications, a boundary value problem is more relevant. As such an example we here consider the turbulent particle ﬂux to a perfectly absorbing spherical surface, which is a realistic physical model for many practical applications. This formulation of the problem serves, for instance, as a model for predator-prey encounters in turbulent waters, and seems to be the application of the problem that has received most attention recently [3, 4]. For small predators, ﬁsh larvae for instance [5], it can safely be assumed that their self-induced motion is small or negligible, and that they are passively convected by the local ﬂow velocity, at least to a good approximation. Similarly, it can be assumed that their food (micro-zooplankton, for instance) is also passively convected by

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the same ﬂow. The feeding process can be modelled by assuming that any individual prey entering a suitably deﬁned “sphere of interception” is captured with certainty. The surface is thus “virtual” in the sense that it does not disturb the ﬂow.
In turbulent waters, the prey ﬂux to a passively convected predator is related to the problem of relative diﬀusion, but now considered as a boundary value problem, with the sphere of interception acting as a perfect absorber of prey. This is the standard model for this particular problem [6,7]. We use this as a terminology in the following, for simplicity and deﬁniteness. The general interest in the problem arises essentially from the simple observation that the food concentration in the near region of a predator will rapidly be depleted, and without any self-induced motion a predator will be starving, unless the prey within its sphere of interception is replaced by turbulent mixing in the surrounding ﬂow. Although the results presented in this work explicitly refer to spherical volumes, the scaling laws that are obtained will have a wider range of applications.
We propose and demonstrate the feasibility of an experimental method for a quantitative study of turbulent transport into an absorbing surface, and present results for varying parameters [8]. It is demonstrated that a relatively simple model equation is adequate for describing the basic features of our observations. The paper is organized as follows; in Sect. 2 we give a short summary of the experimental set-up, and the experimental conditions.
In Sect. 3 we discuss experimental results for particle ﬂuxes to an absorbing sphere where the centre is identiﬁed by a particle which is moving with the ﬂow. Section 4 contains a discussion of a simple analytical model which gives results in fair agreement with observations. Finally, Sect. 5 contains our conclusions. 2 Experimental Set-up
The basic features of the present experiment are described elsewhere [9, 10].
A short summary will suﬃce here. The turbulence is generated by the motion of two plastic grids, in the top and bottom of a tank with 320×320×450 mm3 inner dimensions, see Fig. 1 for a schematic illustration.
√
Typical Taylor micro-scale Reynolds numbers [11], Rλ = λ2 /(η 2 15), are
∼ 100 for the present conditions, using the Taylor micro scale λ = 15νσ 2 / , where ν 0.89 mm2 /s is the kinematic viscosity of the water, the speciﬁc energy dissipation rate, and σ 2 is the variance of one velocity component. The
Kolmogorov length scale η = (ν 3 / )1/4 is less than 1/2 mm for the present conditions, while Kolmogorov time scales τη are in the range 0.05–0.12 s.
The “micro-scale” η represents the length-scales, where the viscous eﬀects become important. A characteristic Eulerian length scale, LE as well as are determined by ﬁtting a von K´rm´n type wavenumber spectrum [9,10] to the a a experimentally obtained data

133

50 cm

Predator-Prey Encounters Studied as Relative Particle Diﬀusion

Fig. 1. Schematic illustration of the experimental set-up, showing the movable grids and the 4 video cameras. A restricted measuring volume of 140×140×120 mm3 is shown by thin lines

E(k) = α

2/3

5/3

LE

(LE k)4
,
[1 + (LE k)2 ]17/6

(1)

where α is the spectral Kolmogorov constant [12]. LE is found to be in the range 20–25 mm. We can interpret LE as the lower limit for separations between ﬁxed frame detection points, where the velocities of ﬂuid elements tend to become uncorrelated. As a working hypothesis we can assume that velocities are also statistically independent for separations larger than LE .
An integral length scale can be deﬁned by the integral of the parallel velocity
∞
component correlation function R (r) as Lint = 0 R (r)dr. A summary of parameters for 8 diﬀerent conditions used in the present work is given in
Table 1.
The motions of small polystyrene particles of size a = 0.5 − 0.6 mm are followed with 4 video cameras, and the simultaneous positions of typically
500–1000 particles recorded at time intervals of 1/25 s. The size of the eﬀective measuring volume is approximately 140 × 140 × 120 mm3 . It is ensured that the particles used in the experiment are approximately neutrally buoyant [9].
By a tracking procedure it is then possible to link the positions of particles
[9], and thus to follow their individual motions in 3 spatial dimensions. In particular also their time varying velocity can be deduced. An illustrative sample trajectory is shown in Fig. 2. The ﬁgure shows a series of small spheres, centred at the particle positions, at individual sampling times. Since the time sequence used here is one of the longer ones obtained, the superposition of the spheres at subsequent sampling times gives rise to an appearance like a grey “band”.
Experiments are carried out for diﬀerent intensities of the turbulent velocity ﬂuctuations, u2 . With the polystyrene particles acting as markers for the

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Table 1. Summary of the parameters derived from the second order structure function and the spectra obtained from it, based on measurements in the restricted volume shown in Fig. 1 α 2/3
(mm4/3 /s2 )

LE
(mm)

σ
(mm/s)

Lint
(mm)

(mm2 /s3 )

τη
(s)

η
(mm)

λ
(mm)

Rλ

45
41
40
45
24
65
56
25

31
27
29
28
29
29
28
27

18
16
16
17
12
21
19
12

23
20
22
21
22
22
21
20

160
140
135
160
62
279
225
65

0.075
0.080
0.081
0.075
0.120
0.056
0.063
0.117

0.26
0.27
0.27
0.26
0.33
0.22
0.24
0.32

5.1
4.9
5.1
4.9
5.8
4.5
4.6
5.6

100
88
93
91
81
104
97
78

Fig. 2. Sample of particle trajectory obtained experimentally with 1/25 s time resolution. The small spheres give the particle position, and the continuous line a numerically interpolated particle trajectory, projected onto three of the bounding surfaces of the box. The distance between the tic-marks on axes is 10 mm. The spheres are here shown enlarged, for the sake of illustration

Predator-Prey Encounters Studied as Relative Particle Diﬀusion

135

local ﬂow velocities, experimental estimates can be obtained for the second order structure function, Ψ2 (y) = (ui (r, t) − ui (r + y, t))2 being independent of t for time stationary conditions. An example is shown in Fig. 3, including also a ﬁt for small separations given by a dashed line, using the universal
Kolmogorov ( r)2/3 law. If we let the separation vector be along the y-axis, we have the longitudinal structure function Ψ2 (y) ≡ (uy (0, t) − uy (y, t))2 given by the dotted line, with CK being the Kolmogorov constant, related to the spectral constant α from (1) by CK ≈ 1.315 α [12]. We note that CK is known with some uncertainty, and a value of CK ≈ 2.5 can be justiﬁed as well as CK ≈ 2.0 [10, 12]. The dash-dotted line in Fig. 3 gives the transverse structure function Ψ2⊥ (y) ≡ (ux (0, t) − ux (y, t))2 . By a general relation [11] we have
1 d(y 2 Ψ2 )
,
Ψ2⊥ =
2y
dy for locally homogeneous and isotropic turbulence. With Ψ2 ≈ CK ( y)2/3 , we ﬁnd Ψ2⊥ ≈ (4/3)Ψ2 in the universal subrange. The full line in Fig. 3 shows
(u(r, t) − u(r + y, t))2 = 2( u2 − u(r, t) · u(r + y, t) ). The purpose of
Fig. 3 is to demonstrate the existence of a universal range, and to indicate the range of its validity, here up to separations of the order of 20 − 25 mm.

Fig. 3. Experimentally obtained second order structure function, as function of separation variable y. The heavy dashed line shows a y 2/3 ﬁt. The dotted line refers to the longitudinal structure function, and the dot-dashed line to its transverse counterpart. The full line gives the structure function Ψ2 + 2Ψ2⊥

The average distance between particles is much larger than their diameter, and particle interactions can be ignored. We estimate a Stokes number [13] as u2 St ≡ (2/9)(a/LE )2 Re with Re being the Reynolds number based on and LE . For typical values [10] of LE = 25 mm and Re ≈ 500 we ﬁnd St ≈
0.05
1. To the given accuracy, we assume that the particles follow the ﬂow as passive tracers [14], and that the particle density is uniform, when

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interpreting ﬂuxes to an absorbing surface. The assumption was substantiated by analysing the particle distributions and comparing the results to a model
Poisson distribution.

3 Particle Flux into a Moving Sphere
With the records for simultaneous particle trajectories being available, we can select one of them to represent the “predator” and label all the others as “prey”. We then select a predetermined radius R in the sphere of interception, and then remove all the particles which happen to be inside this sphere at the initial time. During the subsequent Lagrangian motion of the reference “predator”, we count the number of prey entering its co-moving sphere of interception between successive time steps. Each time a particle enters, it is “eaten” in the sense that it is removed from the database [8].
Of course, if the data analysis is carried out for very long times, all particles representing prey will eventually be removed. Here we are only interested in the time evolution of the prey ﬂux for times up to an eddy turn-over time.
As long as R is much smaller than the size of the measuring volume, we can with negligible error assume the prey concentration to be constant at large distances, corresponding to an ideally inﬁnite system. By choosing a large number of realizations, we can give an estimate for the ensemble averaged
Lagrangian prey ﬂux as a function of time after release.
In Fig. 4 we show, with solid lines, examples of the time varying particle ﬂux to a self consistently moving sphere of interception with a given radius, R.
This ﬂux is the result of a competition between, on one hand, the depletion of the density of polystyrene particles in the near vicinity of the reference sphere

Fig. 4. Time variation of the estimate for the averaged particle ﬂux for unit density
J(t) /n0 , to moving spheres, with radii, R = 10, 20 and 30 mm

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as they “absorbed”, and, on the other hand, inward ﬂux of such particles, due to the turbulent motions in the ﬂow. In each realization, we divide the ﬂux by the particle density for that particular realization. The result thus represents the particle ﬂux for unit particle density, i.e. 1 particle per mm3 .
For small radii, R < LE , we ﬁnd that the ﬂux level is almost constant in time.
A decreasing trend becomes more conspicuous as the radius is increased, and for R > LE we ﬁnd a signiﬁcant ﬂux reduction for times approaching the eddy turn-over time, here estimated by τF ≡ LE /σ. The ﬂux is largest initially, when the concentration of “prey” in the surrounding is largest. At later times there will be a possibility for encountering ﬂuid elements which have already been emptied, and the prey ﬂux becomes smaller. The ﬂux depletion due to this eﬀect increases evidently for increasing radii in the reference sphere.

4 Analytical Results
The problem of turbulent particle ﬂux to a perfect absorber moving with the ﬂow can be studied analytically by allowing for some simplifying assumptions.
Here, an absorbing spherical surface is assumed to have its centre deﬁned by a particle, which is moving with the ﬂow.
4.1 Dimensional Arguments
The present problem is characterized by a few dimensional quantities. With the viscosity, ν, being immaterial for the ﬂow dynamics for scale lengths larger than the Kolmogorov length scale η ≡ (ν 3 / )1/4 , we only have one quantity characterizing the turbulent ﬂow, namely with dimension length2 /time3 , and the length scale R characteristic for the particular problem, here a moving sphere of interception. Out of these quantities the only combination giving a quantity with dimension time is R2/3 / 1/3 , while 1/3 R7/3 gives length3 /time.
The physical dimension of the averaged normalized particle ﬂux J /n0 is length3 /time.
Quite generally it can then be argued, by purely dimensional reasoning, that the turbulent ﬂux for given reference density n0 must have the form
J
= n0 1/3

R7/3 f t

1/3

/R2/3

,

(2)

with f being a dimensionless function of a dimensionless variable. The actual form of f can only be determined by a more detailed model analysis. We can argue that we, in Fig. 4, have determined f experimentally, without reference to any explicit model equations. The arguments do not depend on any speciﬁc shape of the reference volume, and assume only that it scales self-similarly with one length scale, R. The functional dependence f in (2) will, of course, be diﬀerent for diﬀerent shapes of the volume. Note that for t > R2/3 / 1/3 ,

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see Fig. 4, the variation of f (τ ) is rather slow for parameters relevant here.
The observations summarized in Fig. 4 seem to indicate that f approaches a constant value for large times. The constant is assumed to be universal, and we ﬁnd it here to be in the range 5–10, as discussed in more detail later. The observation is not as trivial as it might seem [15].
4.2 A Model Diﬀusion Equation
The particle ﬂux to a perfectly absorbing sphere, which is moving with the ﬂow has been modelled by, for instance, a simple diﬀusion equation with a properly chosen diﬀusion coeﬃcient which depends on the simultaneous mean square velocity diﬀerences obtained at given spatial separations, but independent of time [7]. Essentially, the argument is based on the second order structure function
Ψ2 (r) ≡ (ur (0, t) − ur (r, t))

2

≈ CK ( r)2/3 ,

(3)

with the approximation being valid for separations r smaller than the length scale of the turbulence, see Fig. 3. A diﬀusion coeﬃcient is constructed from a characteristic velocity and a characteristic length. The velocity is taken to be
Ψ2 (r). For the limiting form expressed in (3), the only length characterizing the two particles is their separation r. The resulting diﬀusion coeﬃcient is consequently K(r) ∼ r4/3 1/3 . The proposed diﬀusion equation for the density n is actually identical to the one suggested by Richardson in his study of distance-neighbour functions [16]
1/3
∂ 10/3 ∂
∂
n(r, t) = C 2 r n(r, t) .
∂t
r ∂r
∂r

(4)

The result is written for spherically symmetrical geometry, with r being the radial coordinate, measured from the position of the centre of the reference sphere, and C is a numerical constant, assumed to be universal. While (4) was here argued by dimensional reasoning, it has also an analytical derivation [2].
As a consequence of (4) we have the well-known result for the mean-square separation of two initially close particles r2 = CR t3 , with the Richardson constant being CR ≈ 0.5 [10]. We have the relation C = (3/2)(3CR /143)1/3 , giving C ≈ 0.33. In the present model, the time varying diﬀusion ﬂux of particles to a perfectly absorbing sphere is given by
J(t) = 4πC

1/3

R10/3

dn(r, t) dr , r=R with n(r, t) obtained from (4).
The derivation of (4) assumes that is a deterministic constant, and thereby ignores intermittency corrections [17]. Although the relation (4) had some experimental support from the time when it was ﬁrst proposed [16],

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and also supported more recently [10], its general validity has been criticized [1, 2], as also summarized recently [10]. The range of validity of (4) is thus not fully explored. For large separations, a simple diﬀusion equation, with constant diﬀusion coeﬃcient, is expected to apply, as indicated for instance by experimental results [18], for initial conditions having scales larger than the integral length scale. These cases [10, 18] referred to particle releases considered as initial value problems. It seems that a diﬀusion equation as (4) can indeed be applied for analysing relative two-particle diﬀusion in certain variable ranges [10]. On the other hand, one cannot expect a diﬀusion coeﬃcient depending solely on relative times or spatial separations to be universally applicable for this problem [2]. In general, a Fokker–Planck equation, with (4) being one special example, describes a Markov process, where the probabilities of future states depend solely on the present, and not past ones. Modelling of turbulence as a simple Markov process is known to be rather inaccurate, and a study of the limits of applicability of models like
(4) is therefore worthwhile.
From (4) is easy to derive [7] a steady state ﬂux to a sphere with radius
R as
28π 1/3 7/3
J0
C
=
R
.
(5) n0 3 where n0 is the constant particle density at r → ∞.
4.3 Comparison Between Analytical and Experimental Results
In order to compare our observations with analytical results, we show by open circles in Fig. 5, the ﬂux value at a time t = τF /2, with τF being the eddy turn-over time. This time is suﬃciently short to give a large number of

Fig. 5. The particle ﬂux, J /n0 , to a moving sphere of interception is shown with open circles for diﬀerent radii, as is measured at 1/2 eddy turn over time, t = τF /2.
The full line gives the time-asymptotic result (5). Parameters are σ = 19 mm/s, τF = 1.6 s, and = 225 mm2 /s3 . The ﬂuxes are normalized to unit density

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particle traces for the averaging, and on the other hand, suﬃciently long to give an estimate close to the asymptotic ﬂux value of the particle ﬂux. Vertical lines give the uncertainties on the experimental estimates. For small radii R, this uncertainty is large because we only seldom ﬁnd close particles. For R >
5 mm, on the other hand, this uncertainty is smaller than the size of the circles in Fig. 5. The analytical curve, given by a full line, is the asymptotic limit from
(4), where we used = 225 mm2 /s3 . Taking into account that we have not introduced any free or adjustable parameters, we ﬁnd the agreement between the analytical and experimental results to be satisfactory, although we note a slight, but systematic, reduction of the measured ﬂux as compared with the analytical asymptotic result. The experimental results for the smallest radii give an underestimate, since in this limit a nontrivial fraction of the particles are “glancing”, i.e. they manage to pass through the reference sphere within one sampling time, and are therefore not counted.
The model equations become inadequate for spatial separations larger than the largest eddies in the turbulence, r ≥ LE , although we ﬁnd that the
R7/3 -scaling seems to have a wider range of validity, in particular at early times, t < τF /2. The analysis summarized here refers explicitly to spherical volumes. Qualitatively, the arguments will apply to diﬀerent shapes as well, as long as they scale self-similarly with one characteristic length, R.
We also present results for the ﬂux variation for a ﬁxed value of the radius of the moving sphere of interception, R = 20 mm, and varying , see
Fig. 6. In order to sample each dataset at a consistent time, we present results for a selected time τF /2 used also in Fig. 5, using the proper value of σ.
In this limit, we can in all cases assume that the particle ﬂux is close to its asymptotic, or saturated, level. The circles show the result for obtained by ﬁtting the second order structure function. Other methods for determining can be found, however [9], and these results are used to give the horizon-

Fig. 6. Variation of the normalized ﬂux, J /n0 , with varying for a ﬁxed value of
R = 20 mm. The full line gives the time-asymptotic result obtained from (4). See also Fig. 5

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tal uncertainty bars. The theoretical full line is also in this case obtained as the asymptotic limit of the solution of Richardson’s diﬀusion equation, using the most recent experimental value [10] of Richardson’s constant. Within the range of variability, we ﬁnd the scaling with to agree reasonably well with theoretical predictions based on (4). The numerical agreement between the measurements and analytical results is within a factor of 2, the analysis predicting a slightly larger ﬂux than the observed value, also in agreement with the results shown in Fig. 5. The selected value R = 20 mm can be taken as representative for the length scales smaller than or equal to LE in the experiments. The results summarized in Figs. 5 and 6 refer to ﬂuxes obtained at ﬁxed normalized times. We can also demonstrate a scaling law for the time variations of these ﬂuxes, and compare the results to the results from a model like the one given by (4). In Fig. 7 we show the normalized ﬂuxes for 8 diﬀerent experimental conditions, see Table 1, and radii R = 5, 6, 7, 8, 9, 10, 12.5, 15,
17.5 and 20 mm. The ﬁgure demonstrates the experimental scatter, which is consistent with the uncertainties of the estimates for . Also here we note a
“banded” structure in the ﬁgure. We ﬁnd that the uppermost group of curves originates from the two datasets with the largest -values, see for instance also Fig. 6, where these two datasets also seem to be slightly distinct from the others.
Again, we note that the results have a wider range of applicability, and need not refer explicitly to spherical forms. A change in shape of the reference volume, will only imply a change in the numerical constant. Thus, the scaling law implied in Fig. 6 will apply, for instance, to the prey ﬂux for any predator, independent of its range of vision, when it is exposed to diﬀerent turbulence intensities. Fig. 7. Normalized ﬂux, J /(n0 1/3 R7/3 ), as a function of the normalized temporal variable, t 1/3 /R2/3 . Curves are shown for 8 realizations with diﬀerent turbulence conditions, each with curves for diﬀerent R. The dashed line gives a theoretical result, obtained from (4). The (unphysical) singularity at t = 0 for the full line solution is due to the assumed inﬁnite initial gradient at r = R

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Given the experimental uncertainties, the scaling relations obtained by dimensional reasoning are found to be well satisﬁed when analysing the data from the present experiment. The more speciﬁc diﬀusion equation model (4), is only giving qualitative agreement for the measured Lagrangian ﬂuxes at early times. It seems, however, that the asymptotic limit is well accounted for by the model, in particular also the numerical coeﬃcient obtained by use of the most recent value of the Richardson constant [10]. To some extent, the modest agreement between the model and the experimental results at early times might be surprising, since (4) has given better agreement with estimates of the distance-neighbour functions [10]. We note, however, that in the present case there is an ambiguity in the model for the diﬀusion coeﬃcient: the result (4) uses characteristic eddies being of magnitude comparable to the predator-prey separation [7], which is the most obvious choice when modelling an equation for the distance-neighbour function. For the present case, it could as well be argued that the characteristic eddies should have a size comparable to the separation distance between prey and the surface of interception. Since such models can serve as guidelines only, we shall not pursue the problem any further here.
As particles are absorbed by the surface, with ﬂuxes shown in Fig. 7, the particle density will be depleted in the ﬂow surrounding the moving reference sphere. We can analyse also here the average particle density for r > R, as a function of time, with results shown in Fig. 8. The radius R is chosen to be in the universal subrange. The ﬁrst curve is shown at the ﬁrst sampling time,
i.e. t = 1/25 s. Variations with distance are obtained in “bins” of 1 mm, and the second bin from the surface at r = R is the ﬁrst one shown. To reduce the noise level, we normalized also here the curves with the radial density variation found at t = 0. If we choose a smaller value for R, the noise level increases, while larger R will fall outside the universal subrange.

Fig. 8. Time evolution of the normalized density around an absorbing spherical surface moving with the ﬂow, for R = 15 mm

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5 Conclusions
In this correspondence we investigated the turbulent ﬂux to a perfectly absorbing surface, with particular attention to the problem of predator-prey encounters in turbulent ﬂows. We summarized the basic elements of an experimental method for investigating the prey ﬂux to a moving predator. In the limit of small R, we found evidence for an R7/3 ﬂux scaling (see Fig. 5) in terms of the radius of the sphere of interception. We also found indications of an 1/3 scaling with the turbulent dissipation rate (see Fig. 6), in agreement with the predictions of the model (4). This model agrees to some extent also quantitatively with the observations. In the asymptotic time limit, to the extent it can be reached in the present experiment, the data gives a ﬂux well approximated by (5), provided R < LE . This will in general be the limit relevant for marine environments [19]. We suppose that the observations justify extrapolation to radii, R, smaller than those experimentally accessible. In a general sense, our results provide experimental evidence also for the importance of turbulent motion for the feeding process in marine environments.
We expect that in order to obtain a general analytical model, which can give results for extended time periods and all R, we shall have to allow for a diﬀusion coeﬃcient which depends on time as well as spatial separations, in particular including also memory eﬀects [2].
The turbulent ﬂux to a moving sphere can be signiﬁcantly smaller than the ﬂux to a stationary one. This can be argued simply by noting that the relative mean square velocity of a particle convected past a stationary sphere is u2 , while it is (u(r, t) − u(r + y, t))2 , for a passively convected sphereparticle pair, with separation y. For small separations, y
LE , we have
[7, 20] the result (3), and the relative velocity is small, implying a small ﬂux to the passively convected sphere. For large separations, y
LE , on the other hand, u(r, t) and u(r + y, t) can be supposed to be statistically independent. The mean square relative velocity is then 2 u2 . The ﬂux in this latter case is expected to be larger than that to the stationary sphere
√
with a factor 2, although such large separations cannot be achieved for the present experimental conditions.
We can deﬁne a “gain factor” as the ratio between the ﬂux to a stationary sphere divided by the ﬂux to the passively convected sphere with the same radius, R. In Fig. 9 we show this gain factor for various radii, R. All points are obtained at the reference time τF /2 used before. We ﬁnd that the gain factor is considerable for small spheres of interception, using the length scale LE as a measure. For R ≈ LE this gain factor is close to 1, and the particle ﬂux is the same for a stationary as for the moving sphere. For larger values, R > LE , the ﬂux to a moving sphere exceeds that to a stationary one. The variation of the initial value of the ﬂuxes seen, for instance, on Fig. 4 are consistent with these observations. Heuristically, we can argue for a parameter variation of the gain factor as given by the ratio of the two scaling laws obtained by dimensional arguments, which here gives σ/( R)1/3 . This ratio is shown by

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Fig. 9. Variation of the gain factor for a stationary sphere for various radii of the sphere of interception, R. The ﬁgure refers to a time t = τF /2.

a dashed line in Fig. 9, with a numerical constant not accounted for. We ﬁnd, in this case, that this scaling law does not follow the data points in any convincing way, although the trend seems reasonable. After all, neither the
Eulerian nor the Lagrangian data followed the scaling perfectly, the Eulerian data best at small R, the Lagrangian data best at somewhat intermediate values. To expect a perfect agreement for the ratio of the two quantities may seem somewhat optimistic, in particular also because a small mean ﬂow in the system gives a bias for the Eulerian ﬂuxes. The gain factor shown in Fig. 9 can, for instance, be interpreted as the gain in prey ﬂux for an imagined predator with possibilities for self-induced motion, which it uses to exactly compensate the motions in the surrounding water.

Fig. 10. Lagrangian transit time distribution, shown as a function of normalized temporal variable t 1/3 /R2/3 . The ﬁgure contains 6 experimental conditions with
= 62, 65, 135, 160, 225, and 279 mm2 /s3 , and each of these analysed for radii
R = 5, 10, 15, and 20 mm

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The present analysis, when applied to the predator-prey problem implicitly assumes that prey is captured with certainty. This might be an acceptable assumption for slow motions in the ﬂow, but evidently it might become questionable when the ﬂow is strongly turbulent. Little seems to be known about the capture probability of prey, when the relative velocities are large, although some observations have been quantiﬁed [21]. These results refer seemingly only to relatively large predators, ﬁsh larvae for instance. We can not here add to that discussion, but might note that one possibly essential part of the information, relevant for a detailed discussion, might be the distribution of transit times for prey through the sphere of interception. Also this question can be analysed on the basis of the present experimental data [15].
We can thus obtain the distribution of transit times taken as the time difference from a particle entering a reference sphere until it leaves it again for the ﬁrst time (i.e. ﬁrst passage time distributions). It turns out that also this distribution follows a seemingly universal scaling, as long as the radius of the moving reference sphere is within the universal subrange.
The problem discussed in the present communication is evidently of general interest. It has implications also for coagulation processes in turbulent colloids, for instance. A detailed investigation of this latter problem can, however, not be made by experiments like ours because the volume of the particles change upon coagulation, with a consequent change in their response to the turbulent ﬂow motions. We can not reproduce this eﬀect, for evident reasons.
In standard studies of this problem [22], restricted to diﬀusion by Brownian motion, this eﬀect is in part also ignored. With the same assumption it is possible to perform the relevant studies in experiments like ours, with results having implications for the formation rate of coagulants larger than the
Kolmogorov scale η in turbulent ﬂows.

Acknowledgement
Valuable discussions with Maria P´cseli in the initial stages of the project e are gratefully acknowledged. This work was in part supported by the Danish
Technical Research Council under contracts STVF-9601244 and 26-01-0087.
Two of the authors (HLP and JT) were in part supported by the “Eﬀects of
North Atlantic Climate Variability on the Barents Sea Ecosystem” (ECOBE) project. The present study was completed while two of the authors (HLP and
JT) were aﬃliated with the Norwegian Center for Advanced Studies.

References
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Extinction Dynamics in Lotka–Volterra
Ecosystems on Evolving Networks
Adam Lipowski1 and Michel Droz2
1

2

Department of Physics, Adam Mickiewicz University, 61-614 Pozna´ , Poland n lipowski@amu.edu.pl
Department of Physics, University of Geneva, CH-1211 Geneva, Switzerland
Michel.Droz@physics.unige.ch

Summary. Lotka–Volterra models of interacting populations are typically used on a time scale that is shorter than the lifetime of species. Such long-term behaviour of ecosystems is very often studied but using models that operate at the level of species rather than individuals. Such an approach usually refers to the notion of ﬁtness that is not commonly accepted. In the present paper we show that a
Lotka–Volterra model can be used to describe long-term behaviour of multi-species ecosystems. Interactions among species in our model form a network whose evolution is determined by the dynamics of the model. Numerical simulations show a power-law for the distribution of intervals between extinctions, but only for ecosystems with a suﬃcient variability of species and with networks of connectivity above a certain threshold that is very close to the percolation threshold of the network.
Eﬀect of slow environmental changes on extinction dynamics, degree distribution of the network of interspecies interactions, and some emergent properties of our model are also examined

1 Introduction
Population dynamics, now an important part of biology and sociology, certainly has its roots in mathematics. This is because growth laws, that describe time evolution of the size of a given population, are very often formulated in terms of diﬀerential equations. In the simplest form one assumes that the growth of a population is proportional to the population itself. Such a natural at ﬁrst sight assumption has an immediate and unpleasant consequence: exponentially diverging size of this population. It was a fundamental contribution of Verhulst to provide a remedy for such a drawback, by limiting the growth rate with a term that decreases with the size of the population [1].
Stimulated by such an important discovery, population dynamics became a rapidly developing ﬁeld, aiming to describe also more complicated systems as e.g. interacting populations.
The earliest approach to study interacting populations was initiated by
Lotka and Volterra [2], and their model quickly became a cornerstone of population dynamics [3]. Being inspired by a certain surprising, at that time, behaviour in some prey-predator systems, this model is typically used to describe populations on time scales shorter than the lifetime of species. It

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means that long-term properties of ecosystems (macro-evolution) are usually not captured within such an approach. On the other hand, models used to describe macro-evolution very often use the dynamics that operates at the level of species rather than individuals. Such coarse-grained models usually refer to the notion of ﬁtness of a species that is however not commonly accepted [4].
Recently, there has been some attempts to study macro-evolution using models equipped with dynamics that operates at the level of individuals [5–7].
Taking into account that Lotka–Volterra models are relatively successful in describing many aspects of population dynamics it would be desirable to apply such an approach also to macro-evolution of large ecosystems.
Actually, Lotka–Volterra models were already used by May to study some aspects of the stability of ecosystems [8]. His results, based on the mathematical analysis of an equilibrium state, caused some concern among ecologists, since counter-intuitively, May predicted that complex ecosystems (i.e., those with a large number of species and a large number of interactions between these species) are more likely to be unstable. Ensuing debate tried to reconcile
May’s results with our intuition that is based on observations of e.g., tropical forests (that seem to be complex and stable), and agricultural ecosystems
(simple and unstable). It was suggested that some of his assumptions (as
e.g., the very existence of an equilibrium state of an ecosystem) might not be valid and that ecologically relevant deﬁnition of stability is diﬀerent from the one used in the study of (typical) dynamical systems [9].
In addition to that, Abramson introduced a discrete version of the Lotka–
Volterra ecosystem [10] and studied certain characteristics of extinctions. His model is an example of a one-dimensional food chain with M (∼100) trophic levels and a single species occupying a given trophic level. Since in realistic food webs M ∼4 − 6 with typically many species belonging to a given trophic level [11, 12], these are highly nonrealistic assumptions. Nevertheless, extinction dynamics in Abramson’s model shows some features that are characteristic to Earth biosystem.
In the present chapter we introduce a Lotka–Volterra model that describes a simpliﬁed ecosystem of N species of predators and one species of preys [13].
Our model can thus be considered as a simple food web model with only two trophic levels. The competition between predator species is described by a certain network [14] of interactions whose evolution is governed by the dynamics of the model. Namely, when a certain species becomes extinct (i.e., its density falls below a certain threshold) it is replaced by a new species with a newly created set of interactions with some of the existing species. Despite obvious simpliﬁcations, the model exhibits some properties that are typical of more complicated ecosystems, as for example power-law distributions of the intervals between extinctions. Within our model we can also examine how robust this power-law distribution is. We ﬁnd that under certain conditions, as for example very sparse interactions between species, or too strong domi-

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nance of a small group of species, these power-law characteristics disappear and the model is driven into a regime where extinctions have exponential distributions or where there are no extinctions and the ecosystem enters a steady state. In our opinion, such regimes might be relevant when a restricted
(either in space or time) evolution of an ecosystem or its part is studied. Interestingly, a threshold value of the connectivity that separates power-law extinctions and steady state is very close to the percolation threshold of the random network of inter-species interactions.
According to a large class of statistical physics models of biological evolution, avalanches of extinctions do not require external factors to trigger them, but might be a natural consequence of the dynamics of the ecosystem. As a result, these external factors, as e.g. climate changes, solar activity or impact of a big meteorite, are very often neglected in such studies [15]. But such factors certainly aﬀect the ecosystem and there is a good evidence of it [16].
Let us emphasize that even the basic mechanism that triggers avalanches of extinctions is not known and is a subject of an intensive multidisciplinary debate. One possibility to take external factor(s) into account in our model is to modify the growth rate of the preys. Since the dynamics of the model is nonlinear, such a change might have more dramatic consequences than merely a change of the densities of the species. And indeed we noticed that the dynamics of extinctions is strongly dependent on the growth rate. It turns out, that in our model abundance of preys leads to a larger frequency of extinctions, and in periods of hunger there are less extinctions. This is clearly due to the nonlinearity of the dynamics. A larger growth rate increases the density of preys that in turn increases the densities of predators. With increased densities, the dynamics becomes more competitive and extinctions become more frequent. Such a periodically modulated growth rate leaves also some traces in the probability distribution of the extinctions. It might be interesting to notice that paleontological data also show some traces of periodic events, but their proper understanding is still missing [16, 17].
During evolution some species are favoured and selected at the expense of less fortunate ones. Evolution constantly searches for the best solutions, which resembles an optimization process. For example a large size of organisms of a given species might be of advantage in some situations, but it might cause some problems in other ones. What might be the Nature solution to this problem? Will it be middle-size species or rather two groups of species sitting at the extremes of conﬂicting requirements? In our opinion, this aspect of evolution is also often omitted in models of macroevolution. Within our model we look at such emergent properties of species selected by evolution. It turns out that depending on some dynamical details, our model can reproduce both types of solutions of such an optimization problem.

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2 Model and Numerical Calculations
We study a Lotka–Volterra ecosystem that consists of N species of predators with densities ρi (i = 1, 2, . . . , N ) which are all feeding on one species of preys with density ρ0 . We assume that each predator species i is characterized by a parameter ki (0 < ki < 1) that enters the evolution equations of the model through death and growth terms ρ˙0 = g(t)ρ0 (1 − ρ0 ) −

ρ0
N

N

f (ki )ρi

(1)

i=1

ρi = −d(ki )ρi (1 − ρ0 ) + f (ki )ρi ρ0
˙

1−

ki ρi +

j

kj ρj

ki +

j

kj

,

(2)

where i = 1, 2, . . . , N . In our model we assume that species interact mainly through environmental capacity terms (the last term in (2)). Namely, the growth rate of a given species i is reduced not only due to its density but also due to weighted (with the factor k) densities of a group of randomly selected neighbouring species. In (2) the summation over these neighbouring species is denoted by ( ). A more detailed description of our model can be found elsewhere [13].
The diﬀerential (1)–(2) are solved using a fourth-order Runge–Kutta method. Multi-species Lotka–Volterra ecosystems were subject to intensive studies since the pioneering work of May [8]. It is known that such systems might evolve toward a steady state with positive densities. However, in some cases, in a steady state, the density of some species might be zero. Each time the density of a certain species in model (1)–(2) drops below a threshold value which we ﬁx as ε = 10−7 we consider such a species as extinct. Such a species is then replaced by a new one with a randomly assigned density
(from the interval (0,1)), a coeﬃcient k (0 < k < 1) that is randomly drawn from the distribution p(k), and a new set of neighbours (all links of the ‘old’ species are removed). With such rules the model rather describes N niches, and we assume that the time to create a species that will occupy a niche is relatively short as compared to the typical lifetime of species.
We assume that a newly created species makes z links with randomly selected neighbours. Links are not directional, so a newly created species will also enter the evolution equation of species it is neighbouring. If the extinct species were chosen randomly, the network of interactions would be a random graph. However, it is the dynamics (1)–(2) that determines which species are extinct. Since the extinct species are not selected randomly, the resulting network is in general not a random graph.

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3 Results
In the following we describe numerical results obtained for some particular cases of model (1)–(2).
3.1 Intervals Between Extinctions
Various paleontological data suggest that the dynamics of extinctions has some power-law distributions of sizes or durations [16]. In our model we measured time intervals t between successive extinctions. In this calculation we used a constant growth term of preys g(t) ≡ 1. We examined two cases: (i) model I: f (ki ) ≡ 1, d(ki ) ≡ 1 and (ii) model II: f (ki ) = ki , d(ki ) ≡ 1. Unless speciﬁed otherwise we select ki randomly with a homogeneous distribution on the interval (0,1) (p(k) = 1). Our results are shown in Fig. 1. In the simplest case, model I with z = 4 and ki ≡ 1 (i.e., all species during the evolution have identical ki (= 1)) we obtain an exponentially decaying distribution of intervals between extinctions P (t). Such a decay is also seen for model I
(z = 4) with a linear distribution of ki namely p(k) = 2k. We expect that such a behaviour appears when the distribution of ki in the ecosystem is relatively narrow and shifted toward unity. Such an eﬀect might be due to the small
0

Model I, z=2
Model I, z=4
Model I, z=4, p(k) = 1/2k-1/2
Model I, z=4, p(k) = 2k
Model II, z=4
Model I, z=2,ki=1

-1

0
-3

-1 log 10 [P(t)]

log 10 [P(t)]

-2

-4

-2
-3
-4
-5

-5

-6
0

5000

10000

15000

t

-6
1

1.5

2

2.5

3

3.5

4

4.5

log 10(t)
Fig. 1. Probability distribution of intervals between successive extinctions P (t) calculated for some particular cases of model (1)–(2) for N = 100. Inset shows the same data but plotted on a lin-log scale

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width of the distribution p(k) (i.e., a distribution from which we draw ki ) or might be dynamically generated as in model II. In this case even though ki are chosen from a homogeneous distribution, the dynamics favours large ki species (due to their larger growth rate) and they dominate the ecosystem.
When the distribution of ki in the ecosystem is more uniform (model I with p(k) = 1) our simulations suggest that P (t) decays as a power law. Let us notice, however, that a power-law behaviour is seen only on approximately one decade and we cannot exclude that on a larger time scale a diﬀerent
(perhaps exponential) behaviour appears as was already observed in some other macroevolutionary models [5]. Let us also notice that for model I with p(k) = k −1/2 /2 the power-law distribution P (t) seems to decay as t−2 , i.e., with an exponent consistent with some paleontological data [16] as well as with the predictions of some other models [6]. However, one has to recognise that the error bars on experimental data are rather large and that a nonpower law behaviour cannot be excluded.
However, a power-law decay of P (t) is seen only for suﬃciently large z.
When z is too small, we observed that the ecosystem enters a steady state where all ρi are positive and there are no extinctions. This is probably due to the fact that the competition among predators is too weak (or rather too sparse). To examine the transition between these two regimes in more detail we measured the average time τ between extinctions and the results are seen in Fig. 2. One can see that τ diverges around z ∼ 1.8. Such a value of the threshold parameter suggests that this transition might be related to
0.8

500*τ

0.7

-1

R

500*τ-1, R

0.6
0.5
0.4
0.3
0.2
0.1
0
1.8

2

2.2

2.4

2.6

2.8

3

z

Fig. 2. The inverse average time between extinctions τ −1 and the percolation probability R as a function of z. The plotted results are based on calculations for
N = 100, 200, 300 and 400 and extrapolation N → ∞

Extinction Dynamics in Lotka–Volterra Ecosystems

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the percolation transition in our network of interspecies interactions. To examine such a possibility we measured the average size of the largest cluster of connected links in the network R (normalized by the number of species
N ); the results are shown in Fig. 2. The vanishing of this quantity locates the percolation transition [18]. One can see that the percolation transition takes place at a larger value, namely around z ∼ 2.0. Our results suggest that these two transitions take place at diﬀerent values of z. However the analysis of ﬁnite size eﬀects especially in the estimation of τ is rather difﬁcult and we cannot exclude that these two transitions actually overlap, as might be suggested by their proximity. Such a result would show that the dynamical regime of an ecosystem (i.e., steady state or active with power-law distribution of extinctions) is determined by the geometrical structure of its interactions. 3.2 Eﬀect of a Modulated Growth Rate
Now we examine the role of a modulated in time growth rate of preys. Such a modulation is supposed to mimic the inﬂuence of an external factor like a change of climate. One of the questions that one can ask in this context is how such a change aﬀects the extinction dynamics. We studied model I with p(k) = 1 and d(ki ) ≡ 1. For the growth rate of preys we chose g(t) =
1 + A sin(2πt/T ), where A and T are parameters. A typical behaviour in case of model I with such a growth rate is shown in Fig. 3. One can see that ρ0 ρa
M
g

1

ρ0, ρa , M,g

0.8

0.6

0.4

0.2

0
0

0 10

6

1 10

6

2 10

6

3 10

6

4 10

t

Fig. 3. A time evolution of the density of preys ρ0 , average density of predators ρa = N −1 N , and the number of extinctions M (divided by 20) in the time i=1 interval ∆t = 103 for the model I with N = 100 and z = 4. A rescaled modulated growth rate (g(t) − 1)/10 = 0.09 sin(2πt/T ) (T = 105 ) is also shown

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an increased growth rate increases the density of preys ρ0 that increases the density of predators. However, it increases also the frequency of extinctions.
Such a behaviour, namely an increased extinction rate during abundance of food, might at ﬁrst sight look as counterintuitive. This eﬀect is related to the form of environmental capacity terms in the growth rate in (2), namely
1−(ki ρi + j kj ρj )/(ki + j kj ). Such a term certainly has a larger variability for increased density of predators ρi , and for some species (depending on the distribution of links, coeﬃcients ki and densities) it causes faster extinction.
Let us also notice that since the period of modulation T is quite large, there is no retardation eﬀect between the density of preys and predators. We observed such a retardation for smaller values of T (∼ 1000).
The modulated growth rate of preys also aﬀects the probability distribution P (t) of the intervals between extinctions as shown in Fig. 4. One can see that a period of modulation T is imprinted in P (t). Let us notice that certain paleontological data do show some signs of periodicity but its origin still remains unclear [16, 17].
0.03

T=600, A=0.5, z=4
0.025

P(t)

0.02

0.015

0.01

0.005

0
0

500

1000

1500

2000

2500

3000

3500

4000

t
Fig. 4. Probability distribution P (t) of the intervals between successive extinctions calculated for model I with a modulated growth rate (N = 100)

It is known that slowly changing ecosystems sometimes undergo catastrophic shifts [19]. As a result, the ecosystem switches to a contrasting alternative stable state. It would be interesting to examine whether multi-species ecosystems, as described by our model (1)–(2), might also exist in such alternative states. If so, one can ask whether for example the structure of the network of interspecies interactions or extinction dynamics are the same in such states.

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3.3 Emergent Properties of Species
It might be interesting to ask what are the characteristics of species that are preferred by the evolution in our ecosystem. Since species are characterized only by the number ki it is equivalent to calculate the distribution of ki in the steady state of the model (1)–(2). Of course, due to selection, this distribution will in general be diﬀerent from the distribution p(k), i.e., the distribution from which we draw ki for a newly created species. Some of our results are shown in Fig. 5 (all results for g(t) ≡ 1, z = 4, N = 100).

probability density

Model I
Model II d(ki)=(1-ρ0)-ki d(ki)=ki1/2
2

1

0
0

0.2

0.4

0.6

0.8

1

k
Fig. 5. Distribution of ki in the steady state of some particular cases of model
(1)–(2) (see text)

In the case of model I (f (ki ) ≡ 1, d(ki ) ≡ 1) with a homogeneous initial distribution of ki (p(k) = 1) one can see that the steady state distribution is also approximately homogeneous (with a slight bias favoring small-k species).
We checked that model I shows this behaviour also for other distributions p(k)
(what you put is what you get). A diﬀerent behaviour appears for model II
(f (ki ) = ki , d(ki ) ≡ 1). In this case the growth rate factor f (ki ) of the i-th species is proportional to ki that certainly prefers species with large ki . Numerical results for a homogeneous distribution p(k) = 1 conﬁrm such a behaviour (Fig. 5). We observed similarly strong preferences of large ki species also for Model II with other distributions p(k).
We also examined the selection pattern in the presence of some competing eﬀects. To compensate a strong preference toward large-k species we made
√
simulations for our model with f (ki ) = ki , d(ki ) = ki and p(k) = 1.
Such a term reduces the death rate of small-k species. Our results show
(Fig. 5) that in this case the distribution of ki has two maxima, one at each

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extremity of the interval (0,1). On the other hand with the same model but for d(ki ) = (1 − ρ0 )−ki (that also reduces the death rate of small-k species) we obtain a distribution with a single maximum around k = 0.45. It would be desirable to understand the origin of the qualitative diﬀerence between these two cases.
Actually, there is yet another property of our species that is subjected to evolutionary selection, namely the number of links li (degree) of a given species. Although at the beginning each species acquires z links this number changes during the evolution because some links of a given species might be created or removed due to creation or extinction of another species. And since it is the dynamics of our model and not the random process that determines which species are removed, one can expect that the degree distribution might be diﬀerent from the Poissonian distribution that is characteristic for random graphs (see [20] for a precise deﬁnition of random graphs).
To check the statistical properties of the network of interactions in our model we calculated the degree distribution. Our results for model I with z = 4 and N = 100 are shown in Fig. 6. Let us notice that although each species has z links at the beginning it does not mean that the average number li of links connected to a given site equals z, since the dynamics of the model might preferentially remove sites having a certain connectivity. And indeed, numerical calculations show that in this case li = 2.98 < z =
4, i.e., dynamics preferentially removes sites with a large connectivity. For comparison with the random graph we also plot the Poissonian distribution r(l) = e− li li l /l!, where li = 2.98. It should be emphasized that the
0

Model I
Poisson distribution

-1

log 10 [r(l)]

-2
-3
-4
-5
-6
-7
0

2

4

6

8

10

12

14

16

l
Fig. 6. Probability distribution r(l) of sites with a given connectivity l for model I with z = 4 and N = 100 compared with the corresponding Poissonian distribution
( li = 2.98)

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157

distribution might be approximately ﬁtted using a Poisson distribution, for example with li = 2.65. However, it is not a physically relevant distribution since the average connectivity li = 2.65 diﬀers from the value li = 2.98 obtained from the simulations. In this sense the distribution is not Poissonian.
One can see that for large connectivity the degree distribution decays faster than the Poissonian distribution. This result conﬁrms that the dynamics of the model preferentially removes highly connected species. Such sites are probably more susceptible to ﬂuctuations of the system due to extinctions and creations of new species. On the other hand, poorly connected species are more likely to arrive at a relatively stable state. Similar results concerning the degree distribution were obtained for some other cases of our model.

4 Conclusions
In the present chapter we studied the extinction dynamics of a Lotka–Volterra model for a two-level food web. In our model N species of predators feed on a single species of preys. The competition between predators, that is speciﬁed by a certain network of interactions, leads to their extinction and replacement by new species. The distribution of the intervals between successive extinctions in some cases has power-law tails and thus resembles extinction patterns of real ecosystems. However, when the network of interactions between predators is too sparse the ecosystem enters a steady state. We have shown that such a change of behaviour might be related to a percolation transition of the network. We also examined the inﬂuence of external factors on the evolution of the ecosystem. More speciﬁcally, we studied the evolution of our model in the case when the growth rate of preys is changing periodically in time.
It turns out that such a modulation substantially changes the frequency of extinctions. Counterintuitively, periods with abundance of preys have higher frequency of extinctions than periods with lesser amount of preys. Moreover, we examined some properties of species that are preferentially selected by the dynamics of our model. Under some conditions preferred species are a compromise to the conﬂicting dynamics. Under some other conditions, preferred species form two antagonistic (with respect to the conﬂicting rules) groups. We also examined the degree distribution of the network of interactions between species. It turns out that the dynamics of the model has a slight preference to remove species of high connectivity. As a result the degree distribution shows some deviation from the Poissonian distribution that is characteristic to random graphs.
It would be desirable to examine some extensions of our model. For example one can introduce additional trophic levels or other forms of interspecies interactions. One can also examine a variable number of species that would allow one to create new species using certain mutation mechanism rather than assuming that they appear as soon as a niche becomes empty. Another possibility that is outside the scope of most macroevolutionary models would

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be to make further studies of emergent properties of the species. For example, one can imagine that a group of species in the ecosystem is well adapted and essentially not subjected to evolutionary changes. On the other hand there is a group of ‘newcomers’ for which evolutionary changes are much more frequent. How are the evolution and the properties of the ‘newcomers’ inﬂuenced by the properties of well-adapted species? Such problems might be easily approached within our model. The selection of a certain group of species (with a given value of k for example) can be considered as a selection of a certain strategy. One can examine models of this kind where species have multi-component parameters [k = (k a , k b , . . .)]. Consequently, one can study an evolutionary selection with more complicated traits, strategies, or behaviours. Such an approach would provide an interesting link with certain evolutionary aspects of game theory [21].

Acknowledgements
Some part of our work was done in collaboration with Fran¸ois Coppex (Unic versity of Geneva). One of us (A.L.) was partially supported by the KBN grant 1 P03B 014 27.

References
1. P.-F. Verhulst: Corresp. Math. Phys. 10, 113 (1838)
2. A.J. Lotka: Proc. Natl. Acad. Sci. USA 6, 410 (1920); V. Volterra: Le¸on sur la c Th´orie Math´matique de la Lutte pour la vie (Gauthier-Villars, Paris 1931); e e
V. Volterra: Mem. Accad. Nazionale Lincei 26, 31 (1926)
3. J.D. Murray: Mathematical Biology, (Springer, 1989); J. Hofbauer, K. Sigmund:
The Theory of Evolution and Dynamical Systems (Cambridge University Press,
Cambridge 1988)
4. P. Bak, K. Sneppen: Phys. Rev. Lett. 71, 4083 (1993); B. Drossel:
Adv. Phys. 50, 209 (2001)
5. D. Chowdhury, D. Stauﬀer, A. Kunwar: Phys. Rev. Lett. 90, 068101 (2003)
6. P.A. Rikvold, R.K.P. Zia: Phys. Rev. E 68, 031913 (2003)
7. M. Hall, K. Christensen, S. A. di Collobiano, H. J. Jensen: Phys. Rev. E 66,
011904 (2002)
8. R.M. May: Nature 238, 413 (1972)
9. K.S. McCann: Nature 405, 228 (2000)
10. G. Abramson: Phys. Rev. E 55, 785 (1997)
11. B. Drossel, A.J. McKane: Handbook of Graphs and Networks: From the Genome to the Internet S. Bornholdt and H.G. Schuster (Eds) (Wiley-VCM, Berlin
2002), e-print: nlin.AO/0202034 D. Chowdhury and D. Stauﬀer, e-print: qbio.PE/0311002
12. C. Quince, P.G. Higgs, A.J. McKane. In: Biological Evolution and Statistical
Physics, ed by M. L¨ssig, A. Vallerian (Springer Verlag, Berlin Heidelberg 2002) a Extinction Dynamics in Lotka–Volterra Ecosystems

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13. More detailed exposition of our results can be found in: F. Coppex, M. Droz,
A. Lipowski: Phys. Rev. E 68, 061901 (2004)
14. R. Albert, A.-L. Barab´si: Rev. Mod. Phys. 74, 47 (2002) a 15. B.W. Roberts, M.E.J. Newman: J. Theor. Biol. 180, 39 (1996) However, the model studied in this paper is a variant of the Bak–Sneppen model with dynamics operating at the level of species
16. M.E.J. Newman, R.G.O. Palmer: e-print: adap-org/9908002
17. D.M. Raup, J.J. Sepkoski: Proc. Natl. Acad. Sci. 81, 801 (1984)
18. D. Stauﬀer, A. Aharony: Introduction to Percolation Theory (Taylor & Francis,
London 1982)
19. M. Scheﬀer, S. Carpenter, J.A. Foley, C. Folks, B. Walker: Nature 413, 591
(2001)
20. B. Bollob´s: Random graphs, 2nd edn (Cambridge University Press, Cambridge a 2001)
21. R. Axelrod: The evolution of cooperation (Basic Books, New York 1984);
J. Maynard Smith: Evolution and the theory of games (Cambridge University
Press, Cambridge 1982)

Exact Law of Live Nature
Mark Ya. Azbel’
School of Physics and Astronomy, Ramat Aviv, 69978 Tel Aviv, Israel;
Max-Planck-Institute f¨r Festkorperforschung – CNRS, F–38042 Grenoble u Cedex 9, France

Summary. Exact law of mortality dynamics in changing populations and environment is derived. The law is universal for all species, from single cell yeast to humans.
It includes no characteristics of animal-environment interactions (metabolism etc.) which are a must for life. Such law is unique for live systems with their homeostatic self-adjustment to environment. Its universal dynamics for all animals, with their drastically diﬀerent biology, evolutionary history, and complexity, is also unique for live systems – cf. diﬀerent thermodynamics of liquids and glasses. The law which is valid for all live, and only live, systems is a life speciﬁc law of nature.
Mortality is an instrument of natural selection and biological diversity. Thus, the law which is preserved in evolution of all species is a conservation law of mortality, selection, evolution, biology. The law implies new kind of mortality and adaptation which dominate in evolutionary unprecedented protected populations and, in contrast to species speciﬁc natural selection, proceed via universal stepwise rungs. The law demonstrates that intrinsic mortality and at least certain aspects of aging are disposable evolutionary byproducts, and directed genetic and/or biological changes may yield healthy and vital Methuselah lifespan. This is consistent with experiments. Universality implies that yeast may provide a master key to the cellular mechanism of universal mortality, aging, selection, evolution, and its regulation in all animals. This suggests that one may look for its manifestations in animal cells,
e.g., in their replicative senescence.
Arguably, universal biology emerged in response to major mass extinctions which posed universal threat to diﬀerent species, and is related to disposable genes, which were beneﬁcial for longevity in the wild, but became detrimental in evolutionary unprecedented conditions.
Further theoretical and experimental studies of the universal law and its implications are suggested.

1 Motivation and Approach
Life evolved via selection of the ﬁttest. Selection posed diﬀerent challenges for diﬀerent species, thus yielded enormous biological diversity and complexity of survivors. In contrast, ﬁve major mass extinctions [1] were universal
“rapidly adjust or die” threat to the very existence of large proportion of species (96% perished in the most drastic extinction about 248 to 238 million years ago). Universal threat could yield certain universality in selection.
Indeed, presented physical approach unravels universality which underlies

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enormous diversity of evolutionary branches. Evolutionary data are sparse and largely qualitative. So, study universality of diverse living species. selection proceeded via death of the frail. Thus, quantify selection with mortality.
To amplify universality, consider diﬀerent human [2] and protected laboratory populations of med- and fruit ﬂies [3, 4], nematodes (including mutants and biologically amended) [5–8], yeast [5, 9, 10] in changing conditions. Their protection from elements of nature, predators, shortage of resources, diseases, etc. nearly eliminates extrinsic mortality, and thus selection, which dominate in the wild. Their predominantly intrinsic mortality is well quantiﬁed. It is heterogeneous and non-stationary (e.g., within human lifespan infant mortality increased 30-fold and life expectancy by 50%). Animal populations
(especially genetically homogeneous) are relatively small, and their mortality signiﬁcantly ﬂuctuates1 . So, consider life expectancy e and probability l for a live newborn to survive to a given age x. These quantities are robust to heterogeneity, non-stationarity and ﬂuctuations. Indeed, suppose the population consists of the groups with the number N G (x) of survivors to age x. If
CG = N G (0)/N (0) and lG = N G (x)/N G (0) are correspondingly the ratio of the population and the survivability to x in the group G, then the population survivability l self-averages over population heterogeneity: l= N G (x)
=
N G (0)

CG lG = lG

(1)

( · · · denotes averaging). Since l = p(0)p(1) . . . p(x − 1), where p(x) is the probability to survive from x to (x + 1), so ln l(x) averages ln p over, and ﬂuctuations with, time x. Similarly, life expectancy e = eG averages over population heterogeneity and entire lifespan. Life expectancy changes 20,000 times from yeast to humans. To account for this change, scale age x and life expectancy e with a single species speciﬁc parameter F . Choose F = 1 year for humans, F = 0.5 day for ﬂies and nematodes, F = 0.25 generation for yeast (the choice of F see later). Then Fig. 1 for all animals manifests predominantly universal2 dependence of survivability l on the scaled life expectancy e/F and age x/F .
1

The lifespan of four populations of 623, 662, 248 and 5751 inbred 3×3 male fruit ﬂies in 4-dram shell vials with weekly transfer to fresh medium [3] varied from 18.6 to 34.3 days. In the populations with close life spans (18.6 and 22 days) the probabilities to die on the 38-th day were 18 times diﬀerent. In the largest population the mortality rate of 15 days olds was 17 times lower than of those
4 days younger and 3 days older. Such giant ﬂuctuations may be related to vial diﬀerence and their weekly change. Nematode populations [5–8] include only 50–
100 worms.
2
Actuary Gompertz [11] in 1825 presented the ﬁrst universal law of mortality for human advanced age. Thereafter the search for such law for all animals went on – see [12, 13] and references there. Accurate knowledge of human mortality is important for economics, taxation, insurance, gerontology, etc. So, demographic life tables present millions of mortality data in diﬀerent countries over their history. To

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Fig. 1. Universal dependence of survivability (vertical axis) on scaled life expectancy e/F (horizontal axis) for given scaled ages x/F of 125 Swiss (1876–2001 years, crosses) and 50 Japanese (1950–1999, dashes) female [2]; 17 ﬂy [3, 4] (black) and 15 nematode [5–8] (white); 14 yeast [5,9,10] (circles) populations. Their scaling parameters F and ages x are correspondingly F = 1 year; 0.5 days; 0.25 generations and x = 30, 85, 73 (upper, lower and middle curves) years; 15, 45 (squares),
35 (triangles) days; 7, 21 (white), 16 (black) generations. Each sign presents raw experimental data for a population. Some of them overlap and are indistinguishable for humans and ﬂies, nematodes and yeast. Few accidental deviations are omitted.
Solid lines demonstrate the universal law. The diﬀerence between presented and other human data (e.g. those for, e.g., 252 Swedish female and 159 English male populations) is on the scale of diﬀerence between nematodes and yeast

In all cases total survivability l = l∗ +l , where l∗ is universal, i.e. depends on e/F and x/F only, while non-universal l which depends on all multiple factors aﬀecting mortality is l∗ . (From now on, unless speciﬁed otherwise, only universal variables are considered, and the superscript star is omitted).

2 Universality Law: Derivation
Universality for diﬀerent heterogeneous and homogeneous populations implies that the relation between universal l and e is the same as the relation between their values in any of the groups in the population, i.e. if l = l(e/F, x/F ), then lG = l(eG /F, x/F ). So, by (1), l = lG = l(eG /F, x/F ) and l = l(e/F, x/F ) = l( eG /F , x/F ), i.e. better estimate and forecast mortality, demographers dropped the universal law and developed over 15 mortality approximations [14]. Yet, 180 years after Gompertz, the existence of the universal mortality law remains controversial.

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Fig. 2. Survivability vs scaled age for Japanese females who died in 1999 [2] (black triangles) and yeast [5] (white triangles). Their scaled life expectancies are correspondingly 84 and 89

l

eG x
,
F F

=l

eG
F

,

x
F

(2)

Such equation implies [15] that l is a universal piecewise linear function of e/F with simultaneous for all ages x/F intersections (denote such dependence as the universal law) and that at any age population heterogeneity,
i.e. eG /F in all groups, is restricted to a single interval (“echelon”) of e/F between universal intersections (denote this as a restricted heterogeneity).
The universal law agrees with Fig. 1, and restricted heterogeneity implies that dominant fraction of all its populations reduces to a single echelon.
The knowledge of exact analytical dependence on e allows one to establish species speciﬁc scales F which provide its minimal relative mean square deviation from experimental3 . These scales demonstrate (see Fig. 2) proximity of human (e/F = 84) and yeast (e/F = 89) survival curves vs x/F despite enormous (∼ 20, 000 times) diﬀerence in their life expectancies. Proximity is not always as good as in Fig. 1. Empirical study [16] demonstrates very diﬀerent age dependence in diﬀerent echelons (especially in young and old age), thus high sensitivity to contributions of few echelons. Ages and conditions with low mortality may be more universal (like high e/F in Fig. 2).
3

The number of human data (whose statistics is by far the best) included in the approximations was chosen equal to the average number of data per each class; human e/F were chosen equidistant. Some scaled ages x/F in Fig. 1 are slightly diﬀerent for diﬀerent classes (e.g. 73 human years are compared to 35 rather than to 73/2 = 36.5 ﬂy days) to amplify universality. At certain ages some intersections in Fig. 1 are weakly pronounced and unobservable.

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Poor animal statistics and unknown analytical formula of age dependence
(thus insuﬃciently accurate scaling) do not allow to account for more than a single echelon. Large size and by far the best statistics of human populations allows it.
Consider the period probability d(x) = [l(x) − l(x + 1)] for a live newborn to die between x and (x + 1) years (note that human F = 1 and x/F =
x). Similar to l(x), the value of d(x) self-averages over heterogeneity, but it is more time speciﬁc than l. The most time speciﬁc parameter is “infant mortality” d(0) = q(0) which depends on the time from conception to x = 1 only. Similar to Fig. 1, the dependence of d(x) on q(0) for each human curve, is approximately piecewise linear, also with 5 (as in Fig. 1) intersections (see,
e.g. Fig. 3) which are nearly simultaneous at all ages, but somewhat diﬀerent in diﬀerent countries. Since both d(x) and q(0) are self-averaging variables, previous analysis yields the universal law. Suppose the universal j-th echelon boundaries are
(3)
qj < q G (0) < qj+1 .
Since mortality is never negative, its ultimate minimum is q1 = 0. An arbitrarily heterogeneous population may be distributed at several intervals, and piecewise linear law reduces d(x) to the sum over all intersection crossover values dj (x): cj = 1.
(4)
d(x) = cj dj (x), where

3 Results
The number of population speciﬁc concentrations cj of intersection dj (x) depends on the heterogeneity of the population. If it reduces to a single echelon, thus to two crossovers, then d(0) = q(0) and dj (0) = qj yield the universal law (which maps on coexistence of two phases): d(x) = cj dj (x) + (1 − cj )dj+1 (x);

cj =

qj+1 − q(0)
.
qj+1 − qj

(5)

If a population reduces to two echelons, thus to three crossovers, then, by
(4), d(x) reduces to q(0) and one population speciﬁc concentration. Simple algebra proves that then intersections in all such populations are situated at universal segments of the universal law or their extensions, and allow one to determine the universal law. This is the case in most developed countries (e.
g., in 1948–1999 Austria, 1921–1996 Canada, 1921–2000 Denmark, 1841–1998
England, 1941–2000 Finland, 1899–1997 France, 1956–1999 West Germany,
1906–1998 Italy, 1950–1999 Japan, 1950–1999 Netherlands, 1896–2000 Norway, 1751–2000 Sweden, 1876–2001 Switzerland). The resulting universal law in Fig. 4 is veriﬁed with few percent accuracy with ∼ 3000 human curves [2]
(18 countries, two sexes, ages from 1 to 95) – see Fig. 3, where straight lines

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Fig. 3. (Upper plot). Period probabilities for live newborn Japanese (black) and
Swedish (white) females to die (year of death from 1950 to 1999 and 1751 to 2002) between 60 and 61 (squares), 80 and 81 (triangles), 95 and 96 (diamonds) years of age vs. infant mortality q(0). Japanese relative mean squared deviations from the universal law with two echelons (straight lines) are correspondingly 2.4%, 2.3% and 10%. Signiﬁcant Swedish deviations are related to 1918 ﬂu pandemic in Europe.
(Lower plot). Same for French (diamonds) and Japanese (triangles) females (year of death from 1898 and 1950 to 2001 and 1999) between 80 and 81 years of age. Empty diamonds correspond to 1918 ﬂu pandemic and World Wars. They are disregarded in the universal law with two echelons (straight lines), which yield relative mean square deviations from black signs to generic 5%. When Japanese q(0) = 0, its extrapolation yields d(80) = 0

demonstrate the accuracy of approximations with two echelons. (Deviations are more pronounced when infant mortality signiﬁcantly changes from one calendar year to another due to wars, epidemics, crop failures etc., and/or is relatively large, i.e. when conditions are insuﬃciently protected. The accuracy may be improved if all echelons in Fig. 4 are accounted for). General

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Fig. 4. Universal law (thick lines) of human mortalities d(60), d(80) and d(95) vs q(0)-middle, lower and upper curves. At q(0) < 0.003 it is extrapolated. Thin lines extend its linear segments. Intersections (similar to those in Fig. 3) are exempliﬁed by diamonds and squares for England (two successive intersections), France, Italy and Japan, Finland, Netherlands, Norway, Denmark, France, England. Note that all intersections are close to universal straight lines

universality in Fig. 1 suggests that (properly scaled) law in Fig. 4 is universal for all protected animal populations. Consider its predictions and implications. (Earlier these results were predicted empirically [13,15,16] and derived analytically [17]).

4 Discussion and Conclusions
Derived law is universal for all species, from single cell yeast to humans. At a given age x it depends in Fig. 1 on a single population speciﬁc parameter – life expectancy e – and a single species speciﬁc parameter F . The law includes no characteristics of non-stationary and heterogeneous animal- environment interactions (e.g. via metabolism) which are necessary conditions of life. Such law is speciﬁc for live systems with their homeostatic self-adjustment to environment. Its dynamics which is universal for all animals, with their drastically diﬀerent biology, evolutionary history, and complexity, is also unique for live systems – cf. diﬀerent thermodynamics of liquids and glasses. The law which is valid for all live, and only live, systems is a life speciﬁc law of nature.
Mortality is an instrument of natural selection and biological diversity.
The law which is preserved in evolution of species from humans to yeast is a conservation law of selection, evolution, and biology. It suggests their universal mechanism which dominates in evolutionary unprecedented protected populations with predominantly intrinsic mortality. Then the contribution of all other mechanisms is either relatively small or indirect, via the universal mechanism. Its universality in all animals implies that single cell

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yeast may be a master key to it and its regulation. Universal law demonstrates that species speciﬁc natural selection is replaced in protected populations by predominantly universal adaptation of intrinsic survivability to genotypes, phenotypes, life history, environment, etc., via properly scaled life expectancy. Universal adaptation is stepwise and proceeds via universal
“ladder” of “rungs” with simultaneous for all ages crossovers. The number of its crossovers equals the number of major extinctions. (Note that each live species in the course of its entire history survived all extinctions). Less universal extinctions may yield mini-rungs, and possibly punctuated evolution [18]).
Universal Fig. 1 establishes universal scale of ages for diﬀerent species, and suggests that life expectancy in all existing species is restricted to around 100 human years, while minor directed genetic and/or biological changes increase it to the Methuselah 250 (healthy and vital [8]) human years.
When infant mortality vanishes, the universal law yields, according to
Fig. 4, zero universal mortality till certain age (∼ 80 years for humans), thus correspondingly low total mortality and high life expectancy. Mortality on the scale of stochastic ﬂuctuations, i.e. consistent with zero universal mortality, was indeed observed in humans, ﬂies, nematodes, yeast. In 2001 Switzerland only 1 (out of 60,000) girl died at 5, 9, and 10 years; 5 girls died in each age group from 4 to 7 and from 9 to 13 years; 10 or less from 2 till 17 years; no more than 16 from 2 till 26. Statistics are similar in all 1999–2002 Western developed countries [2]. Similarly, mortality of dietary restricted ﬂies at 8 days was ∼ 0.0004 [19]. Yeast mortality [9] was zero during half of its mean life span
(Jazwinski et al. [9] presented the ﬁrst model which stated that a suﬃcient augmentation of aging process resulted in a lack of aging). The probability to survive from 80 to 100 years increased in Western Europe 20-fold in the last
50 years [20]. Mean life expectancy increased almost three times in the last
250 years with improving (medical included) human conditions [2]; 2.4-fold with genotype change in Drosophila [3]. None of nematodes with changes in small number of their genes and tissues [8] died till 27 days, i.e. during 54 human years on Fig. 1 scale; from 58 till 90, from 126 till 162 “human years”.
25% of amended nematodes survived till 296 and thereafter did not die till 318
“human years”. All human data demonstrate remarkably low non-universal
(in particular, accidental) mortality, whose fraction does not increase with the decrease in total mortality. Zero mortality till certain age implies zero universal mortality at any age (unless it has a singularity at certain age [21]), thus very low total mortality, and the Methuselah life expectancy. Indeed, mean life span of mutant nematodes increased to 90 days [6, 7] and to the
Methuselah 124 days [8] (248 years on “human” scale), with no apparent loss in health and vitality.

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An important implication of the universal law is its plasticity. Universal mortality at any age is related to infant mortality4 (see Fig. 4). Thus, it rapidly adjusts to, and is determined by, current living conditions if they do not signiﬁcantly change in 2 years, from conception till 1 year, for humans; few percent of the life span for any species. So, universal mortality is independent of the previous life history (“short mortality memory” of it) and, together with infant mortality, it may be rapidly reduced and reversed to its value at a much younger age. Indeed, following uniﬁcation of East and West
Germany, within few years mortality in the East declined toward its levels in the West, especially among elderly with ∼ 45 years of their diﬀerent life histories. Mortality of the female cohort, born in 1900 in neutral Norway, at
59 years restored its value at 17 years, i.e. 42 years younger [2]. Note that such mortality decrease, similar to the one in East Germany after its uniﬁcation, is not dominated by death of the frail. The latter alters composition of the cohort, and the resulting change in mortality depends on life history rather than on current conditions only. Thus, it contributes to the deviations from the universal laws (which are relatively small) rather than to the universal mortality. Mortality plasticity is also very explicit in experiments where dietary restriction in rats [22] and ﬂies [19] is switched on. However, when dietary restriction is switched oﬀ and changes to full feeding, their longevity remains higher than in the control group of animals fully fed throughout life.
Also, when ﬂy temperature was lowered from 27 to 18 degrees or vice versa, the change in mortality, driven by life at previous temperature, persisted in these ﬂies compared to the control ones. Such long memory of life history may be related to rapid changes in temperature or feeding, since universal law is valid when infant mortality little changes within a day for ﬂies, a month for rats, a year for humans. This calls for comprehensive tests of mortality adaptation to such conditions. Similar tests may verify a possibility to reverse and reset mortality of a homogeneous cohort to a much younger age.
Restricted heterogeneity implies that at the intersections population homogenizes. This agrees with experimental data [15].

4
Thus, eliminating all deaths before age 50 would not yield just about a 4-year rise in current life expectancy at birth, as it would if mortality at higher ages were little correlated with lower age mortality. Demo- and biodemographers consider the most speciﬁc mortality variable – the probability to die between ages x and (x + 1).
It equals

q(x) =

d(x) l(x) − l(x + 1)
=
, l(x) 1 − d(0) − d(1) − · · · − d(x − 1)

thus its universality is not as explicit as that of d(x) and was unnoticed.

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5 Outstanding Problems
Vanishing and highly plastic universal mortality calls for evolutionary and biological explanation. In the wild competition for sparse resources is ﬁerce, and only relatively few genetically ﬁttest animals survive to their evolutionary “goal”-reproduction. Even human life expectancy at birth was around
40–45 years just over a century ago, and 17.2 years for males in 1773 (crop failure year) Sweden. There are no evolutionary beneﬁts from genetically programmed death and/or aging of tiny number of survivors to old age. Since prior to and during reproductive age (when survival is evolutionary beneﬁcial) mortality, and even aging (thus irreparable damage), may be negligible in protected populations (see above) and there are no evolutionary beneﬁts in switching oﬀ repair mechanisms later, so intrinsic mortality and aging are presumably disposable evolutionary byproducts. Such byproducts may be related to genes, which are beneﬁcial for non-universal longevity in the wild, but are detrimental in evolutionary unprecedented protected conditions where longevity is predominantly universal (new kind of Williams antagonistic pleiotropy). Such genes are relatively easy to alter or switch oﬀ. This is consistent with healthy and vital Methuselah age in nematodes. Universality suggests that its mechanism may be reduced to genetically regulated universal processes in cells, and related to a certain universal genome (cf.
“longevity genes” [23–26]. Single cell yeast may provide a master key to the cellular mechanism of Methuselah age, adaptation, and their regulation, in all animals (see cartoon in Fig. 5). One may look for its manifestations in animal cells, e.g., in their replicative senescence (see review [27] and references therein), apoptosis, possibly even in certain aspects of cancer ( [28] and references therein) and cancer gene therapy (e.g., inhibition of ontogenes and activation of tumor suppressor genes).
In protected populations non-universal mortality is relatively small, thus all other mechanisms are less important or correlated with universal mechanism. Conservation law of universal evolution allows for its quantitative study with current survivors, as well as for accurate deﬁnition of species, families etc. according to their intersection scales in Figs. 1 and 4. Remarkably simple species speciﬁc scales in Fig. 1 suggest the existence of their “quantization law”. Universal piecewise linear dependence on e/F is related to its invariance to restricted population heterogeneity. Invariance which yields analytical formula of the universal dependence on age, remains to be established, as well as accurate species speciﬁc scaling of mortality dependence on age and infant mortality. Universal law presents universal demo- and biodemographic approximation, which may be important for economics, taxation, insurance, gerontology, etc. Interconnection between universal evolution, selection, mortality, aging and its vitality, and presumably mass extinctions, suggests certain univer-

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Fig. 5. The ladder of rungs in the human “bridge of death”. Better social and medical protection at its successive rungs implies higher “protective walls” against, thus delay in, death and aging, but does not shift the precipice of the maximal life span at the bridge end. Biological amendments increase the maximal life span and shift the “bridge of death” end

sality in biology at large and calls for multidisciplinary (evolutionary, biological, demographic, physical and mathematical) study. Universal law, its implications, and predictions may be comprehensively veriﬁed and reﬁned theoretically (with available mortality data) and experimentally. Other outstanding problems include partitioning of universal intrinsic and non universal extrinsic mortality and their relative contribution; “quantization law” of evolutionary scales in Figs. 1, 4; establishing genes, cellular mechanism and the origin of plasticity of universal mortality; physical and biological nature of intersections and echelons.

Acknowledgement
I am very grateful to I. Kolodnaya for assistance and cartoon. Financial support from A. von Humboldt award and R. & J. Meyerhoﬀ chair is highly appreciated. Appendix
Demographic life tables present mortality data in diﬀerent countries over their history. For males and females, who died in a given country in a given calendar year, the data list, in particular, “period” probabilities q(x) (for survivors to x) and d(x) (for live newborns) to die between the ages x and (x+1)

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[note that d(0) = q(0)]; the probability l(x) to survive to x for live newborns; the life expectancy e(x) at the age x. The tables also present the data and procedures which allow one to calculate the values of q(x), d(x), l(x), e(x) for human cohorts, which were born in a given calendar year. Populations, their conditions and heterogeneity are diﬀerent, yet demographic approximations reduce period mortality of any given population to few parameters. Assumption that under certain conditions a dominant fraction of period mortality in all heterogeneous populations is universal is suﬃcient to derive the universal
Fig. 1, as well as (5) and its conditions (3). According to Fig. 4, until 65 years, d(x) decreases when q(0) increases. Beyond 85 years, d(x) increases together with q(0). In between, d(x) exhibits a well pronounced maximum
(naturally, smeared by generic ﬂuctuations). Consider the origin of such dependence on age. The value of d(x) is proportional to the probability for a newborn to survive to x and to die before the age (x + 1). When living conditions improve, the former probability increases, while the latter one decreases. In young age the probability to survive to x is close to 1, so d(x) is dominated by the mortality rate, and thus monotonically decreases together with q(0). For suﬃciently old age, low probability to reach x dominates. It increases with improving living conditions, i.e. with decreasing q(0), thus d(x) increases with decreasing q(0). At an intermediate age, when improving living conditions suﬃciently increase survival probability, d(x) increase is replaced with its decrease. Then d(x) has a maximum at a certain value of q(0). Thus, minor genetic and/or biological changes should yield the d(x) maximum at 95 years and beyond. To quantify the accuracy of the results, consider the number D(x) of deaths at a given age x in each calendar year.
According to statistics, the corresponding stochastic (i.e. minimal) error is
∼ 2/[D(x)]1/2 . At 10 years of age it increases from ∼ 20% in 1976 to ∼ 200% in 2001 Switzerland and leads to large ﬂuctuations in q(10). At 40 years it is
∼ 20%; at 80 years it is ∼ 6% in Switzerland and ∼ 2% in Japan. Universal
Fig. 4 and accuracy of the universal d(x) vs d(0) with two echelons may be reﬁned with larger number of echelons in populations. The total number of
(4) is 2XT , where 2 is the number of sexes, (X −1) is the maximal considered age, T =
Tg , where Tg is the number of calendar years in the period life tables of the country g. The total number of (4) variables with 5 intersections is 10T + 5X. Since T ∼ 2000, X ∼ 100, the number of variables is ∼ 20 times less than the number of equations. So, consider non-universal mortality with the crossover concentrations which change with age (e.g., every ﬁve years) to provide the same number of equations and variables. The resulting change in the concentrations, calculated according to life tables and (4), determines the relative non-universal mortality.

References
1. See, e.g., L. Villier and D. Korn, Science 306, 264 (2004) and Refs. therein

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2. Human Mortality Database, Univ. of California, Berkeley (USA) and MPI of
Demographic Research (Germany), http://www.mortality.org (2003)
3. J.W. Curtsinger, H.H. Fukui, D.R. Townsend, J.W. Vaupel: Science 258, 461
(1992)
4. J.R. Carey: Applied demography for biologists (Oxford university press, New
York 1993)
5. J.C. Jiang, E. Jaruga, M.V. Repnevskaya, S.M. Jazwinski: FASEB J.14, 2135
(2000)
6. K. Houthoofd et al: Exp. Geront. 37, 1371 (2002)
7. K. Houthoofd, B.P. Braeckman, T.E. Johnson, J.R. Vanﬂeteren: Exp. Geront.
38, 947 (2003)
8. A. Arantes-Oliveira, J.R. Berman, C. Kenyon: Science 302, 611 (2003)
9. S.M. Jazwinski, S. Kim, C.-Y. Lai, A. Benguria: Exp. Geront. 33, 571 (1998)
10. S. Shama, C.-Y. Lai, J.M. Antoniazzi, J.C. Jiang, S.M. Jazwinski: J. Exp. Cell
Res. 245, 379 (1998)
11. B. Gompertz: Phil. Tranc. R. Soc. A 115, 513 (1825)
12. S.J. Olshansky, B.A. Carnes, Demography 34, 1 (1997)
13. M.Ya Azbel’: Exp. Geront. 37, 859 (2002)
14. A.J. Coale, P. Demeny, B. Vaughan: Regional Model Life Tables and Stable
Populations, 2nd edn (Academic Press, New York 1993)
15. M.Ya. Azbel’, Phys. Rev E 66, 01607-1 (2002)
16. M.Ya. Azbel’, PNAS USA 95, 9037 (1997); 96, 3303, 15368 (1999)
17. M.Ya. Azbel’, Physica A 329, 436 (2003)
18. S.J. Gould, N. Eldridge: Nature (London) 366, 223 (1993), see also A. Cossing,
Nature (London) 369, 309 (1994).
19. W. Mair, P. Goymer, S.D. Pletcher, L. Partridge: Science, 301, 1731 (2003)
20. J.W. Vaupel, J.R. Carey, K. Christiansen: Science 301, 1679 (2003)
21. M.Ya. Azbel’: Physica A 341, 629 (2004).
22. B.P. Yu, E.J. Masoro, C.A. McMahon: J. Gerontology 40, 657 (1985)
23. S.M. Jazwinski: Science 263, 54 (1996)
24. A.A. Puca et al: Proc. Natl. Acad. Sci. 98, 10505-10508 (2001)
25. G. Atzmon, P. Rabizadeh, R. Gottesman, A.R. Shuldiner, N. Barzilai: Am. J.
Hum. Genetics 73, 1183B (2003)
26. T.S. Nawrot, J.A. Staessen, J.P. Gardner, A. Aviv: Lancet 363, 507 (2004)
27. V.J. Cristofalo et al, Mechanisms of Aging and Development 125, 827 (2004)
28. M.M. Gottesman: Cancer Gene Therapy 10, 501 (2002)

Manifestation of Chaos in Real Complex
Systems: Case of Parkinson’s Disease
Renat M. Yulmetyev1 , Sergey A. Demin1 , and Peter H¨nggi2 a 1

2

Department of Physics, Kazan State Pedagogical University, 420021 Kazan,
Mezhlauk Street, 1 Russia rmy@dtp.ksu.ras.ru rmy@theory.kazan-spu.ru
Department of Physics, University of Augsburg, Universit¨tsstrasse 1, D-86135 a Augsburg, Germany

In this chapter we present a new approach to the study of manifestations of chaos in real complex systems. Recently we have achieved the following result. In real complex systems the informational measure of the chaotic character (IMC) can serve as a reliable quantitative estimation of the state of a complex system and helps to estimate the deviation of this state from its normal condition. As the IMC we suggest the statistical spectrum of the nonMarkovity parameter (NMP) and its frequency behaviour. Our preliminary studies of real complex systems in cardiology, neurophysiology and seismology have shown that the NMP has diverse frequency dependence. It testiﬁes to the competition between Markovian and non-Markovian, random and regular processes and makes a crossover from one relaxation scenario to the other possible. On this basis we can formulate the new concept in the study of the manifestation of chaoticity. We suggest the statistical theory of discrete non-Markov stochastic processes to calculate the NMP and the quantitative evaluation of the IMC in real complex systems. With the help of the IMC we have found the evident manifestation of chaoticity in a normal (healthy) state of the studied system, its sharp reduction in the period of crises, catastrophes and various human diseases. It means that one can appreciably improve the state of a patient (of any system) by increasing the IMC of the studied live system. The given observation creates a reliable basis for predicting crises and catastrophes, as well as for diagnosing and treating various human diseases,
Parkinson’s disease in particular.

1 Introduction
Today the study of the manifestations of chaos in real complex systems of diverse nature has acquired great importance. The analysis of some properties and characteristics of real complex systems is impossible without a quantitative estimate of various manifestations of chaos. The dynamics or evolution of the system can be predicted by the change of its chaoticity or regularity.
The discovery of the phenomenon of chaos in dynamic systems has changed

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the attitude with regard to the functioning of complex systems, a human organism in particular. The chaos is the absence of regularity. It characterizes the randomness and the unpredictability of the changes of the behaviour of a system. At the same time, the presence of chaos in dynamic systems does not mean it cannot be taken under control. Instability of dynamic systems in the state of chaos creates special sensitivity to both external and internal inﬂuences and perturbations. The series of weak perturbations of the parameters of the system allows to change its characteristics in the required direction.
“Chaos” is frequently understood as a determined dynamic chaos, that is, the dynamics depend on the initial conditions, parameters.
Lasers, liquids near the threshold of turbulence, devices of nonlinear optics, chemical reactions, accelerators of particles, classical multipartite systems, some biological dynamic models are examples of nonlinear systems with manifestations of determined chaos. Now manifestations of chaos are being studied in diﬀerent spheres of human activity.
The control of the behaviour of chaotic systems is one of the most important problems. Most of the authors see two basic approaches to solve the problems [1,2]. Both directions envisage a preliminary choice of a certain perturbation. The selected perturbation is used to exert inﬂuence on the chaotic system. The ﬁrst direction relies on an internal perturbation, the choice of which is based on the state of the system. The perturbation changes the parameter or the set of parameters of the system, which results in the ordered behaviour of the chaotic systems. The methods focusing on the choice of such parameters (perturbations) are referred to as “methods with a feedback” [1][6]. They do not depend on the studied chaotic system (model) as these parameters can be selected by observing the system for some period of time.
One also considers that the methods with a delayed feedback [3, 7] belong to the ﬁrst direction. The second approach presupposes that the choice of the external perturbation does not depend on the state of the chaotic system under consideration. By aﬀecting the studied system with the similar perturbation, it is possible to change its behaviour. The present group of methods is an alternative to the ﬁrst one. These methods can be used in cases when internal parameters depend on the environment [1, 8, 9].
Generally, when choosing internal (external) perturbations it is possible to determine three basic stages: the estimation of the initial information, the choice of the perturbation and the bringing of the chosen strategy of control into action (its practical realization). At the ﬁrst step the information on the state of the studied system is collected. At the second stage the received information is processed according to the plan or strategy of the control. On the basis of the achieved results the decision on the choice of the internal
(of the external) perturbation is accepted. After that the chosen strategy of chaos control is put into practice [2].
The initial idea of the present concept was to separate Markov (with short-range time memory) and non-Markov (with long-range time memory)

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stochastic processes. However, the study of real complex systems has revealed additional possibilities of the given parameter. Actually, the non-Markovity parameter represents a quantitative measure of the chaoticity or regularity of various states of the studied system. An increase of the given parameter
1) corresponds to an increase of the chaoticity of the state of the
(ε1 (0) system. A decrease of the non-Markovity parameter means a greater ordering
(regularity) of the state of the system. The given observation allows one to deﬁne a new strategy for estimating the chaoticity in real systems. This new approach in chaos theory can be presented as an alternative to the existing methods. Further analysis of the non-Markovity parameter allows one to deﬁne the degree of chaoticity or regularity of a state of the system.
In this work the new strategy for the study of manifestations of chaoticity is applied to real complex systems. The possibilities of the new approach are revealed at the analysis of the experimental data on various states of a human organism with Parkinson’s disease. Parkinson’s disease is a chronic progressing disease of the brain observed in 1–2% of elderly people. The given disease was described in 1817 by James Parkinson in the book “An essay on the shaking palsy”. In the 19th century the French neurologist Pierre Marie
Charcot called this disease “Parkinson’s disease”. The steady progress of the symptoms and yearly impairment of motor function is typical of Parkinson’s disease. Complex biochemical processes characteristic of Parkinson’s disease result in a lack of dopamine, a chemical substance which carries signals from one nerve cell to another. The basic symptoms typical of Parkinson’s disease form the so-called classical triad: tremor, rigidity of muscles (disorder of speech, amimia), and depression (anxiety, irritability, apathy). The disease steadily progresses and eventually the patient becomes a helpless invalid. The existing therapy comprises a set of three basic treatments: medical treatment, surgical treatment and electromagnetic stimulation of the aﬀected area of the brain with the help of an electromagnetic stimulator. Today this disease is considered practically incurable. The treatment of patients with Parkinson’s disease requires an exact estimate of the current state of the person. The oﬀered concept of research of manifestations of chaoticity allows one to track down the least changes in the patient with the help of an exact quantitative level of description.
Earlier we found out an opportunity for deﬁning the predisposition of a person to the frustration of the central nervous system due to Parkinson’s disease [13]. Our work is an extension and development of the informational possibilities of the statistical theory of discrete non-Markov random processes and the search for parameters aﬀecting the health of a subject.

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2 The Statistical Theory of Discrete Non-Markov
Random Processes. Non-Markovity Parameter and its Frequency Spectrum
The statistical theory of discrete non-Markov random processes [10–12] forms a mathematical basis for our study of complex live systems. The theory allows one to calculate the wide quantitative set of dynamic variables, correlation functions and memory functions, power spectra, statistical non-Markovity parameter, kinetic and relaxation parameters. The full interconnected set of these variables, functions and parameters creates a quantitative measure of chaoticity used for the description of processes, connected with the functioning of alive organisms.
We use the non-Markovity parameter ε as a quantitative estimate of the non-Markov properties of the statistical system. The non-Markovity parameter allows to statistical processes into Markov processes (ε → ∞), quasiMarkov processes (ε > 1) and non-Markov processes (ε ∼1). Besides the non-Markovity parameter we also use the spectrum of the non-Markovity parameter. We deﬁne the spectrum as the set of all values of the physical parameter used for describing the state of a system or a process. Let’s consider the ﬁrst and the nth kinetic equations of the chain of connected non-Markov ﬁnite-diﬀerence kinetic equations [10, 11]:
∆a(t)
= λ1 a(t) − τ Λ1
∆t

m−1

M1 (jτ )a(t − jτ ) ,

(1)

j=0

···
∆Mn (t)
= λn+1 Mn (t) − τ Λn+1
∆t

m−1

Mn+1 (jτ )Mn (t − jτ ). j=0 The ﬁrst equation is based on the Zwanzig–Mori kinetic equation of nonequilibrium statistical physics: da(t) 2
= −Ω1 dt t

dτ M1 (jτ )a(t − jτ ) .
0

Here a(t) is a normalized time correlation function (TCF): lim a(t) = 1,

t→0

lim a(t) = 0 .

t→∞

The zero memory function a(t) and the ﬁrst-order memory function M1 (t) in (1),

Manifestation of Chaos in Real Complex Systems

A0 (0)Am (t) k m+k
, t = mτ,
|A0 (0)|2 k ˆ
ˆ
ˆ
A0 (0)L12 {1 + iτ L22 }j L21 A0 (0)

179

M0 (t) = a(t) =
M1 (jτ ) =

k

k

ˆ ˆ
A0 (0)L12 L21 A0 (0) k k

, M1 (0) = 1.

A0 (0) = (δx0 , δx1 , δx2 , · · · , δxk−1 ), k Am (t) = (δxm , δxm+1 , δxm+2 , · · · , δxm+k−1 ), m+k describe statistical memory in complex systems with a discrete time (A0 (0) k and Am (t) are the vectors of the initial and ﬁnal states of the studied m+k ˆ system). The operator L is a ﬁnite-diﬀerence operator:
ˆ
iL =

∆
,
∆t

∆t = τ ,

ˆ
ˆ
where τ is a discretization time step, Lij = Πi LΠj (i, j = 1, 2) are matrix elements of the splittable Liouville quasi-operator, Π1 = Π, Π2 = P = 1 − Π and Π are projection operators.
Let us deﬁne the relaxation times of the initial TCF and of the ﬁrst-order memory function M1 (t) as follows:
∞

τa = Re

∞

a(t)dt, τM1 = Re
0

∞

M1 (t)dt, . . . , τMn = Re
0

Mn (t)dt .
0

Then the spectrum of the non-Markovity parameter {ε} is deﬁned as an inﬁnite set of dimensionless numbers:
{εi } = {ε1 , ε2 , . . . , εn , . . .} , τa τ M1 τ Mn
, ε2 =
, . . . , εn =
,
ε1 = τ M1 τ M2 τMn+1 τrel ε= . τmem (2)

Note that a(t) = M0 (t). The number εn characterizes the ratio of the relaxation times of the memory functions Mn and Mn+1 . If for some n the value of the parameter εn → ∞, then this relaxation level is Markovian. If εn changes in limits from zero to a unit value, then the relaxation level is deﬁned as non-Markovian. The times τrel (relaxation time) and τmem (memory life time) appear when the eﬀects of the statistical memory in the complex discrete system are taken into account by means of the Zwanzig–Mori method of kinetic equations. Thus, the non-Markovity parameter spectrum is deﬁned by the stochastic properties of the TCF.
In [10] the concept of generalized non-Markovity parameter for a frequencydependent case was introduced: εi (ω) =

µi−1 (ω) µi (ω)

1
2

.

(3)

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Here as µi (ω) we have the frequency power spectrum of the ith memory function: ∞

µ1 (ω) = |Re
0

∞

M1 (t)eiωt dt|2 , . . . , µi (ω) = |Re

Mi (t)eiωt dt|2 .

0

The use of εi (ω) allows one to ﬁnd the details of the frequency behaviour of the power spectra of the time correlations and memory functions.

3 The Universal Property of Informational
Manifestation of Chaoticity in Complex Systems
In our work the discussion of the manifestation of chaoticity is carried out on the basis of a statistical invariant which includes a quantitative informational measure of chaoticity and pathology in a covariant form. The existence of this invariant is very important for taking decisions in the problems related to medicine as well as for analysing a wide area of physical problems related to complex systems of various nature.
In each live organism there is a universal informational property of the following form:
IMC + IMP = Invariant .
(4)
Here IMC is an informational (quantitative) measure of chaoticity for the concrete live system, IMP is an informational measure of a pathological state of a live organism. As an informational (quantitative) measure of the degree of chaoticity (regularity) we propose to use the ﬁrst point of the non-Markovity
1/2
parameter at zero frequency: ε1 (0) = [µ0 (0)/µ1 (0)] . The physical sense of the parameter consists in comparing the relaxation scales of the time correlation function (a(ω)) to the memory functions of the ﬁrst order (µ(ω)).
Depending on the values of this parameter one can discriminate Markov processes (with short-range memory) and non-Markov processes (with longrange memory eﬀects). Thus, the phenomena distinguished by the greatest chaoticity correspond to Markov processes. Non-Markov processes are connected with greater regularity. The informational measure of a pathological state (IMP) deﬁnes the qualitative state of a real live system.
The quantitative estimate IMC of the degree of chaoticity of a system contains the information on a pathological state of the system. It testiﬁes to the close interrelation of the given quantities. A high degree of chaoticity is characteristic of a normal physiological state. In a pathological state the degree of chaoticity decreases. A high degree of regularity is typical of this condition.
Thus, the quantitative estimate of the chaoticity in live systems allows one to deﬁne their physiological or pathological state with a high degree of accuracy.
In the right-hand side of (4) we have a statistical invariant, which reveals the independence of the physical (as well as biophysical, biochemical and biological) laws in the given live organism from the concrete situations as well

Manifestation of Chaos in Real Complex Systems

181

as the methods of description of these situations. The invariance, submitted in (4), is formulated as the generalization of the experimental data. Among other physical laws the properties of invariance reﬂect the most general and profound properties of the studied systems and characterize a wide sphere of phenomena. Equation (4) reﬂects an informational observation. It consists of two informational measures: the measure of chaoticity and the measure of pathology (disease).
Let us use the operator of transformation T (S , S) in both parts of (4). It realizes the transition of the system from one state S to another one S . By taking into account the statistical invariance I(S ) = T (S , S)I(S) = I(S) in the right-hand side of (4) we get:
∆P = P (S ) − P (S) = −∆C = −{C(S ) − C(S)}, ∆P + ∆C = 0 .

(5)

Here the following designations are introduced: I(S) = Invariant, P (S) =
IMP(S) is an informational measure of pathology (disease) for the state S,
C(S) = IMC(S) is an informational measure of the chaoticity for the state of patient S. Besides in (5) we take into account the rules of transformation:
C(S ) = T (S , S)C(S),

P (S ) = T (S , S)P (S) .

(6)

Equations (4)–(6) are rather simple but they make the quantitative description of the state of a patient possible, both during the disease and under the medical treatment. Equations (4)–(6) have a general character. They are true for many complex natural and social systems. It is possible to develop the algorithms of prediction of various demonstrations of chaos in complex systems of diverse nature on the basis of these equations.

4 The Quantitative Factor of Quality of a Treatment
One of the major problems of the medical physics consists in the development of a reliable criterion to estimate the quality of a medical treatment, a diagnosis and a forecasting of the behaviour of real live complex systems.
As one can see from the previous section, such a criterion should include the parameter of the degree of randomness in a live organism. The creation of a quantitative factor for the quality of a treatment QT is based on the behaviour law of the non-Markovity parameter ε1 (0) in the stochastic dynamics of complex systems. The greater values of the parameter ε1 (0) are characteristic of stable physiological states of systems; the smaller ones are peculiar for pathological states of live systems. Thus, by the increase or reduction of the non-Markovity parameter one can judge the physiological state of a live organism with a high degree of accuracy. Therefore the non-Markovity parameter allows one to deﬁne the deviation of the physiological state of a system from a normal state.

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The factor QT deﬁnes the eﬃcacy or the quality of the treatment and is directly connected with the changes in the quantitative measure of chaoticity in a live organism. We shall calculate it in a concrete situation. Let us consider
1 as the patient’s state before therapy, and 2 as the state of the patient after certain medical intervention. Then ε1 (1) and ε1 (2) represent quantitative measures of the chaoticity for the physiological states 1 and 2. The ratio δ of these values (δ = ε1 (2)/ε1 (1)) will deﬁne the eﬃcacy of the therapy. Various jth processes occur simultaneously in the therapy. Therefore the total value of δ can be deﬁned by the following way: n δ= j=1 εj (2)
1
εj (1)
1

,

(7)

where j = 1, 2 . . . n is the number of factors aﬀecting the behaviour of the nonMarkovity parameter. However, the natural logarithm ln δ is more convenient for use.
Then we have: δ > 1, ln δ > 0 ; δ = 1, ln δ = 0 ; δ < 1, ln δ < 0 .
The three values of δ mentioned above correspond to the three diﬀerent qualities of treatment: eﬀective, ineﬃcient and destructive treatment. They reﬂect an increase, preservation and reduction of the measure of the chaoticity in the therapy. Thus, one can deﬁne QT (ε) = ln δ according to (7) as follows: n QT (ε) = ln j=1 εj (2)
1
εj (1)
1

.

(8)

However, the total factor QT is deﬁned both by the quantitative measures of the chaoticity and by other physiological and biochemical data. Now we shall consider the transition of the patient from state 1 into state 2. Then by analogy, one can introduce the physiological parameter k(1), determined for state 1, and k(2) for state 2. In the case of Parkinson’s disease one can introduce the amplitude or the dispersion of the tremor velocity of some extremities (hand or leg) of the patient as this parameter. In other cases any medical data, which is considered for diagnostic purposes, can be used. For greater reliability it is necessary to use the combination of various parameters k j (1) and k j (2).
The value: n εj (2) k j (2)
1
QT = ln
(9)
εj (1) k j (1) j=1 1 will be considered as a generalized quantitative factor of quality of the therapy.

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However in real conditions it is necessary to increase or weaken the magnitude of chaotic, or physiological contributions to (9). For this purpose we shall take the simple ratio: ln (an bm . . .) = n ln a + m ln b + · · ·

By analogy, we can reinforce or weaken various contributions depending on the concrete situation: n QT = ln j=1 εj (2)
1
εj (1)
1

mj

k j (2) k j (1)

pj

.

(10)

If incomplete experimental data are available in some situations, one can assume pj = 1 (attenuation of the physiological contribution). A value mj > 1 can mean an ampliﬁcation of the chaotic contribution. Otherwise, if we want to weaken the chaotic contribution, we should take mj = 1 and if we want to reinforce the physiological contribution we come towards pj > 1. We have presented the results of the calculation of the quantitative factor QT below in Sect. 6

5 Experimental Data
We have taken the experimental data from [15]. They represent the time records of the tremor velocity of the index ﬁnger of a patient with Parkinson’s disease (see, also http://physionet.org/physiobank/database/). The eﬀect of chronic high frequency deep brain stimulation (DBS) on the rest tremor was investigated [15] in a group of subjects with Parkinson’s disease (PD) (16 subjects). Eight PD subjects with high amplitude tremor and eight PD subjects with low amplitude tremor were examined by a clinical neurologist and tested with a velocity laser to quantify time and frequency domain characteristics of tremor. The participants received DBS of the internal globus pallidus (GPi), the subthalamic nucleus (STN) or the ventrointermediate nucleus of the thalamus (Vim). Tremor was recorded with a velocity laser under two conditions of DBS (on–oﬀ) and two conditions of medication (L-Dopa on–oﬀ). All the subjects gave informed consent and institutional ethics procedures were followed. The selected subjects were asked to refrain from taking their medication at least 12 h before the beginning of the tests and were not allowed to have more than one coﬀee at breakfast on the two testing days. Rest tremor was recorded on the most aﬀected side with a velocity-transducing laser [16, 17]. This laser is a safe helium–neon laser. The laser was placed at about 30 cm from the index ﬁnger tip and the laser beam was directed perpendicular to a piece of reﬂective tape placed on the ﬁnger tip. Positive velocity was recorded when the subjects extended the ﬁnger and negative velocity when the subjects ﬂexed the ﬁnger.

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The conditions, counterbalanced across subjects, included the following:
1.
2.
3.
4.
5.

The L-Dopa condition (no stimulation).
The DBS condition (stimulation only).
The “oﬀ” condition (no medication and no stimulation).
The “on” condition (medication on and stimulation on).
The eﬀect of stopping DBS on tremor (time record of the tremor 15, 30,
45 and 60 min after having switched the stimulator oﬀ).

In Fig. 1 the time records of the velocity of changing tremor of the index ﬁnger of the second patient’s hand (man, 52 years old) under various conditions of inﬂuence on the organism are submitted as an example. High tremor velocity is observed: 1) in a natural condition of the patient (a), 2) 15 (45) minutes after the stimulator was switched oﬀ. Lower tremor speed occurs:
1) when both methods (stimulation, medication) are used, 2) when each of
1

1

1

(a)

1

(b)

(c)

(d)

0.5

0.5

0.5

0

0

0

0

−0.5

Velocity [m/τ]

0.5

−0.5

−0.5

−0.5

−1

0

5000

10000

1

−1

0

5000

10000

1

−1

0

5000

10000

1

(e)

−1

0

5000

10000

1

(f)

(g)

(h)

0.5

0.5

0.5

0.5

0

0

0

0

−0.5

−0.5

−0.5

−0.5

−1

0

5000

10000

−1

0

5000

10000

−1

0

−2

Time [τ=10

5000

10000

−1

0

5000

10000

sec]

Fig. 1. The velocity of the change of tremor of the index ﬁnger of the patient’s right hand (the second subject) with Parkinson’s disease under various conditions of the experiment. (a) deep brain stimulation oﬀ, medication oﬀ; (b) the subject was receiving stimulation of the GPi, medication on; (c) deep brain stimulation oﬀ, medication on; (d) the subject was receiving stimulation of the GPi, medication oﬀ; (e)–(h) the recording of rest tremor in the right index ﬁnger of the subject
15 (30, 45, 60) minutes after the stimulator was switched oﬀ, this subject was oﬀ medication for at least 12 hours

Manifestation of Chaos in Real Complex Systems

185

these methods is used separately, 3) 30 (60) minutes after the stimulator was switched oﬀ. Similar results are presented in [15].

6 Results
In this section the results obtained by processing the experimental data for one of the patients (subject number 2) are shown. Similar or related pictures are observed in the experimental data of other subjects.
6.1 The Non-Markovity Parameter as a Quantitative Measure of Deﬁning Chaoticity
In this subsection the technique to calculate quantitative and qualitative criteria under various conditions inﬂuencing the state of a patient is given.
The basic idea of the approach consists in deﬁning the quantitative ratio between chaoticity and regularity of the observed process. It allows one to judge the physiological (pathological) state of a live system by the degree of chaoticity or of regularity. The highest degree of chaoticity in the behaviour of a live system corresponds to a normal physiological state. Higher degree of regularity or speciﬁc ordering is characteristic of various pathological states of a live system. In the given work we use the non-Markovity parameter ε1 (0) as a special quantitative measure deﬁning chaoticity or regularity of the studied process. The examples [10–14], [18] which have been investigated by us earlier serve as a basis for such a reasoning. As one of the examples we shall consider the tremor velocity of the changing of the subject’s index ﬁngers in the case of Parkinson’s disease.
The comparative analysis of the initial time record and the non-Markovity parameter for all the submitted experimental data allows one to discover the following regularity. The value of the non-Markovity parameter ε1 (0) decreases with the increase of the tremor velocity of the patient’s ﬁngers
(deterioration of the physiological state) and grows with the decrease of the tremor velocity (improvement of the state of the patient). We shall also consider the power spectrum µ0 (ω) of the initial TCF under various conditions inﬂuencing an organism, the window-time behaviour of the power spectrum µ0 (ω) and the non-Markovity parameter ε1 (ω), the time dependence local averaging relaxation parameter λ1 (t) as additional sources of information.
Figure 2 represents the power spectrum of the initial TCF for various experimental conditions. One can observe a powerful peak in all the ﬁgures at the characteristic frequency ω = 0.07 f.u.(ω = 2πν, 1 f.u. = 2π/τ, τ =
10−2 s). The amplitude values of this peak for µi (ω) (i = 1, 2, 3) are given in
Table 1. The given peak testiﬁes to a pathological state of the studied system.
A similar picture is observed in patients with myocardial infarction [11]. The comparison of these values reﬂects the amplitude of the tremor velocity at the initial record of time.

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4

300

250

1000

(a)
250

2

(b)

x 10

(c)

200

800

150

600

100

400

50

200

(d)
1.5

200
150

1

100
50

µ (ω) [τ2]

0

0

0.5

0

0

4

5

0

0

0.5

4

x 10

2

0

0.5

0

4

x 10

(e)

0.5

3

0

0.5

4

x 10

4

(f)

x 10

(g)

(h)

2.5

4

1.5

3
2

3
1

1.5

2

2
1
0.5

1
0

1
0.5

0

0.5

0

0

0.5

0

0

0.5

0

0

0.5

ω [2π/τ]
Fig. 2. The power spectrum µ0 (ω) of the initial TCF for the velocity of changing of tremor of subject number 2 under various conditions that inﬂuence an organism. (a) deep brain stimulation on, medication on; (b) deep brain stimulation on, medication oﬀ; (c) deep brain stimulation oﬀ, medication on; (d) deep brain stimulation oﬀ, medication oﬀ; (e)–(h) the power spectrum µ0 (ω) of the initial TCF for the recording of rest tremor in the right index ﬁnger of the subject 15 (30, 45,
60) minutes after the stimulator was switched oﬀ, medication oﬀ. At the frequency ω = 0.07 f.u., with 1 f.u. = 100 Hz (the characteristic frequency), a peak is found.
The presence and amplitude of this peak are determined by the state of the patient
Table 1. The value µ0 (ω) for the initial TCF and µi (ω) (i = 1, 2, 3) for the memory functions of junior orders at the frequency ω = 0.07 f.u. 1: Deep brain stimulation;
2: Medication (subject number 2). For example, OFF OFF – no DBS and no medication
ON ON
ON OFF µ0 µ1 µ2 µ3

75
19
42
37

OFF
ON

OFF
OFF

15
OFF

30
OFF

45
OFF

60
OFF

250
812
1.71 × 104 4.34 × 104 1.53 × 104 2.51 × 104 3.68 × 104
3
52 1.28 × 10 1.17 × 104 3.21 × 104 1.32 × 104 1.82 × 104 2.8 × 104
60
113
71
300
62
137
224
54
141
73
147
74
152
186

Manifestation of Chaos in Real Complex Systems

187

In Table 1 the second patient’s amplitude values µ0 (ω) for the initial TCF and the memory functions of junior orders µi (ω) (i = 1, 2, 3) at the frequency ω = 0.07 f.u. are submitted. The terms of the ﬁrst row deﬁne the conditions under which the experiment is carried out. Under all conditions a power peak at the frequency ω = 0.07 f.u. can be observed. The amplitude values of the given peak (in particular in the power spectrum µ0 (ω)) reﬂect the amplitude of the tremor velocity. For example, the least amplitude 75 τ 2 corresponds to the condition (ON, ON; or: deep brain stimulation on, medication on).
The highest amplitude 4.34 × 104 τ 2 corresponds to the greatest tremor speed
(see Figs. 1e, 2e). Thus, the given parameter can be used to estimate the physiological state of a patient. A similar picture is observed in all other patients. In Fig. 3 the initial time record (the normal state of the subject; OFF,
OFF) and the window-time behaviour of the power spectrum of the TCF (the technique of the analysis of the given behaviour is considered in Ref. [18]) are submitted. In these ﬁgures regions 1, 2, 3, which correspond to the least
0.8

Velocity [m/τ]

0.6

3

2

1

0.4
0.2
0
−0.2
−0.4
−0.6

0

1000

2000

3000

4000

5000

6000

7000

8000

Time [τ=10−2 sec]

µ0 (ω)

3000
2000

3

2

1

1000
0
0
0.1

ω [2 π / τ]

0.2
0.3

0

5

10

15

20

25

30

8

t [2 τ]

Fig. 3. The initial time series and the window-time behaviour of the power spectrum µ0 (ω) of the TCF. Two ﬁgures are submitted to illustrate the case of subject number 2: no stimulation of the brain and no medication. The change of regimes in the initial time series is reﬂected in the decrease of the tremor velocity (regions
1, 2 and 3) and becomes visible as a sharp reduction of the power spectrum µ0 (ω)
(see, the 1st, 12th, 17th windows for more details)

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values of the tremor velocity are shown. The minimal amplitude of the peaks of the power spectrum µ0 (ω) corresponds to the regions with the least tremor velocity. In Fig. 4 the frequency dependence of the ﬁrst point of the non-Markovity parameter ε1 (ω) is submitted for the second subject under various experimental conditions. The value of the parameter ε1 (0) at zero frequency is of special importance for our study of manifestations of chaoticity. It is possible to judge the change of the state of a subject by the increase (or by the decrease) of this value. The comparative analysis of the initial time records allows one to come to similar conclusions. In Figs. 4d–h a well-deﬁned frequency structure of the non-Markovity parameter can be seen. This structure is completely suppressed and disappears only when during the treatment. The characteristic frequency of the ﬂuctuations corresponds approximately to ω = 0.06 f.u.
25

8

3

(a)

2.5

(b)

(c)
2.5

20

(d)
2

6
2

15

1.5
4

1.5

10

1
1
2

ε1 (ω)

5
0

0.5

0.5
0

0.5

2.5

0

0

0.5

2.5

0

0

0.5

3.5

(e)

(f)

0

0

0.5

2.5
(g)

3

(h)

2

2

2

1.5

1.5

2

1.5

1

1

1.5

1

2.5

1
0.5
0

0.5

0

0.5

0

0.5

0.5
0

0.5

0

0

0.5

0

0

0.5

ω [2π / τ]
Fig. 4. The ﬁrst point of the non-Markovity parameter ε1 (ω) for the second subject under various experimental conditions: (a) deep brain stimulation oﬀ, medication on; (b) deep brain stimulation on, medication on; (c) deep brain stimulation on, medication oﬀ; (d) deep brain stimulation oﬀ, medication oﬀ; (e)–(h) the recording of rest tremor in the right index ﬁnger of the subject 15 (30, 45, 60) minutes after the stimulator was switched oﬀ, medication oﬀ. The non-Markovity parameter at zero frequency ε1 (0) plays a special role. These values (6.02 in the ﬁrst case and
1.0043 in the last one) deﬁne the chaoticity or the regularity of the studied states.
The amplitudes of these values also characterize the state of the subject

Manifestation of Chaos in Real Complex Systems

189

These multiple peaks are the most appreciable at low frequencies. At higher frequencies these ﬂuctuations are smoothed out. As can be seen in these ﬁgures, the 2nd subject has a strong peak which remains stable over time.
As our data show, the comb-like structure with multiple frequencies can be observed in all patients with high tremor velocity. In a group of patients with low tremor velocity it disappears, and a wider spectrum that presents some ﬂuctuations over time is observed. The present structure testiﬁes to the presence of characteristic frequency of ﬂuctuations of tremor of human extremities. In Table 2 the dispersion interval of the values and the average value ε1 (0) for the whole group of subjects (16 subjects) are submitted. Let us consider 2 conditions: OFF, OFF and OFF, ON. In the ﬁrst case the dispersion interval and the average value ε1 (0) are minimal. It means the presence of a high degree of regularity of the physiological state of the patient. The degree of regularity is appreciably reduced when applying any method of treatment.
Here the degree of chaoticity grows. The maximal degree of chaoticity corresponds to the condition OFF, ON (medication is used only). The diﬀerence in ε1 (0)av.val with medication and without it (OFF, OFF) is 3.8 times (!).
On the basis of the comparative analysis of the given parameters the best method of treatment for each individual case can be found. It is necessary to note, that the given reasoning is true only for the study of the chaotic component of the quantitative factor of the quality of treatment QT . The most trustworthy information about the quality of treatment can be given by the full quantitative factor QT which takes account of other diagnostic factors. Table 2. The dispersion interval ε1 (0)int of the values and the average value ε1 (0)av.val of the ﬁrst point of the non-Markovity parameter under various experimental conditions for the group of 16 subjects. 1: Deep brain stimulation; 2:
Medication
OFF
OFF
ε1 (0)int ε1 (0)av.val

ON
OFF

OFF
ON

ON
ON

15
OFF

30
OFF

45
OFF

60
OFF

1–1.8
1.41

2–18
4.14

2–22
5.31

1.5–8
3.17

1.5–3
2.43

1.8–5
2.92

1.7–4.5 2–6
2.76
2.93

The results of the calculation of the quantitative factor QT are shown in
Table 3. The data are submitted for a single patient and for the whole group.
Here QT (ε) is the chaotic contribution to the quantitative factor (see (8)). QT is the total quantitative factor (see (10)), where ε(1) (1) and ε(1) (2) are the chaotic contributions for the tremor amplitudes k (1) (1), k (1) (2); ε(2) (1) and ε(2) (2) are the dispersions of the tremor amplitudes k (2) (1), k (2) (2) (physiological contributions). The full factor QT provides detailed information about the quality of the treatment. The present factor includes both the chaotic

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Table 3. The quantitative factor QT (ε) and the total quantitative factor QT for the second patient and for the whole group (16 subjects). 1: Deep brain stimulation;
2: Medication. mj = 1, pj = 1

OFF
OFF

ON
OFF

OFF
ON

QT (ε)
QT

0.758
1.763

2.556
2.013

QT (ε)
QT

1.077
3.661

1.326
2.883

2nd patient ON
ON
1.756
2.654
Whole group 0.810
4.071

15
OFF

30
OFF

45
OFF

60
OFF

0.291 0.438
–0.013 0.883

0.041 0.017
–0.004 0.856

0.544
1.47

0.671
1.624

0.728
1.734

0.731
1.742

component QT (ε), and the physiological contribution QT (k). The calculation of QT (k) is described in Sect. 4. One can deﬁne the quality of a treatment by means of QT . The positive value of the given factor deﬁnes an eﬀective treatment. For a separate patient and for the whole group, QT reaches its maximal value under the conditions ON, ON. The total quantitative factor is supplemented by a diagnostic (physiological) component. It allows one to take into account those features of the system which the chaotic component does not contain. For the second patient under condition 15 OFF (see Table 3) the factor QT has a negative value. It testiﬁes to the negative inﬂuence of the given treatment on the organism of the patient. The best treatment is thus the combination of the two medical methods: electromagnetic stimulation and medication.
Figure 5 reﬂects the behaviour of the parameter ε1 (0) for four diﬀerent subjects. The points lying above the horizontal line testify to an improvement of the state of the subject and the eﬃcacy of the treatment. The points, lying below the horizontal line testify to a deterioration of the state of the subject and the ineﬃciency of the applied treatment. For example, Fig. 5b corresponds to the sevenfold change of the quantitative measure of chaoticity for the 9th patient. In the case of the 8th patient (see Fig. 5c) no inﬂuence could change the measure of the chaoticity. Therefore there was practically no change in the state of the subject either. In some cases (see Figs. 5b, 5d) the DBS or the medication reduces the measure of chaoticity which testiﬁes to a deterioration of the state of the subject. This approach allows one to deﬁne the most eﬀective (or ineﬃcient) treatment in each individual case.
6.2 The Deﬁnition of a Predictor of Sudden Changes of the Tremor Velocity
In this subsection the window-time behaviour of the non-Markovity parameter ε1 (ω) for a certain case (the second patient, two methods of medical

Manifestation of Chaos in Real Complex Systems
7

191

7

(a)

6

(b)

6

5

5

4

4

3

3

2

2

1

1
1

2

3

4

5

6

0

7

1

2

3

4

5

6

7

1

ε (0)

0

2.5

2.5

(d)

(c)
2

2

1.5

1.5

1

1

0.5

0.5

0

1

2

3

4

5

6

0

7

1

2

3

4

5

6

7

Point
Fig. 5. The behaviour of the parameter ε1 (0) for four various patients: (a) the second subject, high amplitude tremor, (b) the 9th subject, low amplitude tremor
(the stimulation of the GPi); (c) the 8th subject, high amplitude tremor, (d) the
15th subject, low amplitude tremor (the stimulation of the STN). The value of ε1 (0) for seven consecutive conditions of the experiment: 1 - both methods are used; 2 treatment by medication is applied only; 3 - the DBS is used only; 4 (5, 6, 7) value of the parameter 15 (30, 45, 60) minutes after the stimulator was switched oﬀ; the horizontal line corresponds to the value of the parameter when no method is used. This representation allows one to deﬁne the most eﬀective treatment for each patient

treatment were used) and the procedure of local averaging of the relaxation parameters are considered. These procedures allow one to determine speciﬁc predictors of the change of regimes in the initial time records.
The idea of the ﬁrst procedure is, that the optimum length of the time window (28 = 256 points) is found ﬁrst. In the studied dependence (in our case the frequency dependence of the ﬁrst point of the non-Markovity parameter) the ﬁrst window is cut out. Then the second window is cut out
(from point 257 to point 512), etc. This construction allows one to ﬁnd the local time behaviour of the non-Markovity parameter. At the critical moments when the tremor velocity increases the value of the non-Markovity parameter comes nearer to a unit value. One can observe that the value of

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1

Velocity [m/τ]

1

2

4

6

5

0.5

0

−0.5

3
−1

0

1000

7

2000

3000

4000

5000

−2

Time [τ=10

1

2

3

4

1

ε (ω)

10

6000

7000

8000

sec]

5

6

7
0

5
0.1
0
0

0.2
5

10

15

20

25

30

ω [2 π / τ]

0.3

8

t [2 τ]

Fig. 6. The initial signal – the change of tremor velocity when the second patient is treated by two methods – and the window-time behaviour of the ﬁrst point of the non-Markovity parameter ε1 (ω). At the time of a sharp change of the mode (sharp increase of the tremor velocity) in the behaviour of the initial time series (regions
1–7) a gradual decrease of the non-Markovity parameter down to a unit value (the
3rd, 6th, 10th, 14th, 17th, 20th, 27th windows) is observed. The decrease of the non-Markovity parameter begins 2–2,5 s earlier than the tremor acceleration on an initial series

the non-Markovity parameter starts to decrease 2–2.5 s before the increase of the tremor velocity (see Fig. 6).
The idea behind the second procedure is the following: one can consider the initial data set and take an N -long sample. We can calculate kinetic and relaxation parameters for the given sample. Then we can carry out a
“step-by-step shift to the right”. Then we calculate kinetic and relaxation parameters. After that we execute one more “step-by-step shift to the right” and continue the procedure up to the end of the time series. Thus, the local averaged parameters have a high sensitivity to the eﬀects of intermittency and non-stationarity. Any non-regularity in the initial time series is instantly reﬂected in the behaviour of the local average parameters. The optimal length of the sample is 120 points. In Figs. 7, 8 the initial time record and the time dependence of the local relaxation parameter λ1 (t) are shown in two cases. The change in the time behaviour of the parameter λ1 (t) begins 2–3 s prior to the change of the regimes of the time record of the tremor velocity.

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0.8

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0.6
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−0.2
−0.4
−0.6

0

1000

2000

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−0.05

−0.1

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−0.2

−2

Time [t=10

sec]

Fig. 7. The change of the tremor velocity for the second patient (stimulation of the brain and medication are not used) and the time dependence of the local relaxation parameter λ1 (t). The localization procedure allows one to ﬁnd sudden changes of relaxation regimes of the system under consideration. The largest amplitude values of the local relaxation parameter are in the region of the lowest tremor velocity.
The change of the time behaviour of the parameter λ1 (t) begins 2–3 s earlier than the sharp change of the regimes in the initial time series appears

The increase of speed of the local relaxation parameter (λ1 (t)) testiﬁes to a decrease of tremor velocity.

7 Conclusions
In this chapter we have proposed a new concept for the study of manifestations of chaoticity. It is based on the application of the statistical non-Markovity parameter and its spectrum as an informational measure of chaoticity. This approach allows one to deﬁne the diﬀerence between a healthy person and a patient by means of the numerical value of the non-Markovity parameter. This observation gives a reliable tool for the strict quantitative estimates that are necessary for the diagnosis and the quantiﬁcation of the treatment of patients. As an example we have considered the changes of

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0
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−2

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sec]

Fig. 8. The change of the tremor velocity for the second patient (15 minutes after the stimulator was switched oﬀ, medication oﬀ) and the time dependence of the local relaxation parameter λ1 (t). The site characterizing the minimal tremor velocity is allocated. The increase and decrease of the local relaxation parameter occur 2.5 s before the decrease or increase of the tremor velocity. The similar behaviour of the parameter λ1 (t) can be explained by its high sensitivity to the presence of nonstationarity of the initial signal

various dynamic conditions of patients with Parkinson’s disease. The quantitative and qualitative criteria used by us for the deﬁnition of chaoticity and regularity of the investigated processes in live systems reveal new informational opportunities of the statistical theory of discrete non-Markov random processes. The new concept allows one to estimate quantitatively the eﬃcacy and the quality of the treatment of diﬀerent patients with Parkinson’s disease. It allows one to investigate various dynamic states of complex systems in real time.
The statistical non-Markovity parameter ε1 (0) can serve as a reliable quantitative informational measure of chaoticity. It allows one to use ε1 (0) for the study of the behaviour of diﬀerent chaotic systems. In the case of Parkinson’s disease the change of the parameter deﬁnes the change of a quantitative measure of chaoticity or regularity of a physiological system. The increase of the chaoticity reﬂects the decrease of the quantitative measure of pathol-

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ogy and the improvement of the state of the patient. The increase of the regularity deﬁnes high degree of manifestation of pathological states of live systems. The combined power spectra of the initial TCF µ0 (ω), the three memory functions of junior orders and the frequency dependence of the nonMarkovity parameter compose an informational measure which deﬁnes the degree of pathological changes in a human organism.
The new procedures (the window-time procedure and the local averaging procedure) give evident predictors of the change of the initial time signal.
The window-time behaviour of the non-Markovity parameter ε1 (ω) reﬂects the increase of the tremor velocity 2–2.5 s earlier. It happens when the nonMarkovity parameter approaches a unit value. The procedure of local averaging of the relaxation parameter λ1 (t) reﬂects the relaxation changes of physiological processes in a live system. The behaviour of the local parameterλ1 (t) reacts to a sudden change of relaxation regimes in the initial time record 2–3 s earlier. These predictors allow to lower the probability of ineﬀective use of diﬀerent methods of treatment.
In the course of the study we have come to the following conclusions:
– The application of medication for the given group of patients proved to be the most eﬃcient way to treat patients with Parkinson’s disease. Used separately stimulation is less eﬀective than the use of medication.
– The combination of diﬀerent methods (medication plus electromagnetic stimulator) is less eﬀective than the application of medication or of stimulation. In some cases the combination of medication and stimulation exerts a negative inﬂuence on the state of the subject.
– After the stimulator is switched oﬀ its aftereﬀect has an oscillatory character with a low characteristic frequency corresponding to a period of about 30 min.
– The eﬃcacy of various medical procedures and the quality of a treatment can be estimated quantitatively for each subject separately with utmost precision. However, if we take both chaotic and physiological components into account, the general estimation of the quality of treatment will be more universal. The combination of two methods (DBS and medication, QT = 4.071) produces the most eﬀective result in comparison with the eﬀect of DBS (3.661) or of medication (2.883) given separately. This is connected with additional aspects of the estimation of the quality of treatment due to the study of both chaotic and diagnostic components of a live system. This conclusion corresponds to the results of [15].
In conclusion we would like to state that our study gives a unique opportunity for the exact quantitative description of the states of patients with
Parkinson’s disease at various stages of the disease as well as of the treatment and the recovery of the patient. On the whole, the proposed concept of manifestations of chaoticity opens up great opportunities for the alternative

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analysis, diagnosis and forecasting of the chaotic behaviour of real complex systems of live nature.

Acknowledgements
This work supported by the RHSF (Grant No. 03-06-00218a), RFBR (Grant
No. 02-02-16146) and CCBR of Ministry of Education RF (Grant No. E 023.1-538). The authors acknowledge Prof. Anne Beuter for stimulating criticism and valuable discussion and Dr. L.O. Svirina for technical assistance.

References
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R.M. Yulmetyev, P. H¨nggi, F.M. Gafarov: Phys. Rev. E 62, 6178 (2000) a R.M. Yulmetyev, P. H¨nggi, F. Gafarov, Phys. Rev. E 65, 046107 (2002) a R.M. Yulmetyev, F.M. Gafarov, P. H¨nggi et al.: Phys. Rev. E 64, 066132 a (2001)
R.M. Yulmetyev, S.A. Demin, N.A. Emelyanova et al.: Physica A 319, 432
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R.M. Yulmetyev, N.A. Emelyanova, S.A. Demin et al.: Physica A 331, 300
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A. Beuter, M. Titcombe, F. Richer et al.: Thalamus & Related Systems 1, 203
(2001); M. Titcombe, L. Glass, D. Guehl et al.: Chaos 11, 766 (2001)
A. Beuter, A. de Geoﬀroy, P. Cordo: J. Neurosci. Meth. 53, 47 (1994)
K.E. Norman, R. Edwards, A. Beuter: J. Neurosci. Meth. 92, 41 (1999)
R.M. Yulmetyev, P. H¨nggi, F.M. Gafarov: JETP, 123, 643 (2003) a Monte Carlo Simulations of Ageing and Speciation
Suzana Moss de Oliveira1,2 and Dietrich Stauﬀer1,3
1

2

3

Laboratoire de Physique et M´canique des Milieux H´t´rog`nes, Ecole e ee e
Sup´rieure de Physique et de Chimie Industrielles de la Ville de Paris, 10, rue e Vauquelin, 74231 Paris Cedex 05, France.
Visiting from Instituto de F´ ısica, Universidade Federal Fluminense, Av.
Litorˆnea s/n, Boa Viagem, Niter´i 24210-340, RJ, Brazil a o suzana@if.uff.br Visiting from Institute for Theoretical Physics, Cologne University, D-50923
K¨ln, Euroland o stauffer@thp.uni-koeln.de

Summary. We present results from computer simulations of ageing and speciation, using mostly the Penna bit-string model which is based on the mutationaccumulation hypothesis.

1 Introduction
Verhulst would be 200 years old now, and the combined age of the present authors is more than half that age. Therefore, it is appropriate if we apply
Verhulst death probabilities to simulations of biological ageing. But there is more to life than eating, drinking, giving conference talks, and dying. We also want to be better than others, like the Brazilians from S˜o Paulo would like a when compared to those from Rio de Janeiro, or Homo Neanderthalis (living in D¨sseldorf) would like to be better than Homo Sapiens from Cologne. u This diﬀerentiation is called speciation if members from one group can no longer have viable children with members of the other group. (The opposite is not true: Both authors count themselves among homo sapiens and are male and female but have no children together [1]). We concentrate here on widespread simulations of ageing and on the few speciation theories which take into account the ageing of the individuals. In particular we want to know which ageing simulations explain the exponential growth of adult mortality with increasing age, i.e. the Gompertz law from about the same time as the Verhulst probability, and which theories explain the branching into two species without the help of any river or mountain.
The Chowdhury model assumes ageing and then puts it into a foodweb model. This model is reviewed elsewhere [2] and thus will not be given in this chapter. Sections 2.1 and 2.2 deal with the asexual Penna model of biological ageing and some of its applications and in Sects. 2.3 and 2.4 we present its

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sexual variant including some important results. Recent reviews are published in [3–5], and an earlier exhaustive review including complete Fortran programs and more biological literature is given in [6]. In Sect. 2.5 we present alternative computer simulations performed in order to explore some other possible reasons for biological ageing, diﬀerent from mutation accumulation.
Finally, Sect. 3 is dedicated to more recent versions of the sexual Penna model adapted to the study of sympatric speciation.

2 The Penna Model
2.1 Asexual Version
According to the mutation accumulation-theory, Darwinian selection pressure tries to keep our genomes as clean as possible until reproduction starts. For this reason we age: bad mutations that appear early in life are not transmitted and disappear from the population, while those that become active late in life can accumulate, decreasing our survival probability but without risking the extinction of the species. The Alzheimer disease is a good example of such a mechanism: although its corresponding defective gene is present in the genetic code since birth, its eﬀects generally appear at old ages (around 80). As a consequence, this disease is a very common one. The Penna model’s dynamics can be regarded as a computational realization of the mutation accumulation theory. In the original asexual version of the model [7] the chronological genome of each individual is represented by a bit-string of 32 bits. Each bit corresponds to one “year” in the individual lifetime, and consequently each individual can live at most for 32 “years”. A bit set to one means that the individual will suﬀer from the eﬀects of a deleterious inherited mutation (genetic disease) in that and all following years. As an example, an individual with a genome 10010... would start to become sick during its ﬁrst year of life and would become worse during its fourth year when a new disease appears. One step of the simulation corresponds to reading one bit of all chronological genomes. Whenever a new bit of a given genome is read, we increase by one the individual’s age.
Selection pressure is modelled by the introduction of a threshold T for the number of deleterious mutations that can be simultaneously active in a living individual’s genome. Usually, the probability of death because of genetic causes is assumed to be given by a step function Θ(Nmut − T ), with
Θ(x) = 1 if x ≥ 0 and = 0 otherwise; Nmut corresponds to the number of accumulated diseases up to the current individual’s age. If T = 2, for instance, the individual of the example above would die at age 4. Smoother functional forms have also been used [8] but, except at very old ages, the fundamental results of the Penna model do not appear to be sensitive to this choice.

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Death for non-genetic causes, representing the outcome of intra-speciﬁc competition for the limited resources of the environment, is modelled by a density- and time-dependent quantity, the Verhulst factor. This is a meanﬁeld death probability, given by
V (t) =

N (t)
,
Nmax

where N (t) is the total population at the beginning of time step t and Nmax is the carrying capacity, that is, the maximum population size the environment could support if no other causes of death would exist. We usually consider
Nmax ten times larger than the initial population N (0). At each time step and for each individual a random number between zero and one is generated and compared with V : if it is greater than V , the individual dies independently of its age or genome. The larger the population size is, the higher is the probability of an individual to die due to this death roulette.
The introduction of the Verhulst factor, or some equivalent form of limiting factor for the total population, is a necessity in simulational models to avoid population overﬂow. The one just presented has been criticized by some authors, since really random deaths in nature are hardly observed.
In [9], for instance, the authors adopted the alternative strategy of killing randomly only the newborns. In [10], simulations of the Penna model on a lattice present an equivalent strategy.
If the individual survives until a minimum reproduction age R, it generates b oﬀspring in that and all following years. The oﬀspring genome is a copy of the parent’s one, except for M mutations at randomly chosen genome positions introduced at birth. Although the model allows good and bad mutations, generally we consider only the bad ones. The reason is that harmful mutations are around 100 times more frequent than the backward ones – reverse mutations deleting harmful ones [11]. In this case, if a bit position holding a bit 1 is randomly tossed in the parent’s genome, it remains 1 in the oﬀspring genome; however, if a bit zero is randomly tossed, it is set to 1 in the mutated oﬀspring genome. In this way, for the asexual reproduction, the oﬀspring is always as good as or worse than the parent. Even so, a stable population is obtained, provided the birth rate b is greater than a minimum value [12].
2.2 Applications of the Asexual Version
A crucial test of ageing theories is the Paciﬁc Salmon, which dies shortly after producing oﬀspring. Alternative theories of ageing, based on wear and tear, on damage due to oxygen radicals, or on telomere depletion leading to the
Hayﬂick limit, can hardly explain this sudden death of previously healthy animals. Within the mutation accumulation hypothesis as implemented in the
Penna model, it is explained very simply by introducing a maximum reproduction age: Random mutations acting for ages above that age are not weeded

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out by Darwinian selection pressure, and thus the bits for ages above the cessation of reproduction are all mutated (if back mutations are prohibited).
Thus the animals die necessarily shortly after producing oﬀspring. [13, 14].
In the opposite direction, a good ageing model should also explain the
Gompertz law of the 19th century, which is well obeyed by modern humans of industrialized countries: q µ ∝ exp(ba) ,

where q is the mortality at age a, i.e. the fraction of people dying between ages a and a+1, but alive at age a. By deﬁnition this q cannot become bigger than one and, for q > 1/2, it is better to use the mortality function µ − ln(1−q), also called the hazard factor [15]. This mortality function can increase above unity and is deﬁned as −d ln S(a)/da ln[S(a)/S(a + 1)], where S(a) is the probability to survive from birth to age a. Figure 1 shows that the mortality function for Germany around the year 2000 agrees, for adult men, very well with the Gompertz law, and Fig. 2 shows that the Penna model reproduces this law. For children, the standard Penna model, the Gompertz law, and reality all diﬀer from each other.
For old age, the last dozen years has raised the hope that the mortality of the oldest old shows a maximum, a plateau, or at least a mortality deceleration (i.e. a negative second derivative d2 µ(a)/da2 with respect to age.) For ﬂies and some other animals this deviation is widely believed to hold [16]; in the case of humans, for which demographic statistics are the most complete,
Yearly mortality function, German males 2001; Gompertz exp(0.094(x-107)); Robine+Vaupel (112)
1

mortality

0.1

0.01

0.001

0.0001
0

20

40

60

80

100

age in years

Fig. 1. Mortality function µ for German men (from www.destatis.de) compared with a Gompertz ﬁt in this semilogarithmic plot

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mu(x) in asexual Penna model, 10^9 animals, 5000 < t < 10000; and 0.001 exp(0.52*age)
10

mortality function

1

0.1

0.01

0.001
0

2

4

6

8

10

12

14

16

age

Fig. 2. Mortality function µ for the standard asexual Penna model, counting (upper data) or not (lower data) the deaths from the Verhulst factor. From [5]

lots of exaggerations are published in [14] which contradict existing reliable data for Swedish or German men, Fig. 1. (Women usually obey less strictly the Gompertz law.) Only above the age of 110 years does a plateau possibly exist [15, 17] but the statistics there are sparse, and systematic errors due to age exaggeration seem still to play an important role in many industrialized countries [17]. Perhaps with better statistics and better estimates of systematic errors the deviations of human mortalities from the Gompertz law will have vanished in years to come. Nevertheless, several modiﬁcations of the
Penna model were published to give downward deviations from the Gompertz law, ﬁrst in [18] and perhaps best in [19]. For ecosystems with prey and predator, He et al. [20] found strong oscillations with time on a lattice, similar to those found in the solutions of the Lotka–Volterra equations.
The reason for ageing in the mutation accumulation theory is the high selection pressure before the minimum age of reproduction is reached, and the lack of such selection after the oﬀspring is produced. This does not mean, however, that only the minimum age of reproduction ﬁxes the maximum lifespan. Figure 3 shows that it also depends slightly on the length of the genomic bit-string. Only if besides the minimum age of reproduction also the mutation rate per bit is kept constant, do the results for long bit-strings become independent of the bit-string length. There seem to be no simple scaling laws how the mortality curves depend on this bit-string length [21].
Their simulations with sex are complemented here by our asexual simulations in Fig. 4.

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Age at which all individuals are dead, versus length of bitstring, minimum reproduction age = 6
22

20

18

lifespan

16

14

12

10

8
10

100

1000

length

Fig. 3. Lifespan versus genome length at ﬁxed minimum age of reproduction
10*10^5 asexuals, L=8,16,32,64,128,256,512,1024,2048,4096; minage=fmut=L/8; babykillers

survivors

1000

100

10

1
1

10

100

1000

age

Fig. 4. Number of asexual individuals as a function of age, for widely varying lengths of the genomic bit-string. 10 samples with 104 individuals were simulated up to mostly 20,000 iterations (longer for the longest genome). Four births were given at each iteration by each adult

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This ﬁgure shows that for the longer genomes, the survival curve becomes more rectangular than for the shorter genomes. This compression of mortality was actually observed during the medical progress in most of the last two centuries, though perhaps no longer in the last few decades in Europe. In the extreme case it means that we remain healthy until our sudden death, similar to Paciﬁc Salmon [13]. But the genetic reasons for this rectangular survival curve are diﬀerent: Now nearly all bits become mutated to one, independent of their position (age), for long genomes. Thus the individuals die if their age reaches the threshold of tolerated diseases at which they die.
This threshold, as well as the mutation rate per genome and the minimum age of reproduction, were taken as 1/8 of the genome length in this scaling test. Following [9] we applied the Verhulst deaths only to the babies.
These simulations used the program given below, based on the one published and explained in [6]. The bit-string consists of w computer words of nbyte bytes each such that the overall genome lengths is 8wnbyte bits. This parameter nbyte is deﬁned in line 6 and the same number must be put into line 7 after the integer*, except that integer*1 does not exist in usual
Fortran and needs to be replaced by byte. Thus the word length varies from
1 to 8 bytes. (The random integers ibm always use 8 bytes.) c asexual version (females only); variation of genome size c many words w for each genome; Verhulst\index{Verhulst} variable or constant c
Verhulst at birth only implicit none integer popdim,nbyte,nbit,nshift,w,nbiw parameter(w=2,nbyte=8,popdim=100000,nbit=8*nbyte,nbiw=nbit*w) integer*8 gen1f(popdim,w), gene1, p, bit(0:nbit) c byte gen1f(popdim,w), gene1, p, bit(0:nbit) integer popmax,inipop,maxstep,medstep,minage,fage,lim,fmut,
1 n6,t,i,seed,fa,n,imut,age,nmut,fpop,j,k,irun,nrun,agm,birth,
2 fnumber(0:nbiw),dataf(popdim) parameter(popmax=popdim/10, inipop=popmax/10, maxstep=20000,
1 medstep=maxstep/2, minage= nbiw/8, fage=nbiw, lim=nbiw/8,
2 fmut=nbiw/8, birth=4, seed=2, nrun=1 ) real factor integer*8 ibm,verhu,mult,fymed(0:nbiw),count(0:nbiw),ict data n6/’00007FFF’X/ if(nbyte.eq.8) nshift=58 if(nbyte.eq.4) nshift=59 if(nbyte.eq.2) nshift=60 if(nbyte.eq.1) nshift=61 c ict=0 ibm=2*seed-1 factor=(0.5/2147483647)/2147483647 mult=13**7 204

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2

28

6 c c

c

9

c

21

mult=mult*13**6 print *, popmax,inipop,maxstep,medstep,minage,
1
fage,lim,fmut,birth,seed,nbit,w,nrun bit(0)=1 do 2 i=0,nbiw if(i.gt.0.and.i.lt.nbit) bit(i)=ishft(bit(i-1),1) fymed(i)=0 fnumber(i)=0 count(i)=0 ibm=ibm*16807 do 29 irun=1,nrun print *, irun call flush(6) do 28 i=0,nbiw fnumber(i)=0 fpop=inipop fnumber(0)=fpop do 6 i=1,fpop dataf(i)=0 do 6 k=1,w gen1f(i,k)=0 dataf: age at bits 0 to 14, mutations at 15 to 29 do 7 t=1,maxstep if((t.le.100.and.t.eq.(t/10)*10).or.t.eq.(t/1000)*1000.or. 1
(t.le.1000.and.t.eq.(t/100*100)).or.t.le.10) then if(irun.eq.1) print *, irun, t,fpop call flush(6) end if verhu=2147483648.0d0*(fpop*4.0/popmax-2.0)*2147483648.0d0 i=1 fa=fpop age =iand(n6,dataf(i)) verhu=2147483648.0d0*(fpop*4.0/popmax-2.0)*2147483648.0d0 k=1+age/nbit agm=mod(age,nbit) nmut=iand(n6,ishft(dataf(i),-15)) n=iand(1,ishft(gen1f(i,k),-agm)) if(n.ge.1) nmut=nmut+1 fnumber(age)=fnumber(age)-1 age=age+1 if(nmut.ge.lim.or.age.eq.nbiw) then death if(fpop.le.1) goto 1 do 21 k=1,w gen1f(i,k)=gen1f(fpop,k) dataf(i)=dataf(fpop) fpop=fpop-1 Monte Carlo Simulations of Ageing and Speciation

c

c

23

c c 13
12
c

c c 10

26

if(fpop.ge.fa) then i=i+1 else fa=fa-1 endif else survival fnumber(age)=fnumber(age)+1 dataf(i)=ior(age,ishft(nmut,15)) if(age.ge.minage.and.age.le.fage) then do 12 n=1,birth ibm=ibm*16807 if(ibm.lt.verhu) goto 12 birth fnumber(0)=fnumber(0)+1 fpop=fpop+1 if(fpop.gt.popdim) goto 1 dataf(fpop)=0 do 23 j=1,w gen1f(fpop,j)=gen1f(i,j) do 13 imut=1,fmut ibm=ibm*16807 k=iabs(ibm)*factor*w+1 gene1=gen1f(i,k) ibm=ibm*16807 p=bit(ishft(ibm,-nshift)) mutations in mother gen1f(fpop,k)=ior(gene1,p) gen1f(fpop,k)=ior(gen1f(fpop,k),ior(gene1,p)) continue continue if(female suitable) then endif i=i+1 endif if(death) .. else (survival, birth) .. if(i.le.fa) goto 9 end of selection and birth, now start averages if(t.lt.maxstep-medstep .or. t.ne.(t/10)*10) goto 7 do 10 i=0,nbiw fymed(i)=fymed(i)+fnumber(i) do 25 i=0,nbit do 26 j=1,w do 26 k=1,fpop if(i+j.eq.1) ict=ict+1 gene1=gen1f(k,j) age=i+nbit*(j-1) count(age)=count(age)+iand(1,ishft(gene1,-i)) 205

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25
7
29
100
1

continue continue continue print100,(i,fymed(i)*1.0/medstep,count(i-1)*1.0/ict,i=1,nbiw) format(1x,i5,2f13.5) stop print *, ’error’ ,t, fpop stop end

2.3 Sexual Version
In the sexual version of the Penna model [22, 23] the population is divided into males and females. Individuals are diploids, with their genomes represented by two bit-strings that are read in parallel. Each bit-string contains the genetic information inherited from one of the parents. In order to count the accumulated number of mutations and compare it with the threshold T , it is necessary to distinguish between recessive and dominant mutations. A mutation is counted if two bits set to 1 appear at the same position in both bit-strings (homozygous) or if it appears in only one of the bit-strings (heterozygous) but at a dominant position (locus). The dominant positions are randomly chosen at the beginning of the simulation and are the same for all individuals. After reaching the minimum reproduction age R, a female randomly chooses a male with age ≥ R to breed. To construct one oﬀspring genome ﬁrst the two bit-strings of the mother are cut in a random position (crossing), producing four bit-string pieces. Two complementary pieces are chosen to form the female gamete (recombination). Finally, Mf deleterious mutations are randomly introduced. The same process occurs with the male’s genome, producing the male gamete with Mm deleterious mutations (see Fig. 5). These two resulting bit-strings form the oﬀspring genome. The sex of the baby is randomly chosen, with a probability of 50% for each one. This whole strategy is repeated b times to produce the b oﬀspring. The Verhulst killing factor works in the same way as in the asexual reproduction.
2.4 Applications of Sexual Version
From the example of Paciﬁc Salmon one could ﬁrst guess that women should not survive menopause, or more generally, females should not survive the end of their reproductive phase (which is observed in many animals: [24], Austad in [14]). Simulations [6] show that this is not so: Men are assumed to be able to reproduce until their death, and then they also help women to survive after menopause since the sex of the child is selected randomly; thus, within this model, mutations relevant for old age accumulate equally for both men and women, prohibiting to selectively kill women but not men.

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mother
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father

1

2

4

0

1 0 1

0 0 1 1

1

2

3

4

0 0 0 1
1 0 1

0

offspring

1 0

1 1

1 1 0 1

Fig. 5. Schematic representation of sexual reproduction in the Penna model using bit-strings of only four bits (individuals would live at most for four “years”). Vertical lines indicate crossover positions. Arrows indicate where random mutations were introduced in each gamete. The union of the two gametes forms the oﬀspring genome

These simulations only explain why menopause is possible and not why it arises. For humans, cultural eﬀects may be important [25]: Grandmothers help in raising their grandchildren instead of giving birth. For animals, in general, this explanation does not hold. Fortunately, simulations of the sexual
Penna model, with a lethal risk of giving birth increasing with age, and with a certain age interval where oﬀspring can survive only if their mother takes care of them, have shown that cessation of female reproductive ability emerges by itself, without being put in at the beginning [26], Fig. 6. This self-organisation explains menopause or its analogues without recourse to any traits special to humans; these computer simulations [26] were, to our knowledge, the ﬁrst quantitative explanations of menopause analogues for animals. Correlations like those in Fig. 1 of [25] are also found in the Penna model (to be published).
Why do women live longer than men, in nearly all countries at present?
One of the authors claims that men drink too much wine and eat too many steaks, as opposed to her. The other claims to be oppressed by women in general and Mother Nature in particular, since he got only one X chromosome while women got two. Only the second claim thus far is conﬁrmed by computer simulations [27]: A bad mutation in one X chromosome does not reduce the health if the other X chromosome is not mutated and if the bitposition is recessive; for men with only one X chromosome this helpful eﬀect is impossible. However, the actual diﬀerence between male and female life expectancies at birth in Sweden (no war since nearly two centuries) shows drastic variations in time, Fig. 7, though always favouring women. Thus presumably both genetic and life-style reasons are important and may even be coupled [28].

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fraction with age of menopause

0.20

0.15

0.10

0.05

0.00

10

12

14

16

18
20
22
24
age of menopause

26

28

30

32

Fig. 6. Fraction of females with a given age of menopause as a function of the age of menopause. Circles: parental care for 5 years; Diamonds: parental care for 4 years; Line: no parental care

If men are slightly defective versions of women, why has Mother Nature invented them? Why not follow bdelloid rotifers and some fungi [29], who have lived without males since ∼ 108 years. Redﬁeld [30] already questioned the usefulness of males and was largely ignored by fellow biologists [31], while her paper triggered many simulations of physicists reviewed in [3]. By assuming that mutations only kill the individual when enough of them have accumulated, but that already before they reduce the survival probabilities in addition to the Verhust deaths, an advantage of sexual versus asexual reproduction was found [32], in spite of the loss of births by a factor two (since men don’t become pregnant). Much clearer is the situation for bacteria where, without a separation of males and females, a clear advantage was found if sometimes they exchange parts of their genome [33]. Also changes in the environment favours sex [34]. However, no explanation was found yet explaining why hermaphroditic reproduction is not clearly dominating over other forms of life [3]. A good review of the biological questions is given in [35].

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Female minus male life expectancy in Sweden, from statistiska centralbyran web site www.scb.se
6

5.5

5

difference

4.5

4

3.5

3

2.5

2
1800

1850

1900

1950

2000

calendar year

Fig. 7. Diﬀerence between the female and male life expectancies at birth for Sweden, over more than two centuries

2.5 Other Models
The Dasgupta model [36] uses no bit-string as a genome but instead works with survival probabilities. Originally it was thought not to give a large number of diﬀerent ageing steps, but years later minor modiﬁcations solved this problem. It even was successfully generalized to sexual reproduction.
Mueller and Rose [37] tried to ﬁnd a mortality plateau for the oldest old; it seems however that their simulations were not run long enough to get into a good equilibrium. Charlesworth made simulations with the same aim, but not many follow-up simulations are known to us [38]. Promislow [39] has very recently published a senescence theory based on pleiotropic protein networks, using the scale-free networks investigated by physicists. More similar to the
Penna model is the one of [40].
More follow-up was published for the Stauﬀer model [4] which revived the century-old Weismann idea that we die to make place for our children. This idea works if one assumes that the total number of oﬀspring during the life of an individual remains constant [41]. Thus random mutations which increase the life span simultaneously reduce the birth rate per unit time. Normally this model gives a mortality increasing linearly (not exponentially) with age, as for mayﬂies (but not for humans). It was used for another conﬁrmation
[42] that menopause and its analogues are not necessarily connected with

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grandmothers and human civilization: Death risk increasing with age was suﬃcient to let menopause emerge by itself.
All these theories including the Penna model use the accumulation of mutations over many generations, sometimes combined with antagonistic pleiotropy, which has good eﬀects at young age and bad eﬀects at old age.
There may be entirely diﬀerent reasons for ageing, like the wear and tear which often ends the careers of athletes. Or oxygen radicals produced by normal metabolism may slowly damage the cells and their DNA; perhaps the polyphenol resverotrol in red wine helps to scavenge these dangerous molecules. But we are not aware of quantitative theories allowing a comparison with real mortality functions.
The most recent quantitative theory known to us uses telomeres [43].
These are end segments of the DNA string; at each cell division, the duplication of the DNA causes the loss of one of these segments. If none of them is left the cell no longer divides and has reached its Hayﬂick limit in vitro.
The enzyme telomerase may restore lost telomeres in vivo. Aviv et al suggested, against widespread opinion, that telomere loss may be the reason for ageing [44]. Masa et al [43] combined telomeres with cancer and showed that genetic manipulation for longer telomeres may increase the risk of dying due to cancer, which can prevail over the positive eﬀect of longer telomeres on the longevity. Figure 8 shows one of the resulting mortality functions, in striking similarity with reality, Fig. 1.

3 Sympatric Speciation
3.1 Model with a Single Phenotypic Trait
Speciation involves the division of an already well-adapted species, so that each part moves onto a new adaptive peak. This process is easily understood if a species becomes subdivided by a physical barrier, like a river: Each part experiences diﬀerent mutations, population ﬂuctuations and selective forces, in what is called allopatric speciation. In contrast, conceiving the division of a single population and radiation onto separate peaks without geographical isolation, in what is called sympatric speciation, is intuitively more diﬃcult.
Through which mechanism can a single population be converted into two reproductively isolated segments in the absence of spatial barriers to prevent gene exchange?
Many papers have appeared during the last 5 years, mainly produced by biologists (see for instance [45, 46] and also [47] for a review), proposing diﬀerent mechanisms to explain the origins of sympatric speciation. We concentrate here on simulations including ageing. The ﬁrst successful attempt to simulate this phenomenon through the Penna model [48] was performed using the same ingredients proposed in [45, 46], which are: competition for resources (related to some ecological phenotypic trait of the individuals) and

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Simulations of M. Masa for ageing through telomere loss and cancer mutations
1

mortality function

0.1

0.01

0.001

0.0001

0

20

40

60

80

100

age

Fig. 8. Mortality function in a telomere model with cancer, from [43]. Note the similarity with the plot for German men, Fig. 1

sexual selection (related to some phenotypic trait that determines sexual preferences). In the simulations presented in [48], the ecological and the sexual selection traits were chosen to be the same. A sudden change of the food resources provokes a change in the individuals’ phenotypes, favouring two substantially diﬀerent phenotypes (disruptive selection). Assortative mating (non random mating) then leads to reproductive isolation between these two phenotypes, followed by the elimination of all intermediate phenotypes (speciation). The biological motivation of these simulations was to mimic a real phenomenon observed in the Galapagos Islands [49–51], where the size of the available seeds is dictated by the amount of rain. Depending on the season, the distribution of these seeds changes from a broad one, centred at middle-sized seeds, to a double-peaked distribution, of only small and large seeds. It has a great impact on the morphology of beak sizes in the population of ground ﬁnches that feed on these seeds: the distribution of beak sizes follows that of the seeds, in a very fast process of adaptation.
The ﬁrst step to implement the above ingredients was to introduce a phenotype into the Penna model. It was done by adding to the chronological genome an extra-pair of non-age structured bit-strings of 32 bits each, representing the beak. The dynamics of reproduction and mutations are the same

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mother
1

2

3

4

0

1 0 1

0 0 1 1

0 0 1 1

0 0 1

1 0

0 0 0 1

1 1

0

Fig. 9. Schematic representation of gamete formation in sexual reproduction, adding now a non age-structured part to the chronological genome. Arrows indicate where random (good and bad) mutations occurred. The same scheme applies to the father’s genome (who cares about males?)

for both the age-structured and the new strings – for the latter, a mutation that changes a bit from 1 to 0 is also allowed (Fig. 9).
The beak size is determined by counting, in this non-structured pair of bit-strings, the number of recessive bit-positions (chosen as 16) where both bits are set to 1, plus the number of dominant positions with at least one of the two bits set. It will be a number n between 0, meaning a very small beak, and 32, for a very large one. The second step was to make use of the
Verhulst factor to attribute a selective character to the beak size and to change the competition between individuals from a general one to an intraspeciﬁc competition, where animals with large (small) beaks dispute only among them.
The new Verhulst factor is given by:
V (t) =

Nn (t)
,
Nmax F (n)

where Nn (t) accounts for the population that competes for resources available to individuals of beak size n. The ﬁtness function F (n) is a measurement of the individual’s ability to make use of food resources, depending on the current distribution of seeds and its beak size. Nmax is the same carrying capacity of the original Verhulst factor and so the practical role of the function F (n) is to increase or decrease this carrying capacity perceived by each individual, depending on its beak size.
The simulations were done with two diﬀerent functional forms for the function F (n). At the beginning of the simulations, F (n) is a single-peaked function with a maximum at n = 16, representing large availability of medium-sized seeds. The whole population of ﬁnches will compete for the

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same resources and Nn (t) is the total population. After some number Nstep of iterations, the function F (n) changes to a two-peaked shape, with maxima at n = 0 and n = 32, and now the food resources concentrate on either small or large seeds, with a vanishing number of medium-sized ones. Only small(large)-beaked individuals – those with n < (>)16 – can compete for the small(large) seeds. For that reason, the death probability V (t) of an individual with n < (>)16 is computed by assigning to Nn (t) the number of individuals with n < (>)16 plus half of the population that has n = 16. An individual with n = 16 competes either for small or large seeds, and this choice is random.
Finally, in order to simulate sexual selection a single locus was introduced into the genome that codes for this selectiveness, also obeying the general rules of the Penna model for genetic heritage and mutation. If it is set to
0, the individual will not be selective in mating (panmictic mating), and it will be selective (assortative mating) if this locus is set to 1. The mutation probability for this locus was set to 0.001 in all simulations. Females that are selective will choose mating partners that satisfy some criterion related to the beak size.
At the beginning of the simulations, when the ﬁtness function F (n) is single-peaked, there is no selective pressure for mating selectiveness, and the population mates randomly: all females are non-selective. After F (n) becomes double-peaked, females that mutate into selectiveness will choose mating partners that have beak sizes similar to their own: if a female has n < (>)16 and is selective, she will only mate with a partner that also has n < (>)16.
In Fig. 10 we show the distribution of beak sizes of the populations at time step Nstep = 12 000, up to which the ﬁtness function was single-peaked, and at time step 50 000, after it has been double-peaked for 30 000 time steps. A stable polymorphism has clearly been established as a result of the duplicity of food resources. The fraction of selective females in the population, which was
0 at the start of the run, also increases to nearly 1.0 after the establishment of a double-peaked F (n). Now there are two distinct populations, each of which does not mate with a partner from the other.
3.2 Model with Two Phenotypic Traits
Not always the ecological trait, determining individual ﬁtness to the environment, is the same used by females to choose a partner. In case of ﬁsh, for instance, sexual selection is related to the colour, while ﬁtness depends on the size [52]. In order to make the simulations more realistic a second phenotypic trait – another pair of non age-structured bit-strings – was added to the Penna chronological genome [53].
Dynamics of death and birth follow the same rules as before, and the phenotype space for this new sexual trait is mapped again onto an integer between 0 and 32. For mating, a female chooses, among a random selection

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0.2
seeds middle-sized seeds extreme-sized

relative frequency

0.15

0.1

0.05

0

0

5

10

15 beak size (n)

20

25

30

Fig. 10. Distribution of beak sizes when middle-sized seeds are abundant (circles – time step = 12,000) and when only small and large-sized seeds are available
(squares – time step = 50,000). We show the fraction of the population with each beak size

of a ﬁxed number of males, chosen to be 6 in the simulations, a suitable mating partner for whom the phenotype for this second trait (colour, say) matches her own. Let us call f the phenotype of the female and m the one for the male, for the sexual selection trait. Then, if the female has mutated into selective, it follows the rules:
If f < 16 then it selects the male with the smallest m;
If f > 16 then it selects the male with the largest m;
If f = 16 then the female chooses randomly to act as one of the above.
If we think about this mating trait as colour, for instance, and assign f < (>)16 to a blue (red) character, a blue (red) female will choose the male that lies deepest into the blue (red) region.
Figure 11 shows that, like in the former case, the distribution of the ﬁtness trait is single-peaked at n = 16 up to step Nstep , as a consequence of the number of loci (16) where the 1 allele is dominant, and moves into a polymorphism after the ecology becomes bi-modal. The sexual selection trait also shows a single peak until step Nstep , and splits the population in two groups afterwards. But now a strong correlation develops between these traits, and the individuals with sexual selection phenotype < (>)16

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0.2 size colour
0.18
0.16

relative frequency

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

5

10

15

20

25

30

number of bits 1

Fig. 11. Distribution of ﬁtness (size = squares) and sexual (colour = *) traits at the end of the simulation. These traits are correlated, and the population with ﬁtness trait to the left has its sexual trait also to the left of the plot

have their ﬁtness trait < (>)16. Sexual selectiveness also develops as a result of evolutionary dynamics, and at the end of the simulation all females are selective. 3.3 Model Without Phenotypic Traits
A much simpler strategy, without considering phenotypes but adopting the same ingredients of competition and sexual selection, has also succeeded in obtaining speciation through the Penna model. In this case a single bit position, which was taken as position 11 in the Penna chronological genome, determines the mating [53]. Each diploid individual has k = 0, 1, or 2 bits set at this position. A female with k such bits at position 11 selects only males with the same number k of such “speciation” bits to mate. The simulation starts with a single species, that is, all individuals have k = 0. Due to the randomness of mutations and crossover, oﬀspring do not necessarily have the same k speciation bits set to one as their parents, and this randomness allows the emergence of new species out of the original one. At every time step t three populations Nk , depending on the number k = 0, 1, 2 of speciation bits set to one, are now co-evolving, and each of these three sub-populations is half male and half female.

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Co-existence is again obtained by adopting intra-speciﬁc competition, that is, by replacing the standard Verhulst factor by three separate Verhulst factors for the separate populations k = 0, 1, 2. Suppose, for instance, that the original population k = 0 is vegetarian, and that the second population k = 2 emerging out of it consists of carnivores. Both populations are limited by the amount of food, but their food sources are completely diﬀerent; thus, there is no competition between the two diﬀerent populations, but the meateating females will not select any herbivore males for mating, and vice versa.
The small population with k = 1 can be regarded as one that feeds in both niches. It is added half to k = 0 and half to k = 2 for the evaluation of the two intra-speciﬁc Verhulst factors,
V0 =

(N0 + N1 /2)
(N2 + N1 /2) and V2 =
,
Nmax
Nmax

and has the arithmetic average of these two Verhulst factors as its own foodlimiting Verhulst factor.
Figure 12 shows how males of the new species N2 emerges from the old species N0 . For the females the results are about the same.
Similar results were found using a square lattice without Verhulst factors; intra-speciﬁc competition is introduced through some rules that determine whether a place for a newborn exists [54].
2e+06
1.8e+06
1.6e+06

population size

1.4e+06
1.2e+06
1e+06
800000
600000
400000
200000
0
100

1000

10000

time

Fig. 12. Variation in time of N0 (dashed line, original species) and N2 (+, new species). Note linear, not logarithmic, vertical scale. The intermediate population
N1 , that competes both with N0 and N2 , is only about one percent of the total and is not shown

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The inﬂuence of recombination rate in speciation has also been investigated making use of the Penna model [55].

Acknowledgments
We thank ESPCI/PMMH for the warm hospitality during the period when this manuscript was produced. We also thank many people, in particular
P.M.C. de Oliveira, T.J.P. Penna, A.T. Bernardes, S. Cebrat, J.S. S´ Martins a and A.O. Sousa for helpful discussions over many years. S.M.O. thanks J.S.
S´ Martins and K. Luz-Burgoa for the collaboration in the results presented a in Sects. 3.1 and 3.2 and the Brazilian agency CNPq for ﬁnancial support.

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Part III

Econophysics

Inﬂuence of Information Flow in the Formation of Economic Cycles
J. Mi´kiewicz1 and M. Ausloos2 s 1

2

Institute of Theoretical Physics, University of Wroclaw, pl M. Borna 9, 50-204
Wroclaw, Poland jamis@ift.uni.wroc.pl SUPRATECS, B5, University of Li`ge, B-4000 Li`ge, Euroland e e marcel.ausloos@ulg.ac.be 1 Introduction
Despite the fact that Verhulst’s idea [1] of regulated biological populations is 200 years old, it is still very useful since it allows to investigate features of various systems. Here an eight order logistic map is applied in modelling the inﬂuence of information ﬂow delay onto the behaviour of an economic system. The delay of information ﬂow is an internal feature of all economic systems, because continuous monitoring of such systems both on macro and microeconomic scales is either extremely diﬃcult or even impossible. The more so since the data is not easily available nor even reliable as it could be in physics laboratories. Macroeconomic parameters such as Gross Domestic
Product, Gross National Product, inﬂation, demographic data etc. are announced in well deﬁned time intervals (monthly, quarterly or annually). The same situation is observed in the case of various companies. They announce their ﬁnancial statements about their economic results at speciﬁc dates and for given time intervals – according to internal or external rules (usually according to law regulations). Sometimes some “warning” is issued. However the tendency is that intervals between announcements are rather long, e.g. the value of a dividend is announced annually or at various trimester ends.
It seems obvious that only very small companies are able to perform continuous monitoring. But even then, the process of collecting information from a signiﬁcant (on a macroscopic scale) number of such companies inhibits or makes it impossible to perform continuous monitoring. In view of the data collecting procedure it is clear that every economic decision is based on some information describing a past situation. It is also important to notice that the time delays between information gathering, decision taking, policy implementation, and subsequent data gathering are not constant, nor are a fortiori continuous variables, as that was considered in [2]; indeed the information about the system is updated at the end of discrete time intervals.
Therefore econophysics-like modelling of such features encounters some diﬃculty, surely at the testing level. Recently a microscopic-like approach has been presented, through a model [3–5] including some measure of a company

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J. Mi´kiewicz and M. Ausloos s ﬁtness with respect to an external ﬁeld, and a birth-death evolution, according to some business plan, and the local company close range environment.
The information ﬂow was however considered to occur instantaneously.
In order to investigate the discrete information ﬂow time delay and its eﬀect, a model, hereby called the ACP model [3–5], has been modiﬁed by splitting the information about the system into two parameters. One is monitored continuously (is updated at every iteration step) and is known to the system itself; the second, like oﬃcial statements of the system, is announced at the end of discrete time intervals and is used by companies for calculating their strategies. Therefore the strategy of a company depends on the delay time information and the information itself. As it is shown in Sect. 3 the length of the time delay (td ) inﬂuences quite strongly and in a nontrivial way the behaviour of the overall system.
Detailed description of the ACP model is given in Sect. 2 and the properties of the system as a function of time delay and initial concentration are investigated (Sect. 3) in the case of short, medium and long time delays.

2 ACP Model
For the sake of clarity the basic ingredients of the ACP model are recalled here below. The main problem was to simulate the behaviour of economic systems in spatio-temporally changing environmental conditions, e.g. political changes and destruction of economic barriers. The model was set in the form of a
Monte Carlo simulation. Notice that the ACP model [3–5] contains among its variants an adaptation of the Bak–Sneppen model and was built in order to answer economic questions3 . The model consists of
1. space – a square symmetry lattice,
2. companies, which are initially randomly placed on the lattice, in an
3. environment characterized by a real ﬁeld F ∈ [0, 1] and a selection pressure sel,
4. each company (i) is characterized by one real parameter fi ∈ [0, 1] (its so-called ﬁtness).
The following set of actions was allowed to companies:
1. companies survive with the probability pi = exp(−sel|fi − F |)

(1)

2. companies may move on the lattice horizontally or vertically, one step at a time, if some space is available in the von Neuman neighbourhood.
3. if companies meet they may
3

Let us recall that the Bak–Sneppen model was originally built in order to investigate the coevolution of populations [6]

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a) either merge with a probability b,
b) or create a new company with the probability 1 − b.
The ACP model may be described in a mean ﬁeld approximation [7, 8] by introducing the distribution function of companies N (t, f ), which describes the number of companies having a given ﬁtness f at time t. The system is then additionally characterized by the concentration of companies c(t).
The present report of our investigations is restricted to the case of the best adapted companies (f = F ), so that the selection pressure has no inﬂuence on the survival of companies. So the only factor which could alter the number of companies is the strategy, i.e. the decision to merge or create a new entity.
The ideas behind the mean ﬁeld approximation [7, 8] are applied here and developed by introducing a strategy depending on the system state and the discrete time of the oﬃcial announcement about the state of the system.
The introduction of the strategy depending on the state of the system reﬂects the idea of Verhulst [1], when replacing the constant Malthus grow rate by the function 1 − x, which introduced a limit for the system to grow.
In the present investigation it is assumed that the strategy should depend on the state of the system. Moreover the company board takes its decision knowing information announced about its environment. The generation of new entities is more likely in the case of a low concentration of companies than when this concentration is high. The merging parameter describes the reversed dependency, i.e. merging is more likely to occur in the case of a high density of companies than if the density is low. The simplest function which fulﬁls this condition is 1 − c, the same as in Verhulst original work [1].
The additional ingredients to the ACP model are thus
1. the merging parameter b is replaced by a strategy (1 − c),
2. the companies know the value of the concentration c according to oﬃcial statements announced after the time delay td .
The evolution equation of the system with companies, using the state dependent strategy is:
1
ct = ct−1 + ct−1 (1 − c8 )(1 − (1 − ct−1 )8 )(2ST (c(g(t))) − 1) , t−1 2

(2)

where ST (c) = 1 − c, g(t) = k[t/k] and [ ] denotes the procedure of taking a natural number not larger than the one given in the brackets. The time is measured in iteration steps IS.

3 Results
Numerical methods were used in order to investigate properties of the system.
Because the coevolution equation (2) is given as an iteration equation the time is discrete and counted in iteration steps (IS). The following features of the system were examined:

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J. Mi´kiewicz and M. Ausloos s 1. the coevolution of c(t) as a function of the initial concentration,
2. the stability time deﬁned as the time required to achieve a unique stable solution; because of numerical reasons the criterium applied here is |cn+1 − cn | < 10−10 ,
3. the crash time tc , such that ctc < 0 (it is understood as the time when all companies are wiped out from the system),
4. the stability intervals – the intervals of initial values for which the evolution of the system is longer than a given time ts
5. the complex Lyapunov exponent
1
N →∞ N

N

λ = lim

log2 n=1 dxn+1 dxn .

(3)

The Lyapunov exponent calculated in its complex form (3) gives also some information about the oscillations of the system. Using the properties of logarithm: a < 0 ⇒ log a = log(−1 · |a|) = log(−1) + log |a| .

(4)

The imaginary part of log2 (dxn+1 /dxn ) gives some information on whether the distances between consecutive iterations are monotonic.
The numerical iterations were performed for the initial concentration in the interval c0 ∈ (0, 1), at consecutive values distant of 0.02. Therefore 500 histories of evolution were investigated.
It is possible to observe three types of coevolution: a unique, a periodic and a chaotic solution. In the case of a unique solution the system may approach this solution as a “damped” coevolution or “damped oscillation”.
The damped coevolution happens if ∀t > 0: c(∞) − c(t) > 0 or if ∀t > 0: c(∞) − c(t) < 0 and |c(t − 1) − c(t)| ≥ |c(t) − c(t + 1)|, where c(∞) is the asymptotic state of the system. This means that the distance between concentration and asymptotic concentration is decreasing in every iteration step and the concentration is either smaller or bigger than the asymptotic concentration. The damped oscillations are observed if |c(t − 1) − c(t)| ≥
|c(t)−c(t+1)| and ∃t0 such that ∀t > t0 : c(t) > c(∞) and c(t+1) < c(∞). This means that the distance between consecutive concentrations of companies is decreasing. In the case of a periodic solution for t > t0 there exists a n-tuple of concentrations which is repeated for t > t0 , where t0 is the time required by the system to reach the stable or periodic solution. The length of the ntuple is deﬁned as the period of oscillations. The system is chaotic if the real part of the Lyapunov exponent is positive: Re (λ) > 0.
The coevolution of the system is presented either as a function of time
(Figs. 1, 4, 7, 10, 13, 15, where the coevolution is plotted for chosen initial concentrations) or as a function of initial concentration (Figs. 2, 5, 8, 11, 14,
16, where the coevolution of the system is plotted in one vertical line so the plot is a set of coevolutions for 500 diﬀerent initial concentrations.

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3.1 Stability Window
The short time delay (td ) is deﬁned by 2 IS ≤ td ≤ 4 IS. In this case the system evolves to the unique stable solution c = 0.5. Within this time delay the Lyapunov exponent is equal to zero; no chaotic behaviour is seen. td = 2 IS
The time delay td = 2 IS means that the information about the system is updated every two iteration steps. The evolution of the system is presented in Fig. 2 and is plotted as a function of initial iteration. For every 500 initial concentrations 103 iteration steps have been used. The history examples are presented in Fig. 1 as a function of concentration in time. In the case of the shortest time delay considered here the system has a unique solution c = 0.5.
The stability time as a function of initial concentration is shown in Fig. 3.
0.01 a long time (ts ≥ 47 IS)
For a very low initial concentration 0 < c0 is needed in order to achieve the stable state. It is also illustrated in Fig. 1, where in the case of low initial concentrations c0 = 0.002 the stability time is quite long (about 100 IS). However except for very small initial concentrations
(c0 > 0.1) the stability time is short, i.e. ts ∈ (10 IS, 20 IS).
0.6

concentration

0.5
0.4
0.3
0.2
0.1
0
0

50

100

150

200

time c0 = 0.002 c0 = 0.01

c0 = 0.02 c0 = 0.03

Fig. 1. Coevolution of the system for chosen initial concentrations; delay time td = 2 IS

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J. Mi´kiewicz and M. Ausloos s 1
0.9
0.8

concentration

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4
0.5
0.6 initial concentration

0.7

0.8

0.9

1

Fig. 2. Coevolution of the system as a function of initial concentration. The coevolution of a system is represented by a vertical series of dots; delay time td = 2 IS

80
70
60

time

50
40
30
20
10
0
0

0.2

0.4

0.6

0.8

1

initial concentration
Fig. 3. The time required for the system to achieve a stable concentration as a function of initial concentration; delay time td = 2 IS

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td = 3 IS
Increasing the time delay by one unit, up to three iteration steps, induces important changes in the system. In the evolution of the system, damped oscillations become observable, e.g. for c0 = 0.002 damped oscillations are observed for t ∈ (140 IS, 155 IS) (Fig. 4 and Fig. 5). The maximum time required for the system to achieve a stable state extends to ts ≥ 220 IS as compared with ts ≥ 47 IS for td = 2 IS. For most initial concentrations
(c0 > 0.05) the stability time is in the interval ts ∈ (70 IS, 100 IS). Therefore the system requires a longer time to achieve a stable state. However there are some “stability points” for which the system achieves a stable state markedly faster. These can be found on Fig. 6; these points are: c0 = 0.074, ts = 76 IS; c0 = 0.136, ts = 73 IS; c0 = 0.284, ts = 61 IS; c0 = 0.5, ts = 1 IS; c0 =
0.826, ts = 58 IS; c0 = 0.952, ts = 67 IS.
Comparing the results obtained in the case td = 2 IS and td = 3 IS it can be noticed that the stability times is signiﬁcantly extended and new features become visible (damped oscillations). Therefore we can conclude that the system is very sensitive to the ﬂow of information and that increasing td to td + 1 IS changes the behaviour of the system quite signiﬁcantly.
0.6

concentration

0.5
0.4
0.3
0.2
0.1
0
0

50

100

150

200

time c0 = 0.002 c0 = 0.01

c0 = 0.02 c0 = 0.03

Fig. 4. Coevolution of the system for given initial concentrations; delay time td =
3 IS

230

J. Mi´kiewicz and M. Ausloos s Fig. 5. Coevolution of the system as a function of initial concentration. The coevolution of a system is represented by a vertical series of dots; delay time td = 3 IS

250

200

time

150

100

50

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

initial concentration
Fig. 6. The time required for the system to achieve stable concentrations as a function of its initial concentration; delay time td = 3 IS

Inﬂuence of Information Flow in the Formation of Economic Cycles

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td = 4 IS
For a time delay td = 4 IS, the features seen above (damped oscillations) are also present as it can be observed on both ﬁgures showing the coevolution for the considered initial concentrations and for chosen histories presenting explicitly the time evolution of the system – Fig. 5 and Fig. 8 respectively.
It is worth noticing that the damping of the oscillations is much weaker than in the case td = 3 IS (compare Figs. 5 and 8). The oscillation amplitude is decreasing signiﬁcantly more slowly for the case td = 4 IS than for td = 3 IS.
However in all considered cases td = 2 IS, td = 3 IS, td = 4 IS, the system has one stable solution, but the stability time depends on the delay time; it is the longest in the case td = 4 IS (3200 IS ≤ ts ≤ 4200 IS). The time required for the system to achieve a stable state is presented in Fig. 9. As in the previous case td = 3 IS there are initial concentrations for which the system reaches the stable state signiﬁcantly quicker, e.g. c0 = 0.23.
0.6

concentration

0.5
0.4
0.3
0.2
0.1
0
0

50 c0 = 0.002 c0 = 0.01

100 time 150

200

c0 = 0.02 c0 = 0.03

Fig. 7. Evolution of the system for given initial concentrations; delay time td = 4 IS

3.2 Medium Time Delay td = 5 IS and td = 6 IS
The ﬁve iteration step delay time (td = 5 IS) is very interesting, because this is the shortest time for which cycles of concentration can be observed. For

232

J. Mi´kiewicz and M. Ausloos s Fig. 8. Coevolution of the system as a function of initial concentration. The coevolution of a system is represented by a vertical series of dots; delay time td = 4 IS

4400
4200

time

4000
3800
3600
3400
3200
3000
0

0.2

0.4

0.6

0.8

1

initial concentration
Fig. 9. The time required for the system to achieve stable concentrations; delay time td = 4 IS

Inﬂuence of Information Flow in the Formation of Economic Cycles

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this time delay the system has several solutions. Despite the fact that the real part of the Lyapunov exponent is still equal to zero, its imaginary part is not.
According to (4) this shows that the system has a periodic solution. These solutions can be seen in Fig. 11 and Fig. 10. In the case of Fig. 11 the evolution is shown as a function of its initial concentration, whereas Fig. 10 presents the system evolution as a function of time for chosen initial concentrations.

1

concentration

0.8

0.6

0.4

0.2

0
0
c0 = 0.002

50

100 time c0 = 0.02

150

200 c0 = 0.03

Fig. 10. Evolution of the system for chosen initial concentrations; delay time td =
5 IS

Oscillating solutions can also be found in the case td = 6 IS; the imaginary part of the Lyapunov exponent, as in the previous case (td = 5 IS) is negative
(Fig. 12).
3.3 Long Time Delay td ≥ 7 IS
Extending the delay time above six iteration steps leads to a possible collapse of the system. For td ≥ 7 IS the system may crash. The crash is deﬁned when the concentration of companies becomes negative or zero. Examples of such evolutions which lead to a crash are presented in the cases td = 12 IS and td = 15 IS in Fig. 14 and Fig. 16 respectively. The crash of the system is presented in such plots as a white band containing very few points in the vertical direction. It is also seen in Fig. 17, where for several intervals on

234

J. Mi´kiewicz and M. Ausloos s Fig. 11. Coevolution of the system as a function of initial concentration. The coevolution of a system is represented by a vertical series of dots; delay time td =
5 IS

0.6

Re

Lyapunov exponenet

0.5
0.4
0.3
0.2
0.1
0
-0.1
0

0.2

0.4
0.6
initial concentration

Fig. 12. Lyapunov exponent for td = 6 IS

0.8

1

Inﬂuence of Information Flow in the Formation of Economic Cycles

235

the initial concentration axis, e.g. td ∈ (0.15; 0.2) ∪ (0.34; 0.36) ∪ (0.53; 0.61) the crash of the system occurs very quickly. However there are some initial concentrations for which the evolution of the system before crash time is quite long (up to 400 IS). Additionally in the case td = 15 IS, the system may evolve toward a stable state, with a full occupation of the environment by companies. Examples of such an evolution as a function of time for given initial concentrations are presented in Fig. 13 and Fig. 15 for the cases td =
12 IS and td = 15 IS respectively.

1

concentration

0.8
0.6
0.4
0.2
0
0

50 c0 = 0.002 c0 = 0.01

100 time 150

200

c0 = 0.02 c0 = 0.03

Fig. 13. Coevolution of the system as a function of time for chosen initial concentrations; td = 12 IS

4 Conclusions
Economic cycle causes and occurrences are fascinating and relevant subjects of interest in many economic questions [9, 10]. The problem has also been studied by means of sociology techniques [11], showing that opinions about recession or prosperity undergo drastic changes from one equilibrium to another, both having ﬂuctuations in stochastically resonant systems. In the present investigation, an information ﬂow, typical of economic systems, has been incorporated into the ACP model [3–5]. This has led to observe diﬀerent forms of so-called cycles, through concentration oscillations. In the case of a short delay time td ∈ (2 IS, 4 IS), between data acquisition and policy

236

J. Mi´kiewicz and M. Ausloos s Fig. 14. Coevolution of the system as a function of its initial concentration. The coevolution of a system is represented by a vertical series of dots; td = 12 IS

1

concentration

0.8
0.6
0.4
0.2
0
0

50

100

150

200

time c0 = 0.002 c0 = 0.01

c0 = 0.02 c0 = 0.03

Fig. 15. Coevolution of the system as a function of time for chosen initial concentrations; td = 15 IS

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Fig. 16. Coevolution of the system as a function of its initial concentration. The coevolution of a system is represented by a vertical series of dots; td = 15 IS

1000
900
800
700

time

600
500
400
300
200
100
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

initial concentration
Fig. 17. The crash time of the system as a function of its initial concentration; td = 15 IS

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J. Mi´kiewicz and M. Ausloos s implementation by a company, the system evolves toward a unique stable equilibrium state. This situation can be highly welcomed in some economic systems. Indeed this indicates that, through an information control, a system can insure the existence of a high number of companies, whence not threatening the system of a collapse.
In the case of medium size delay times td = 5 IS or td = 6 IS, the system undergoes oscillations: stable concentration cycles appear in the system. This form of evolution is often observed in economy, e.g. agricultural markets, where without external control the level of agricultural production oscillates between over- and underproduction. Since the enlarging of the delay time leads to the possibility of the system to crash, such a system may require some external (governmental) control, for its stability. In reality, the delay of information ﬂow and policy implementation may also ﬂuctuate. For long information ﬂow delay times, td ≥ 7 IS , the systems may crash for most initial concentrations. However, despite the frequent possibility of the system to crash the situation is not hopeless because the crash time in many cases is long enough to allow for some particular control and to avoid the collapse of the company concentration. It is also possible to observe an “economic resonance” where despite a long delay time the system evolves for a long time or can even reach a stable state, which insures its existence. This latest observation is especially interesting for market control purposes, because it points to the existence of initial conditions for which the system may evolve during a very long time, which is vital for the possibility of creating and applying some control procedures.

5 Acknowledgement
JM stay in Li`ge was partially supported by a STSM grant funded by the e COST P10 Physics of Risk program.

References
1. P.-F. Verhulst: Nouveaux M´moires de l’Acad´mie Royale des Sciences et e e
Belles-Lettres de Bruxelles 18, 1 (1845)
2. D. Huber, L.S. Tsimring: Phys. Rev. Lett. 91, 260601 (2003)
3. M. Ausloos, P. Clippe, A. P¸kalski: Physica A 324, 330 (2003) e 4. M. Ausloos, P. Clippe, A. P¸kalski: Physica A 337, 269 (2004) e 5. M. Ausloos, P. Clippe, A. P¸kalski: Physica A 332, 394 (2004) e 6. P. Bak, K. Sneppen: Phys. Rev. Lett. 71, 4083 (1993)
7. J. Mi´kiewicz, M. Ausloos: Physica A 336, 206 (2004) s 8. M. Ausloos, P. Clippe, J. Mi´kiewicz, A. P¸kalski: Physica A 344, 1 (2004) s e
9. M. Kalecki: Econometrica 3, 327 (1935)
10. M. Aoki: Macroeconomic Dynamics 4, 487 (2000)
11. M. Hohnisch, S. Pittnauer, S. Solomon, D. Stauﬀer: Physica A 345, 646 (2005)

Logistic Function in Large Financial Crashes
G. Rotundo
Department of Business, Technological and Quantitative Studies, Faculty of
Economics, University of Tuscia, Viterbo, Italy 01100
Giulia.Rotundo@uniroma1.it

Summary. This chapter examines the signatures that arise after large ﬁnancial crashes in order to evidence the presence of logistic growth in indices of markets that validates the hypothesis of symmetry on the trend before and after the crash. It is also shown how the probability meaning of the logistic function can be exploited in order to set up a bayesian analysis model. The large crash in the NASDAQ 100 composite index which occurred in April 2000 is presented as a case study.

1 Introduction
The aim of this chapter is to show how the logistic function can occur in ﬁnancial markets and to explore its relevance when dealing with large crashes produced by the collapse of speculative bubbles due to endogenous causes.
The theory about crashes has already assessed the validity of a scaling property in the proximity of large ﬁnancial crashes ( [5,7–9,11,19–22,24–26]). The studies examine also the behaviour of ﬁnancial data after the occurrence of a large drawdown. The analysis of signatures in ﬁnancial indices that undergo some large crash is important in order to estimate the time necessary for the market to exit from the descent that follows the fall and thus for driving decisions about investment policies. Empirical evidences show that the symmetry between pre and post crash behaviours can be established only in the presence of log-periodic corrections to scaling [18]. After the occurrence of a crash a market experiences an oscillatory deﬂating. The time of the exit of the market from the anti bubble and its return to the fundamental values can be obtained from the symmetry with respect to the rise of the bubble.
However the mirroring is not perfect and the analysis of deviations from symmetry can be useful also for the estimate of the expected deﬂating behaviour of prices.
A logistic sigmoidal law [1] can be detected by suppressing the oscillations through the patterns that arise after the occurrence of a crash. This is based on symmetry hypotheses. The case of the large crash during April 2000 of the NASDAQ index is deeply studied here.
The outline of the chapter is the following: the next section resumes the main results of the theory of large ﬁnancial crashes. Section 3 examines the logistic function for a description of post crash behaviour. In Sect. 4 the large

240

G. Rotundo
5000

4500

4000

NASDAQ

3500

3000

2500

2000

1500

1000

500
1997

1998

1999

2000

2001

2002

2003

2004

trading days

Fig. 1. The NASDAQ 100 composite index from January 1st, 1997 to June 5th,
2003

crash which occurred during April 2000 in the NASDAQ 100 composite index is examined. In the last section it is shown how the detection of the logistic function allows to set up a bayesian analysis model for crashes.

2 Large Financial Crashes Models
A series of papers about speculative bubbles proposes an explanation of large ﬁnancial crashes due to causes endogenous to the market. Similarities with critical phenomena like earthquakes [4] and the sound emission in materials close to the rupture point led the research to the detection of cooperative underlying phenomena evidenced through discrete scale invariance in ﬁnancial data close to large crash [15] that can thus be well described as critical points [7]. Market indices have been widely studied through the literature.
Let F be a market index and τ the interval of time to the crash. Then a general form for an evolution equation for F close to the crash time is obtained by the Landau expansion of F around τ = 0: dF (τ )
= αF (τ ) + β | F (τ ) |2 F (τ ) · · · d log τ

(1)

where in general the coeﬃcients may be complex [10]. In this expansion only terms that are invariant with respect to a change of phase φ → φ + C where

Logistic Function in Large Financial Crashes

241

F ≡ Beiφ are kept because a phase translation corresponds to a change of time units.
Equation (1) is independent from the sign of τ and can be useful for both the description of the ascent of a bubble and the relaxation dynamics of after crashes market prices. Thus before the crash τ = tc − t (t < tc ), where t is the time and tc is the time of the crash, and (1) describes the bullish behaviour of a bubble; after the crash τ = t − tc (t > tc ) and (1) characterizes the bearish phase of an anti bubble.
The research of foreshock and aftershock patterns through the same equation relies on the hypothesis that self-aggregation phenomena of the market are suitable for its description through hierarchical structures that drive the system to the crash. A further taxonomy that distinguishes the causes that generate crashes can be found in [13] and conﬁrms the detection of logperiodic power law signatures as a hallmark of speculative bubbles due to causes endogenous to the market. An important remark is that the crash time tc is the expected time for the end of the bubble. It corresponds to the most probable time for the occurrence of a crash [15], but it could either occur before, albeit with small probability, or do not happen at all, leading the bubble to deﬂate smoothly. If the amplitude of the crash is proportional to the bubble part of the crash then F (t) actually is the value of the index at time t. However if the amplitude is proportional to the total price then the best quantity to model is the logarithm of the index value [12, 27].
2.1 First Order Solution
The maintenance of only the ﬁrst term in the r.h.s. of (1) gives the following evolution equation
F (t)

A + B(tc − t)m + C(tc − t)m cos[ω ln(tc − t) − φ] where t < tc . (2)

tc is the crash time and A, B, C, m , ω, φ, tc are parameters to be estimated via numerical optimization, for example by a least square method.
1 (2)
In the limit m → 0 and under the hypothesis ω ln(1 − t/tc ) becomes [15]
F (t)

A + B ln(tc − t) + C ln(tc − t) cos[ω ln(tc − t) − φ] where t < tc . (3)

Setting C = 0, (3) becomes
F (t)

A + B ln(tc − t) for t < tc .

(4)

The symmetry hypothesis leads to the following aftershock behaviour
F (t)

A + B(t − tc )m + C(t − tc )m cos[ω ln(t − tc ) − φ] where t > tc . (5)

Hence in the limit m → 0 and under the hypothesis ω ln(1 − t/tc ) becomes 1, (5)

242

F (t)

G. Rotundo

A + B ln(t − tc ) + C ln(t − tc ) cos[ω ln(t − tc ) − φ)] where t > tc (6)

leading to the following approximation
F (t)

A + B ln(t − tc ) where t > tc

(7)

if C = 0. Empirical studies available through the literature state that the values of m and ω belong to a narrow range of values [13]. Such a behaviour is a robust feature of the bubbles that collapse into a large crash and can be obtained by models of cooperative behaviour among investors [15]. As an alternative to ω the values of λ = e2π/ω have been studied. The value of λ describes the diﬀerence equation of successive either maxima or minima in the oscillations and thus it is of immediate use. Available results about the range of λ are consistent with the ones about ω [25].
2.2 Second Order Solution
The second order solution of (1) is obtained by keeping the two ﬁrst terms in the r.h.s.:
F (t)

A+

τα τ 1 + ( ∆t )2α

B + C cos ω log τ +

∆ω
2α

1+

τ
∆t

2α

+φ

(8) where τ =| t − tc |. This equation doesn’t change form for the description of pre and post crash data.

3 The Logistic Function
The logistic function, also known as sigmoidal function because of its shape, σ : R → R , σ(x) =

1
, x∈R
1 + e−a(x)

was born for the description of population dynamics [2], but is now widely used in probability theory, statistics and neural networks. The aim of this section is to show how it can occur through functions that describe large ﬁnancial crashes.
3.1 The First Order Solution
The ﬁrst order solution can lead to the logistic function for the description of after crashes behaviour. In fact from (7): tc t

1
.
1 + e[F (t)−A]/B

(9)

Logistic Function in Large Financial Crashes

243

Thus the after crash behaviour can be described through the logistic function σ(·) tc σ[F (t)] ,
(10)
t where a(x) = −ax + b, with a = −1/B and b = A/B + log tc . Therefore when using the ﬁrst order solution the sigmoidal function is suitable for the description of data after the crash, but not before it.
3.2 The Second Order Solution
When setting C = 0 (8) can be approximated by
A +B

F (t)
F (t) − A
B

2

1
∆t

τα τ 1 + ( ∆t )2α

(11)

2α

σ(log τ ) where a(x) = 2α(log τ ) − 2α log ∆t .

(12)
This equation does not change form for the description of pre and post crashes data τ =| t − tc |. In this case the logistic function relies over the logarithm of the time and describe the rescaled quantity of interest.
3.3 Probabilistic Approach
The logistic function is well known in probability and especially in bayesian analysis. The probabilistic meaning of the logistic function allows to provide an interpretation of the r.h.s. of (10) as a probability estimate for the occurrence of the crash at time tc , that can be calculated relying on the level of F (t). Moreover, the approximation states that this probability decays like the inverse of the time to crash, under an appropriate rescaling of the time.
This allows to remark the analogy with Omori formula [23] for a law for the rate Q(t) of the shocks following a main earthquake, that decays from the main shock as Q(t) ∝ 1/tp where p is close to 1.
The sigmoidal function occurring in (10) has the interesting feature to be well suited for the description both before and after a crash. Moreover it depends on the logarithm of time, thus keeping trace of the log-periodic structure embedded in the market.

4 Numerical Results
The crash of the index NASDAQ 100 composite which occurred in April 2000 has been deeply examined. The data set collects index values of each trading day since January 1st, 1997 to September 26th, 2002.

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As also pointed out in Sect. 2 a rational bubble model proposed in [14] shows that if the magnitude of the crash is proportional to the increase of values exclusively produced by the rise of the bubble [12] then the right quantity to study is the data time series, but if the depth of the crash is proportional to the values, then it is necessary to study the logarithm of data. As evidenced in [13], in the case of NASDAQ data, the logarithm of the index values must be chosen and thus the data {y(t)}t used here below are the logarithm transformation of the NASDAQ 100 composite index.
The time is expressed as a fraction of the year, according to the series of papers about crashes. Diﬀerent unit measures for the time lead to diﬀerent numerical values of parameters. One year contains approximately 260 trading days, depending on the overlap between holidays and weekends, but it is normalized to 100 days. As an example 2000.18 is March 10th, 2000.
A minimum least square method was used in order to perform the ﬁt of functions (2–11) to data. Constraints are mandatory over the scope of the logarithm function (tc > t before the crash and t > tc after the crash).
Moreover lower and upper bounds for each parameter were ﬁxed across all the examples by relying on the literature for the ﬁt of data before the crash [11],
[13]. In (2) parameters A, B, C, φ represent the rescaling of measure units, whilst parameters m and ω contain structural information about the market dynamics because they quantify the acceleration in the price increase and the oscillations, respectively [13]. In the ﬁt of (2, 3, 5, 6, 8) the optimal value of ω is searched by taking into account the narrow range across the series of speculative bubbles due to endogenous causes. The same remark holds for the exponent m. In fact the values of m and ω were found to belong to a short range through several analyses of speculative bubbles observed on a wide variety of markets. As an example in [13] for more that thirty crashes on the major ﬁnancial indices ω ≈ 6.36 ± 1.56 and m ≈ 0.33 ± 0.18. The bounds on λ ∈ [2.3, 2.5] are consistent with the ones on ω [25]. Thus the bounds used in the optimization procedures related to the ﬁt of (2) are ω > 4.5 and ω < 8.5.
Another bound over the time is obtained from the empirical evidence that the estimate of the most probable crash time is in a range of approximately at most three months from the actual date of the beginning of the index drop [16]. The end of the bubble is deﬁned as the date at which the market reached its maximum. The bounds for each ﬁt after the crash are obtained from the ones stated before the crash by using the symmetry hypothesis.
The optimal set of parameters is very sensitive to the choice of the initial point. Thus optimization procedures were performed by choosing a grid of
100 initial points for each parameter, equally spaced in the interval between the related bounds. For each ﬁt the value of the parameters that gave rise to the lowest value of the residuals without active constraints is reported in
Tables 1 and 2.

Logistic Function in Large Financial Crashes

245

Table 1. Analysis of NASDAQ 100 composite precursory patterns of the crash of
April 2000. Data set contains daily index values from January 1st, 1997 to March
10th, 2000
Formula

Parameters

Figure

(13)

A = 9.36, B = −1.60, tc = April 11th, 2000, m = 0.26.

(2)

(15)

A = 9.36, B = −1.60, tc = April 11th, 2000, m = 0.26,
C = 0.05, ω = 6.32, φ = −0.83.

(3)

(4)

A = 7.91, B = −0.54, tc = July 4th, 2000.

(4)

(3)

A = 7.93, B = −0.60, tc = August 10th, 2000, φ = 2.96, ω = 6.26, C = −0.07.

(5)

(18)

1
A = −18.60, B = 34.65, tc = June 21st, 2000, ∆t = 144.01, α = −0.034.

(10)

(19)

1
A = −18.60, B = 34.65, tc = June 21st, ∆t = 144.01, α = −0.03,
C = 0.08, ω = 7.32, ∆ω = 1.02, φ = 0.10.

(11)

Table 2. Analysis of NASDAQ 100 after crash signatures. Crash of April 2000. Data set contains daily index values from September 13th, 2000 to September 26th, 2002
Formula

Parameters

Figure

(16)

tc = August 8th, 2000 A = 10.57, B = −3.00, m = 0.11.

(6)

(17)

tc = August 8th, 2000, A = 10.57, B = −3.00, m = 0.11,
C = 0.097, ω = 9.713, φ = −1.82.

(7)

(7)

A = 7.62, B = −0.40, tc = July, 20th, 2000.

(8)

(6)

A = 7.58, B = −0.33, tc = July 31st, 2000, φ = −2.47, ω = 8.38, C = 0.10.

(9)

(18)

A = 9.17, B = 21.85, tc = July 8th, 2000,

(19)

A = 9.17, B = 21.85, tc = July 8th, 2000,
C = 0.13, ω = 6.86, ∆ω = 4.62 , φ = 3.99.

1
∆t
1
∆t

=179.58, α = −0.03.

(12)

=179.58, α = −0.03,

(13)

The next paragraphs describe the methods and the results both for the ﬁt to ﬁrst and second order solutions, each of them before and after the crash.
The data set used in order to study the ascending speculative bubble ending to the crash uses index data available for the trading days since January 1st,
1997 till March 10th, 2000, for a total amount of 833 points [11]. The ﬁrst point can be identiﬁed as the lowest values of the index prior to the onset of the bubble while the last point is that of the all time high of the index.
Moreover this data set gives rise to the best estimate of the crash time when using (4) [16]. For the post crash study data range since September 13th, 2000 till September 26th, 2002. Data between March 10th, 2000 and September
13th, 2000 are not considered in order to allow the market to overcome its

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most unstable phase and to start a descent with the same slope as that of the last month before the crash. The results are collected in Tables 1, 2.
4.1 The First Order Fit
Data {y(t)}t were at ﬁrst ﬁtted by y(t) A + B(tc − t)m

(13)

by the least square estimate, whence the residuals y(t) − A − B(tc − t)m
(tc − t)m

(14)

should be a pure cosine function y(t) − A
− B = C cos[ω ln(tc − t) − φ]
(tc − t)m

(15)

and were ﬁtted again by least square methods. This procedure splits the optimization of (2) into two phases and allows avoid numerical problems due to the number of variables.
Before the Crash
These ﬁts are necessary in order to get the values of the parameter in order to compare them with the after crashes ones. The results of the ﬁt are collected in Table 1. The ﬁrst function ﬁtted is (13). The lower bound for (A, B, tc , m) is (5, −10, 2000.19, 0), the upper is (10, −0.1, 2001.21, 10), and the optimal values are consistent with the ones reported in [13] (see Fig. 2). The ﬁt of
(13) is necessary in order to proceed with the ﬁt of (15). The bounds for
(C, ω, φ) are (−10, 4, −1) and (10, 8, 1), so they are not tight. The results obtained are consistent with the one reported in [11] (see Fig. 3). The next function to ﬁt is (4) and the bounds for (A, B, tc ) are (7, −1, 2000.19) and
(9, −0.1, 2000.52). The procedure used for the best ﬁt of (2) evidences its limits when ﬁtting (3). In this case it gave rise to poor ﬁtting and it was necessary to perform a simultaneous optimization over all the parameters.
The bounds on (C, ω, φ) given by (−10, 1, −1) and (10, 10, 0).
After the Crash
The data set contains NASDAQ data since September 13th, 2000 to September 26th 2002. The best results with no active constraints are collected in
Table (2). Formula (5) was tested starting with the best ﬁt of y(t) A + B(t − tc )m .

(16)

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247

5500

5000

4500

NASDAQ

4000

3500

3000

2500

2000

1500

1000
1997

1997.5

1998

1998.5
1999
trading days

1999.5

2000

2000.5

Fig. 2. NASDAQ 100 before the crash. The continuous curve is the best ﬁrst order ﬁt according to (13)
5500

5000

4500

NASDAQ

4000

3500

3000

2500

2000

1500

1000
1997

1997.5

1998

1998.5
1999
trading days

1999.5

2000

2000.5

Fig. 3. NASDAQ index before the crash. The continuous curve is the best ﬁrst order ﬁt according to (15)

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5500

5000

4500

NASDAQ

4000

3500

3000

2500

2000

1500

1000
1997

1997.5

1998

1998.5
1999
trading days

1999.5

2000

2000.5

Fig. 4. NASDAQ index before the crash. The continuous curve is the best ﬁrst order ﬁt according to (4)
5500

5000

4500

NASDAQ

4000

3500

3000

2500

2000

1500

1000
1997

1997.5

1998

1998.5
1999
trading days

1999.5

2000

2000.5

Fig. 5. NASDAQ index before the crash. The continuous curve is the best ﬁrst order ﬁt according to (3)

Logistic Function in Large Financial Crashes

249

The bounds for (A, B, tc , m) are (9, −3, 2000.06, 0) and (20, −1, 2000.70, 1).
The bounds are chosen relying over the results got for data that precede the crash they are not tied with respect to these values, in order to allow a wide search for the minimum value. A remark can be made about the estimate of tc : for (16) the estimate of the crash time is approximately 20 days after the last day used for the ﬁt and in (16) tc is approximately 20 days before the initial day. This gives a conﬁrmation about the symmetry of the process.
4000

3500

NASDAQ

3000

2500

2000

1500

1000
2000.5

2001

2001.5

2002

2002.5

2003

trading days

Fig. 6. NASDAQ index after the crash. The continuous curve is the best ﬁrst order ﬁt according to (16)

Formula (5) was ﬁtted by using the values of A, B, tc , and m already got with the ﬁt of (16) and by using the least squares method over the residuals y(t) − A
− B = C cos[ω ln(t − tc ) − φ]
(t − tc )m

(17)

for the estimate of C, ω, φ. The bounds for (C, ω, φ) are (−1, 3, −20),
(1, 10, 20).
There is a wide basin of attraction also for (C, ω, φ) = (0.06, 4.04, −4.94), but no convergence has been evidenced to values of ω around 2π. The bounds for (17) are again not strict (−1, 3, −2) ≤ (C, ω, φ) ≤ (1, 10, 0) and the ﬁt of the residuals leads to the estimate of (5).
The estimate of (6) was made over all the variables simultaneously, as done for (3)

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4500

4000

NASDAQ

3500

3000

2500

2000

1500

1000
2000.5

2001

2001.5

2002

2002.5

2003

trading days

Fig. 7. NASDAQ index after the crash. The continuous curve is the best ﬁrst order ﬁt according to (17)

4.2 The Second Order Fit
The ﬁt of the (8) has also been performed in two steps. At ﬁrst the parameters
A , B , tc , ∆t were estimated by the following approximation: y(t) A +B

τα
.
τ
1 + ( ∆t )2α

(18)

Then searching for the minimum of the residuals has been performed by using y(t) − A
−B
√ τ ατ 2α
1+( ∆t )

C cos ω log τ +

∆ω τ 1+
2α
∆t

2α

+φ

(19)

Before the Crash
The best ﬁt of (18) was made with lower and upper bounds for (A, B, tc ,
1/∆t , α) given by (−100, 0, 2000.19, −10, −20) and (100, 100, 2000.70, 1000, 0), respectively. The results are given in Table 1.
After the Crash
The bounds are the same ones used for the best ﬁt reported in the previous paragraph. Logistic Function in Large Financial Crashes

251

4500

4000

NASDAQ

3500

3000

2500

2000

1500

1000
2000.5

2001

2001.5

2002

2002.5

2003

trading days

Fig. 8. NASDAQ index after the crash. The continuous curve is the best ﬁrst order ﬁt according to (7)

4.3 The Logistic Approach
As shown in Sects. 3 and 4 from the approximations to ﬁrst and second order solutions it is possible to get the logistic function. The values of the parameters of the logistic can be calculated immediately from the ﬁts in
Tables 1 and 2. Some important remarks must be made. The ﬁrst one is about the fact that this kind of aftershock behaviour relies over the structure of the market that is detected through the log-periodic oscillations. Thus it is necessary to perform the ﬁt of the formula that contains oscillations, too, in order to test for the presence of the periodic part. Moreover the description of the after crash behaviour is interesting in order to give an estimate of the deﬂate of the bubble whilst it is still at its beginning. An empirical strategy that can be held by investors can be the one of performing the data ﬁt as soon as possible. However a raw application of this procedure can lead to huge instabilities and high volatility over the parameter set because the overall function is not stationary.
An important remark is that the residuals on the left hand side of each
(15) after the crash, and (17) before and after the crash are stationary logperiodic processes with zero mean. Also (7) that gives rise to (10) can be ﬁtted from (6) taking into account the residuals

252

G. Rotundo
5000

4500

4000

NASDAQ

3500

3000

2500

2000

1500

1000
2000.5

2001

2001.5

2002

2002.5

2003

trading days

Fig. 9. NASDAQ index after the crash. The continuous curve is the best ﬁrst order ﬁt according to (6)
5500

5000

4500

NASDAQ

4000

3500

3000

2500

2000

1500

1000
1997

1997.5

1998

1998.5
1999
trading days

1999.5

2000

2000.5

Fig. 10. NASDAQ index before the crash. The continuous curve is the best second order ﬁt according to (18)

Logistic Function in Large Financial Crashes

253

5500

5000

4500

NASDAQ

4000

3500

3000

2500

2000

1500

1000
1997

1997.5

1998

1998.5
1999
trading days

1999.5

2000

2000.5

Fig. 11. NASDAQ index before the crash. The continuous curve is the best second order ﬁt according to (19)
4000

3500

NASDAQ

3000

2500

2000

1500

1000
2000.5

2001

2001.5

2002

2002.5

2003

trading days

Fig. 12. NASDAQ index after the crash. The continuous curve is the best ﬁrst order ﬁt according to (18)

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G. Rotundo
4500

4000

NASDAQ

3500

3000

2500

2000

1500

1000
2000.5

2001

2001.5

2002

2002.5

2003

trading days

Fig. 13. NASDAQ index after the crash. The continuous curve is the best ﬁrst order ﬁt according to (8)

F (t) − A
+B
ln(t − tc )

C cos[ω ln(t − tc ) − φ] where t > tc

(20)

that exhibits the usual log-periodic structure.
This remark provides a guideline for the estimate of the parameters of the left hand side that allows to improve the stability of their estimate. For trading purposes the optimization task can be performed by adding subsets of data, as soon as they are available and thus depending on the time. In this dynamic setting it is necessary to take into account the accelerating oscillations cos[ω ln(tc − t) − φ] which lead to local periods converging to zero according to a geometrical progression with scale factor λ ≈ e2π/ω .
Therefore a dynamic ﬁtting procedure that can ensure the best stability over the parameters is a mixed online and batch one where data points are added only when a sequence as long as one period at least is available. The length of sequences changes because the period changes as the logarithm of the time. When more than one frequency is detected over the data before the crash then the best choice is to use the theoretical results and to try the theoretical value ω 2π for the after crash data.
Because of the fact that the function is periodic with the logarithm of time then the length of the sequences to be added is not ﬁxed, but goes as the logarithm of the time. Thus when working with respect to the variable

Logistic Function in Large Financial Crashes

255

−4

11

x 10

10

NASDAQ

9

8

7

6

5
900

1000

1100

1200 trading days

1300

1400

1500

Fig. 14. NASDAQ index after the crash, rescaled through (10). The continuous curve is a detail of the best ﬁrst order ﬁt according to (10). Please refer to
Table 2, line corresponding to the (7) for the values of parameters

“time” it is not possible to work over vectors of ﬁxed size, but it is necessary to change their length.
The point symmetric to January 1st, 1997 with respect to the initial crash time is June 5th, 2003. At that time the market has stopped its descent and it is going to start a longer rise, but at that point it is still in the bounds of log-periodic oscillations.

5 Bayesian Analysis
Large ﬁnancial crashes can be classiﬁed on the causes that generate them.
Only endogenous crashes exhibit scaling. It is interesting to understand how strong is the mirroring between the progressively accelerating power law up to the crash and the subsequent approximately symmetrical power law decay, that is lower for exogenous peaks, in order to detect the end of the deﬂating period of the anti-bubble and the restart of the market. The parameters from the ﬁt of either the ﬁrst or the second order solutions or both of them can be used in order to write the proper form of the logistic functions. Thus it is possible to get an immediate estimate of them. An interesting question is to

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understand how the parameters of the logistic function that describe the pre crash growth have to change in order to describe the after crash decay. In order to perform the addressed tasks it is possible to rely over some bayesian analysis results that can be useful for the ﬁt of the parameters of the logistic
¯
function [3]. Let tc be the crash time estimated over data before the crash.
¯
Deﬁne a binary variable Iδ (tc ). Iδ (tc ) = 1 if |tc − tc | < δ, and 0 otherwise. It is a measure of the stability of the estimate of tc . The estimate of the crash is correct when Iδ (tc ) = 1. The value of δ that describes the range of the estimate of tc can be chosen from numerical results over the data before the crash. The estimate of parameters depends on data y t = {yt }t . It is possible to ﬁx an a priori probability distribution over the parameters that occur in the logistic function in a = a(y t ). Usually the prior distribution over them is given as a gaussian one, centered on the expected values of parameters. In this case it is possible to use as expected values of the parameters the ones obtained from the best ﬁts before the crash, thus relying on the symmetry hypothesis. The variance can be ﬁxed and gives the spread of the numerical instabilities. With the usual notations of conditional probability it is possible to deﬁne
1
.
(21)
p(Iδ (t) = 1|y, a(y)) =
1 + exp−a(y)
This probability assignment is consistent with the detection of the logistic law (10). The estimate of the probability to get the crash in the given interval can be performed when more data are available and thus considering y t+T instead of y t . As remarked in the previous paragraph the interval time T must be equal to a period in order to get some stability in the numerical estimates. The best ﬁt of the parameters of the sigmoidal function proceeds via the usual least square method with the observation of successive data y t+T . After each observation there is an a posteriori distribution p(Iδ (t) = 1|y t+T ) =

P (Iδ (t) = 1|a(y t+T ))P (a(y t+T )|y t+T ) da(y t+T )
(22)

where
P (a(y t+T )|y t+T ) =

P (a(y t+T )|y t+T ) da .

And the joint a posteriori distribution is given by
N

p(a(y t+T )|y t+T ) =

1
P (a(y t+T ))
P (Iδ (t) = 1 | a(y t )) ,
Z
n=1

where Z is a normalizing constant. Theoretical upper and lower bounds are available [3,6] in order to approximate (22), that provides a framework for the description of the adjustment of the probability when new data are available.

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257

6 Conclusions
This chapter shows the occurrence of the logistic function in the description of large ﬁnancial crashes due to speculative bubbles. It can be accomplished for both the ﬁrst and second order approximation to the solution of the Landau equation. In the case of ﬁrst order expansion its probabilistic meaning is also on line with Omori law. The study of after crashes signatures is important in order to estimate the time necessary to the market to absorb the shock and for taking decisions about after crashes investment policies. Whereas the speculative bubbles exhibit symmetry, the logistic function can be used also for the description of after crash data and for the deﬂating of bubbles. A further observation is that the use of a logistic function instead of the traditional solutions to the Landau equation allows the use of bayesian learning results that give an estimate over the ﬁt of the function to data. The large crash of the index NASDAQ 100 composite that has occurred during April
2000 is presented as a case study. The null hypothesis is that post crash perfectly symmetric to pre crash, thus giving a description of the deﬂating of the bubble. As soon as time passes and new data arrives it is possible to detect whether the symmetry is respected, thus conﬁrming the estimated end of the bubble, or allowing a diﬀerent classiﬁcation of the post crash behaviour.
Further examples can be found in [12]. The Argentina stock market bubble of 1992 is well ﬁtted by (4) and the logarithm of data if its anti-bubble is well ﬁtted by (7). The same happens for the Venezuela bubble ending in 1997 and its related anti-bubble [12]. In these cases however the descent is more rapid, even if the log-periodic structure is maintained and allows to keep the diﬀerence with the case of crashes where the values return quickly to their fundamentals looking close to the behaviour of a damped harmonic oscillator [25,26]. As an example in [20] the S&P500 dumps like a single dissipative harmonic oscillator and thus the symmetry after crash is not maintained and it is interesting to study the fastness of the rejection of the after crash symmetry. Moreover it is also interesting to give some bounds [17] in order to understand if the decay is violating the mirroring, thus getting a diﬀerent classiﬁcation of the crash and consequently a diﬀerent estimate for the end of the anti-bubble. A further development of this study could be to investigate a taxonomy among crashes in order to detect the minimal time which is necessary after the occurrence of the crash in order to give a stable estimate for the post crash logistic function and thus to determine some further information about the type of crash towards the forecast of the end of the bearish period and for the start of a new phase of the market.

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Agent Based Approaches to Income
Distributions and the Impact of Memory
Przemyslaw Repetowicz, Peter Richmond,
Stefan Hutzler, and Eimear Ni Dhuinn
Department of Physics, Trinity College Dublin 2, repetowp@tcd.ie, richmond@tcd.ie, shutzler@tcd.ie, nidhuine@tcd.ie
Summary. Agent based models have been used to study the dynamics of wealth or income distributions in populations. In this chapter we develop a generic theory of interacting agents where the property of memory may be included. The model may be analysed both analytically and numerically via computer simulation. Particular regimes are analysed and compared with empirical data for both the UK and the
Republic of Ireland. We also demonstrate that with the inclusion of memory the predicted wealth distribution can exhibit Pareto tails with values of the exponent that are greater than unity unlike the model without memory where the exponent is always unity.

1 Introduction
Reverend Thomas Malthus [3] is generally recognized as the person who made the ﬁrst serious attempt to understand and quantify economic growth. However his predictions of everlasting exponential growth were quickly recognized as being incorrect. The issue was resolved by Verhulst [1] in a series of seminal papers [2]. A statistical as opposed to deterministic approach to economic modelling was ﬁrst made by Italian economist Pareto [4] who examined the distribution of wealth in Italian society. He found that the high end (tail) of the wealth distribution was described by power law, f1 (w) ∼ w−1−α . Pareto noted at the time that “The problem of pure economics bears a striking likeness to that of rational mechanics... It therefore appears quite legitimate to appeal also to mathematics for assistance in the solution of the economic problem.” Recently it has been shown via numerous empirical studies [5–8] that this power-law tail is temporarily stable over a wide range of wealth values. Studies of income data in Sweden in [5], Revenue data concerned with the net capital of individuals at death in the United States [6], income distributions in Japan in [8], the UK [18] and Australia [19]) all support Pareto’s initial conclusions. In addition, distributions of sizes, cities, Internet ﬁles, biological taxa, gene families and protein family frequencies in [7]) also suggest that
Pareto’s conclusions are of a general nature and suggest that this high end

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tail that exhibits the power law is characterized by several multiples or even tens of multiples of the average wealth. Note that in this paper we use words income and wealth interchangeably. This corresponds to the starting assumption that these economic systems are in equilibrium ((quasi-)stationary) and the plausible assertion here that income of an individual is roughly equivalent to wealth in his or her possession in the long term i.e. at death.
Models based on stochastic processes with multiplicative noise (with the updated value of process depending on a product of the present value and a random variable) that yield a power-law distribution of the value of the process have been known for years from studies as diverse as studies of economic growth [12], city populations [13], wealth distributions [14], stochastic renewal processes [15] and of frequency of words in text [11]. Recently Di
Matteo [19] published a model based on additive stochastic processes with individuals (agents) interacting through a network. In this model the distribution that is a weighed sum of Gaussians with averages proportional to the connectivity of the network, equal variances, and weights depending on the degree distribution of the network appears to be a power law in the high end and a log-normal distribution in the low end. Other models [20] simulate in a simplistic way the eﬀect of tax. However all these models relate essentially to a Generalized Lotka–Volterra equation that combines a multiplicative stochastic process with a process that redistributes a fraction of the total money to ensure the money possessed by an agent is never zero. The resulting distribution f1 (v) has the form: e(1−α)/v (1) v 1+α where α − 1 is a positive number that is, it is suggested related to social security and some random investments.
A diﬀerent class of models [21–24] are based by analogy with the kinetic theory of dilute gases (so called wealth exchange models). In these models, agents exchange money via pairwise interactions and the rate of exchange depends on the number of agents. The models yield a power-law tail as a result of the breaking of the conservation of wealth in exchange processes
(inelastic scattering).
We have examined in detail elsewhere the basic model due to Patriarca,
Chakraborti and Kaski (PCK) [9] and shown that aspects of the model [10] can be solved analytically. The model yields a form for the wealth distribution that appears reasonable for low values of wealth. At the high end, the model admits a power law but the value of α for the model is precisely unity whereas it is known that for empirical data that α may be 1.5 or higher. Here we propose a generalized model (RRH model) that includes the possibility of agents with memory. The model is developed in section (2). We then proceed to examine various limits. The generalization clearly encompasses the PCK model as a special case therefore and continues to ﬁt reasonably well the low end of the wealth distribution. This is now demonstrated in section (5) f1 (v) ∼

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using data from both the UK and the Republic of Ireland. In the following sections, we examine the possibility that multi point and opposed to pairwise interactions might allow a change in the tail index, α. The result is negative.
We ﬁnally show that when memory is included, the power law describing the higher end tail can take on values that diﬀer from unity and that are more in agreement with empirical data. This result is demonstrated both analytically and supported via numerical simulation.

2 Kinetics of Wealth Distributions
Elsewhere [16] we developed the solution of the PCK model where agents have no memory. Here we consider a more generalized model that includes the possibility of agents with a memory. As with the simple PCK model the time evolution of the distribution of wealth is described through dynamical rules that involve exchanges between two agents (two-agent exchanges) in two consecutive times t and t + δt, viz.:
ˆ
ˆ
ˆ
wi (t + δt) = λi wi (t) + [(1 − λi )wi (t) + (1 − λj )wj (t)] wj (t + δt) = λj wj (t) +
ˆ

1

[(1 − λi )wi (t) + (1 − λj )wj (t)]
ˆ
ˆ

(2)

where λi and λj describe the amount of money saved during the exchange process by agents i and j respectively (saving propensities). The random number ∈ [0, 1] is uniformly distributed with
= 1/2 and 1 = 1 − . The
ˆ
variables wi (t) and wj (t) are computed from values of wealth at time t and
ˆ
at some random time q ≤ t that is sampled from a distribution pt (q), that we call the memory horizon distribution, prior to the start of the dynamical process. wi (t) = wi (t) + γwi (q)
ˆ
wj (t) = wj (t) + γwj (q)
ˆ

(3)
(4)

Thus our process is not Markovian and is measured by some decay rate of the distribution pt (q) (to be speciﬁed in next paragraph). Notice that with this deﬁnition of memory, the agents tend to accumulate money in the long term with a rate that is speciﬁed by the parameter γ (the money accumulation parameter). Since the model involves two-agent exchange processes at two times it is convenient to describe it in terms of a four-agent distribution function f4 (v, w, v , w ; t, q). This function speciﬁes the probability density of an event in which two randomly chosen agents have wealth values Vt = v and Wt = w at time t and values Vq = v and Wq = w at time q < t. This means that f4 (v, w, v , w ; t, q) := P (Vt = v, Wt = w, Vq = v , Wq = w ). The twoagent function f2 (v, w; t) is obtained by integrating the four-agent function over values v , w ≥ 0 at time q. The kinetic equation for the two-agent

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distribution function is constructed by counting all states of agents (v , w ) at time t and (v , w ) at time q, states that give rise to a predeﬁned state (v, w) at time t + δt. We assume that the waiting times between wealth exchanges are distributed exponentially with some rate r. Therefore conditioning on the occurrence of the exchange process in time interval [t, t + δt] for small δt yields: f2 (v, w; t + δt) = (1 − r δt)f2 (v, w; t)
+(r δt)
4
R+

dv dw dv dw f4 (v , w , v , w ; t, q)

,
×P (Vt+δt = v, Wt+δt = w |Vt = v , Wt = w , Vq = v , Wq = w ) (5) where the integration in (5) runs from zero to inﬁnity, the conditional probability in the integrand in (5) is expressed via a delta function (no dissipation of wealth assumption) as follows: δ (v − λ(ˆ)ˆ − {[1 − λ(ˆ)]ˆ + [1 − λ(w)]w}) v v v v
ˆ ˆ
× δ (w − λ(w)w − 1 {[1 − λ(ˆ)]ˆ + [1 − λ(w)]w}) ,
ˆ ˆ v v
ˆ ˆ

(6)

where v = v + γv
ˆ

and w = w + γw
ˆ

(7)

and the saving propensities λ(v) may depend on the values of wealth v. In order to change (5) into an algebraic equation we make use of a generalized
Laplace transform:
(λ,γ)
˜
L(λ,γ) : f (v1 , v2 ) → f (λ,γ) (x) := Lv1 ,v2 [f (v1 , v2 )] (x)
∞

:=

∞

dv1
0

dv2 e−(x1 +λ(v1 +γv2 )x2 )(v1 +γv2 ) f (v1 , v2 )dv

(8)

0

where x = (x1 , x2 ). Note that (8) depends on the functional form λ(ˆ). For v ˜ λ(ˆ) = λ = const(ˆ) the generalized transform f (λ,γ) (x) = Lv1 ,v2 [f (v1 , v2 )] v v v v
(x1 +λx2 , γ(x1 +λx2 )) and in general (λ(ˆ) = const(ˆ)), as we will see in next paragraph, the generalized transform is expressed via partial derivatives of the Laplace transform itself. Multiplying both sides of (5) by exp(−xv − yw) and integrating over v , v ∈ [0, ∞] and w , w ∈ [0, ∞] we get:
˜
˜ f2 (x, y; t + δt) − f2 (x, y; t)
˜
+ f2 (x, y) r δt
˜(λ,γ) ( x + 1 y
= f4
1 (x − y)

x + 1y
− (x − y)

; t, q)

(9)

˜(λ,γ) are the Laplace and generalized Laplace transforms of
˜
where f2 and f4 the two-agent and the four-agent functions respectively. Note that the ﬁrst and the second vector on the right hand side of (9) corresponds to values

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of wealth of the ﬁrst agent and of the second agent at times t and time q respectively. In order to solve (9) we need to express the generalized Laplace transform through the Laplace transform. We assume that values of wealth of diﬀerent agents at the same time are independent whereas values of wealth of the same agent at times t and q are not. This is a plausible assumption since in real societies densities of agents are small but agents possess a memory horizon even though its distribution decays fast with the time lag. This means that the four point function in (9) factorizes as follows:
˜
˜
˜(λ,γ) (x1 , x2 ; t, q) = f (λ,γ) (x1 ; t, q)f (λ,γ) (x2 ; t, q) f4 2
2

(10)

We assume the saving propensity λ(v) to be a periodic function λ(v) = λ(v + 2L) of the wealth v. This means that λ(v) can be expanded in a Fourier series: ∞ nπ λ(v) = λ0 +
(11)
λ(n) sin( v)
L
n=1
Inserting (11) into the deﬁnition of the generalized Laplace transform (8), factorizing the exponential in the integrand into two parts that depend on x1 and on x2 only and expanding the exponential with x2 in a Taylor series we obtain:
˜
˜(λ,γ) (x) = L(λ,γ) [f2 (v1 , v2 )] (x) = 1 + Ax ,s(x) f2 (s(x), γs(x)) f2 v1 ,v2
2

(12)

where s(x) = x1 + x2 λ0 and
∞

Ax1 ,x2

(−x1 )n
:=
n! n=1 ∞

λ m=1 (m)

π
∂−x2 sin( m∂−x2 )
L

n

(13)

is a pseudo-diﬀerential operator that is deﬁned by means of a Taylor expansion in partial derivatives of higher order. Inserting (12) into the kinetic equation (9) we obtain in the limit δt → 0 the following pseudo-diﬀerential equation: ˜ r−1 ∂t + 1 f2 (x, y; t) =
˜
1 + A (x−y),P (x,y) f2 (P1 (x, y), γP1 (x, y); t, q)
1

× 1 + A−

1

(x−y),P2 (x,y)

˜ f2 (P2 (x, y), γP2 (x, y); t, q)

(14)

1 λ1 y, (λ0

(15)

where
P(x, y) = ((λ0 + λ1 )x +

+

1 λ1 )y

+ λ1 x)

we deﬁned a vector whose components P1 (x, y) and P2 (x, y) are invariant with respect to interchanging ( , x) with ( 1 , y).
A further analysis of (14) is left for future work.
In the following we now assume λ(v) = const(v) = λ and consider the following particular cases:

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– Agents that have no memory (γ = 0) and the saving propensity λ is the same ﬁxed number for all agents
– An extension of the model to include three-agent exchange processes.
– Agents have a memory (γ > 0) and the savings propensity for agents may be diﬀerent and characterized by a distribution ρΛ (λ). Note that in this case agents can accumulate wealth during an exchange process.

3 Lack of Memory and Equal Savings
In the absence of memory i.e. γ = 0, wealth is conserved during the exchange processes and the two-agent distribution function reaches a stationary solution at large times. The model has been solved numerically by Chakrabarti
[17], who found that to a good approximation the distribution of wealth could be ﬁtted well by a heuristic function: f1 (v) =

nn n−1 v exp (−nv)
Γ (n)

(16)

where Γ (n) is the gamma function and the parameter n is related to the saving propensity, λ as follows: n (λ) = 1 +

3λ
1−λ

(17)

This relation (17), in eﬀect a conjecture by Patriarca, Chakraborti and Kaski
(PCK) [10], can in fact be derived analytically. Since the PCK conjecture is concerned with the one-agent distribution we may assume that its explanation can be achieved using the mean ﬁeld approximation
˜
˜
˜
f2 (x, y; t) = f1 (x; t)f1 (y; t)

(18)

using this approximation, (30) takes the form:
˜
˜
˜
f1 (x) = f1 ((λ + λ1 )x)f1 (( λ1 )x)

1
=
λ1 x

λ1 x

˜
˜
f1 (λx + φ)f1 (φ) dφ

(19)

0

where the average in the ﬁrst equality in (19) is over = Uniform (0,1) and
˜
in the second equality we substituted for λ1 x. In this case a solution f1 (x) =
∞
(−1)n mn xn that is analytic in x exists. The moments v n = mn · n! n=0 satisfy recursion relations:
(1−λ)

˜ (p) mq mp−q Cq (λ)

mp = q=0 q

(λ + η) η p−q dη

p

where

˜ (p)
Cq (λ)

=

0

1−λ

(20)

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265

˜ (p) where the constants Cq are computed from following recursion relations:
˜ (p)
(1 − λ)p−q−1 − (q + 1)Cq
˜ (p)
Cq+1 = p−q with

(1 − λ)p
˜ (p)
C0 = p+1 (21)

Solving the equations of moments (20) with initial conditions m0 = 1 and m1 = 1 recursively, i.e. expressing, via the pth equation, mp as a function of λ and all previous values of m one obtains: λ+2 λ+2 m3 =
2(1 + 2λ)
2(1 + 2λ)2
72 + 12λ − 2λ2 + 9λ3 − λ5 m4 =
24(1 + 2λ)2 (3 + 6λ − λ2 + 2λ3 )

m2 =

(22)
(23)

The ﬁrst three moments m1 , m2 and m3 coincide with the moments mconj ,
1
mconj and mconj of PCKs function (16) if the relation between the parameters
2
3 n and λ is given by (17). Indeed the coeﬃcients mconj (n) of a series expansion i of the Laplace transform
˜
f1 (x) =

n x+n ∞

n

(−1)j mconj (n)xj j =

(24)

j=0

of the function (16) agree with moments (23) up to the third order, subject to (17) being satisﬁed. This is shown in a nice way in Fig. 1. The deviation
˜
∆f1 (x) between the exact solution of (20) and the Ansatz (16) has a leading fourth order:
˜
∆f1 (x) =

(n − 1)(n + 1)(n + 8) x4 + O x5
8n3 (10n3 + 30n2 + 45n − 4)

(25)

It is hard to say whether a more general class of functions than (16) would fulﬁll (20) to higher expansion orders.

4 Three-Agent Exchange Processes
Elsewhere [16] we have shown that the two-agent exchange process in the mean ﬁeld approximation does not provide a full explanation of the model.
Obviously many-agent distribution functions fm (x1 , . . . , xm ) may not be produced correctly within this approach but more importantly the value of α for the high end wealth distribution where the savings are chosen according to a random distribution is always unity. Might this change if we include m-point interactions: ⎡
⎤
wi (t + δt) = λi wi (t) +

⎣ i m

j=1

(1 − λj )wj (t)⎦

(26)

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∆(λ;n)
0.25

Num. n = 2
Num. n = 3
Num. n = 4
Num. n = 5
Anal. n = 2
Anal. n = 3
Anal. n = 4
Anal. n = 5

0.2
0.15
0.1
0.05

0.2

1/4

0.4

0.6

2/5 3/6 4/7

0.8

1

λ

Fig. 1. Deviations ∆(λ; n) = 10 (mp (λ) − mconj (n))/p! of the exact moments p p=0 mp of the wealth distribution from the moments mconj derived from the conjecture p plotted as a function of λ for n = 2, 3 . . . , 9. Solid lines (dot symbols) correspond to analytical (numerical) solutions of the moment (20). We see that the minima λ = (n − 1)/(n + 2) = {1/4, 2/5, 3/6, 4/7, 5/8, 6/9, 7/10, 8/11} of the deviations do correspond to the PCK conjecture (17) m where i ∈ [0, 1] and i=1 i = 1. This means that at every time step, exchange processes involving any number of agents can happen with a certain likelihood. We perform the analysis for m = 3 in order to establish the type of mathematical diﬃculties we might encounter. Now the kinetic equation for the two-agent distribution function in the Laplace domain is expressed through three-agent distribution functions and takes the form:
1 ˜
˜
∂t f2 (x, y; t) + f2 (x, y; t) r ˜
= (1 − σ) f2 (λx + λ1 [ x + (1 − )y] , λy + λ1 [ x + (1 − )y] ; t) σ ˜ f3 (λx + λ1 ( 1 x + 2 y) , λy + λ1 ( 1 x + 2 y) , λ1 ( 1 x +
2
σ ˜
+
f3 (λ1 ( 2 x + 3 y) , λx + λ1 ( 2 x + 3 y) , λy + λ1 ( 2 x +
2

+

2 y) ; t)
3 y) ; t)

(27)

where λ + λ1 = 1, + 1 ≤ 1 and σ and (1 − σ) denote likelihoods of threeagent and two-agent exchange processes respectively. The ﬁrst (second and third) term(s) on the right-hand side in (27) account(s) for two-(three-)agent exchange processes respectively. Setting y = 0 and σ = 0 we obtain the kinetic equation of two-agent exchanges (19) Setting x = y = 0 we obtain an identity because f2 (0, 0) = f3 (0, 0, 0) = 1.
Now the transcendental equation derived from the kinetic equation (27) has the following form (compare with (41)):

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1 = (1 − σ) ( (λ + λ1 )α + ( λ1 )α ) σ + ( (λ + 1 λ1 )α + (λ + 2 λ1 )α + 2 ( 1 λ1 )α + 2 ( 2 λ1 )α ) (28)
2
where , 1 , 2 = Uniform(0,1) and 0 < 1 + 2 < 1 are sampled independently from . Note that since
= 1/2 and 1 + 2 = 2/3 both sides of (28) coincide for α = 1. For α > 1 the right hand side is a strictly decreasing function of α because it is a sum of powers functions aα where a < 1.
Thus α = 1 is once again the only solution of this equation for arbitrary saving propensity distributions ρΛ (λ) and for any likelihood σ ∈ [0, 1] of three-agent exchange processes. This result is conﬁrmed by numerical simulations.

5 Comparison of the Model to Empirical Data
We have analysed distributions of yearly incomes in the United Kingdom within 1993–2002 and in the Republic of Ireland within 2001–2004 and ﬁtted the Patriarca, Chakraborti and Kaski model with a non-random saving propensity to the data (see Fig. 2). The ﬁtted values of saving propensities for both countries are plotted versus time in Fig. 3. Since the number of yearsdata available is small it is hard to see any trend in the saving propensities as a function of time. However, the accuracy of the ﬁt, in terms of the error bars, is reasonable. Most of the data that we analysed were characterized by an exponential decay of the distribution of wealth in the high end of the distributions. Hallmarks of power laws were vaguely visible in few data sets and set in only starting from a couple of multiples of the average income.
This may be due to the fact that high-income data are underestimated due to the fact that rich people (tax-evaders) are not willing to disclose their full earnings to the revenue commissioner.

6 Presence of Memory and Random Savings
For λ(v) = const(v) the pseudo-diﬀerential operators on the right hand side in (14) are equal to zero. Therefore (14) takes the form:
˜
r−1 ∂t + 1 f2 (x, y; t)
˜
˜
= f2 (P1 (x, y), γP1 (x, y); t, q)f2 (P2 (x, y), γP2 (x, y); t, q)

(29)

˜
˜
˜
Setting y = 0 in (29) and using f2 (x, 0) = f2 (0, x) = f1 (x) we obtain an equation for the one-agent function
˜
r−1 ∂t + 1 f1 (x; t)
˜
˜
= f2 (P1 (x, 0), γP1 (x, 0); t, q)f2 (P2 (x, 0), γP2 (x, 0); t, q)

(30)

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1.2
income distribution UK 1995

distribution of income (area normalised)

1

0.8

0.6

0.4

0.2

0
0

0.5

1

1.5

2

2.5

3

3.5

4

income in multiples of average income

Fig. 2. A ﬁt of the PCK function (16) to the distribution of yearly incomes in £ in the United Kingdom in 1995 for a saving propensity λ = 0.54 ± 0.05 overlayed with numerical simulations of the two-agent wealth exchange lack of memory model (2) for the same value of λ
United Kingdom
Republic of Ireland
0.65

savings parameter lambda

0.6

0.55

0.5

0.45

0.4

0.35

0.3

1992

1994

1996

1998

2000

2002

2004

year

Fig. 3. Yearly saving propensities λ for the United Kingdom and for the Republic of Ireland derived from the PCK model. The error bars are derived from the least square deviations ﬁt errors

Agent Based Approaches to Income Distributions

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We diﬀerentiate (30) with respect to x at x = 0 and obtain the time dependence of the mean wealth
˜
v (t) := − ∂x1 f2 (x1 , 0; t)

x=0

˜
= − ∂x2 f2 (0, x2 ; t)

(31) x=0 We have: r−1 ∂t + 1 v (t) = (λ + 2 λ1 ) ( v (t) + γ v (q)) = v (t) + γ v (q) (32) where in the second equality in (32) we have averaged over using
= 1/2.
Now we assume that the memory horizon distribution pt (q) = p(t − q) is homogenous in time. This allows us to solve (32) as follows: v (t) = L−1 s v (0)
(t)
s − γrp(s)
˜

(33)

where p(s) := Lt [p(t)] (s) is the Laplace transform of v (t) with respect to
˜
time t. We assume for simplicity that the memory horizon decays exponentially:
(34)
p(t) = θe−θt with a mean 1/θ (memory horizon), and that v (0) = 1. This yields s2 + θ s1 + θ exp(s1 t) − exp(−|s2 |t) s1 − s2 s1 − s2

v (t) =

(35)

where

s1,2
.
(36) θ Since the wealth is not conserved we assume that the two-point function has a quasi-stationary solution of the form: rγ = s1,2 1 +

˜ f2 (x, y; t, q) = Φ2 (xˆt,q , yˆt,q ) , v v

(37)

where vt,q := v (t) − C v (q) and C is a constant that will be determined
ˆ
later. Inserting (37) into (30) and replacing x by x/(ˆt,q ) we obtain: v r−1 s1 x∂x + 1 Φ2 (x, 0) = Φ2 ((λ + λ1 )x, γ(λ + λ1 )x)Φ2 ( λ1 x, γ λ1 x) ,
(38)
where
.

s1 :=

v t,q
ˆ
vt,q
ˆ

t

= s1 q ∞

Cn

= s1 1 + n=1 0

θe−θτ dτ
1 − Ce−s1 τ

1 − e−(θ+ns1 )t
(1 + n sθ1 )

∞

= s1 1 +

t→∞

Cn
(1 + n sθ1 ) n=1 (39)

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For small values of x and y we expand the function
Φ2 (x, y) = 1 − x − y + A(xα + y α ) in a series, insert it into (38), and compare coeﬃcients at powers of x. This yields: x: 1 + r−1 s1 = 1 + γ xα : 1 + αr−1 s1 = ( (λ + λ1 )α + ( λ1 )α ) (1 + γ α )

(40)
(41)

where the average on the right hand side in (41) is over the distributions of λ and of . We insert (39) into (40) and we realize that (40) is equivalent to
(36) if C = rγ/θ and C ≪ 1.
From (40) we see that (41) is fulﬁlled for α = 1 and we seek for a diﬀerent solution α ∈ [1, 2]. Since for γ > 1 the derivative of the right hand side of
(41) with respect to α changes sign from minus to plus for some α ≥ 1 there will be another solution for α = 1 of (41). The analysis of these solutions and their comparison to the numerical calculations is left for future work.
We emphasize that this analysis is valid in following limits:
(a) The money accumulation parameter per wealth exchange rγ is much smaller than the inverse memory horizon θ.
(b) The money accumulation parameter γ > 1.
(b) The time in the simulation is large enough t → ∞.
We have also performed numerical simulations with the rate of wealth exchanges r = 1, and with γ > θ. Results of these simulations, shown in
Fig. 4, clearly show that the distribution has a slope diﬀerent from unity in the high end and the corresponding value of α can take on values that conform with empirical values.

7 Conclusions
We have presented a generalised model of agent interactions that incorporates agent memory and shown that both analytic and numerical solutions for the subsequent wealth distributions may be obtained. The model includes the approach of PCK as a special case. Using data from both the Republic of Ireland and the UK we show that the model describes reasonably well the low end of the wealth distribution. Furthermore contrary to the PCK model our generalised model including agent memory can account for the observed values of the Pareto exponent α that characterises the high end wealth distribution. A more detailed analysis of these results and other aspects will be given elsewhere.

Agent Based Approaches to Income Distributions

271

1

cumulative distribution of money P(m)

0.1

0.01

0.001

0.0001 power law, exponent -1 memory, uniform distribution of savings parameter fit to power law, exponent -1.29 memory, Gaussian distribution of savings parameter
1e-05
0.01

0.1

1

10

100

money m in multiples of average money

Fig. 4. Numerical calculations of the distribution of wealth with presence of memory and with random saving propensities. Here the rate of exchanges of wealth r = 1, the money accumulation parameter γ = 1 and the inverse memory horizon θ = − log(0.63). It is readily seen that the slope of this distribution |α| becomes larger than unity once memory is present

Acknowledgments
The authors are grateful to Science Foundation Ireland for ﬁnancial support via Basic Award 04/BR/P0251). E. Ni Dhuinn in particular acknowledges support via UREKA Supplement 04/BR/PO251UR04. Participation “Verhulst 200” at the Royal Military Academy in Brussels, BELGIUM was aided by COST Action P10 “Physics of Risk”. Irish income data was provided by the Irish Revenue Commissioners http://www.revenue.ie. We are grateful to
Geoﬀ Willis for giving us the income data from the United Kingdom data.

References
1. A. Quetelet, Pierre-Fran¸ois Verhulst, Annuaire de l’Acad´mie royale des scic e ences de Belgique 16, 97 (1850)
2. P.-F. Verhulst: A note on the law of population growth, 1835. In: D. Smith,
N. Keyﬁtz, Mathematical Demography. Selected Papers (Springer Verlag, 1977);
P.-F. Verhulst: Mem. Acad. R. Belg. 18, 1 (1845)

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3. T. Malthus: An Essay on the Principle of Population, as it Affects the Future Improvement of Society with Remarks on the Speculations of Mr. Godwin, M. Condorcet, and Other Writers, LONDON,
PRINTED FOR J. JOHNSON, IN ST. PAUL’S CHURCH-YARD, 1798,
WWW:http://www.ac.wwu.edu/∼stephan/malthus/malthus.0.html
4. V. Pareto: Cours d’Economie Politique (F Rouge, Lausanne 1897)
5. M. Levy, S. Solomon: Physica A 242, 90 (1997)
6. A. Dragulescu, V.M. Yakovenko: Physica A 299, n 1-2, 213 (2001); A. Dragulescu, V.M. Yakovenko: Eur. Phys. J. B 20, 585 (2001)
7. W.J. Reed, B.D. Hughes: Phys. Rev. E 66, 067103 (2002)
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9. A. Chatterjee, B.K. Chakrabarti, S.S. Manna: Phys. Scripta T 106, 36 (2003); preprint cond-mat/0311227; A. Chatterjee, B.K. Chakrabarti, S.S. Manna:
Physica A 335, 155 (2004)
10. M. Patriarca, A. Chakraborti, K. Kaski: Phys. Rev. E 70, 016104 (2004)
11. June-Yule Lee, Signal processing of chaotic impacting series, IEE Colloquium on Signals Systems and Chaos, p 7/1-6 (1997)
12. R. Gibrat: Les In´galit´s ´conomiques (Librairie du Recueil Sirey, Paris 1931) e e e
13. G.K. Zipf: Human Behavior and the Principle of Least Eﬀort (Addison-Wesley,
1949)
14. Y. Ijiri, H. Simon: Skew Distributions and the Sizes of Business Firms (NorthHolland, New York 1977)
15. H. Kesten: Random Diﬀerence Equations and Renewal Theory for Products of
Random Matrices, Acta Mathematica, CXXXI: 207-248, 1973
16. P. Repetowicz, S. Hutzler, P. Richmond: preprint cond-mat/0407770, submitted to Physica A
17. A. Chatterjee, B.K. Chakrabarti: Physica Scripta T 106, 36 (2003) and condmat/0311227; A. Chatterjee, B.K. Chakrabarti, S.S. Manna: Physica A 335,
155 (2004)
18. T. Cranshaw: poster presentation at EPS meeting Applications of Physics in
Financial Analysis (APFA) 3, London December (2003)
19. T. Di Matteo, T. Aste, S.T. Hyde: preprint cond-mat/0310544
20. S. Solomon et al: Phys. Rev. E 66031102 (2002); S. Solomon, P. Richmond:
Eur. Phys. J. B 27, 257 (2002); P. Richmond, S. Solomon: Int. J. Mod. Phys.
C 12, 333 (2001)
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23. E. Ben-Naim, P.L. Krapivsky: Phys. Rev. E 61, R5 (2000)
24. P.L. Krapivsky, E. Ben-Naim: J. Phys. A: Math. Gen 35, L147 (2002)

Agglomeration/Aggregation and Chaotic
Behaviour in d–Dimensional Spatio–Temporal
Matter Rearrangements Number–Theoretic
Aspects
Adam Gadomski1 and Marcel Ausloos2
1

2

U.T.A. Bydgoszcz, Institute of Maths & Physics, Bydgoszcz PL–85796, Poland agad@atr.bydgoszcz.pl University of Li`ge, SUPRATECS, Li`ge B–4000, Euroland e e marcel.ausloos@ulg.ac.be Summary. Matter gets organized at several levels of structural rearrangements.
At a mesoscopic level one can distinguish between two types of rearrangements, conforming to diﬀerent close-packing or densiﬁcation conditions, appearing during diﬀerent evolution stages. The cluster formations appear to be temperature- and space-dimension dependent. They suﬀer a type of Verhulst-like saturation (frustration) when one couples the growing (instability) and mechanical stress relaxation modes together. They manifest a chaotic behaviour both in space and time domains. We pretend to oﬀer a comprehensive and realistic picture of a material or mega-cluster formation in d dimension.

1 Introduction
Matter organisations at a mesoscopic (molecular–cluster) level typically manifest a multitude of microstructural rearrangements. Cluster–cluster aggregations of proteins and/or colloids, phase separations, ﬂocculation–coagulation phase transformations, sol-gel systems, (wet) sand or rice piles, etc., are manifestations of loosely-packed rearrangements, typically occurring under moderate or high temperature conditions. In contrast ripened polycrystals, sintered powders, soap froths and bubbles, and other cellular systems, constitute a type of rearrangement that usually emerges in a (relatively) low temperature limit and under certain (“ﬁeld dependent”) matter close–packing constraints.
Beside such an agglomeration, fracture, desaggregation, desorption, dissolution, and alike, can be thought to be the “inverse process”, ﬁnding its place in the opposite part of the relevant phase diagram [1].
In all of them spatial as well as temporal signatures of chaotic behaviour, due to matter reorganisations, can be detected: They are temperature and space-dimension dependent. In particular, one can show rigorously that in the limit of the spatial dimension going to inﬁnity loosely-packed agglomerations become non–chaotic by suppressing totally their instability growing mode since it is related to the nonequilibrium agglomerate’s entropy [2], while their closed-packed counterparts are not. When the growing mode is coupled to a

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mechanical stress relaxation mode as a power law via some phenomenological relation of Hall–Petch–Griﬃth (H-P-G) [3] type (an Onsager-type conjecture
[4] of the present study), certain marks of Bethe–lattice frustration, related to a spatial overcrowding of the Cayley–tree branches, appear in the (meanﬁeld) approach – a kind of frustration qualitatively of very similar type than that observed in Verhulst-type systems in an adequate time and parametric zone. In the speciﬁc cases discussed in this review, however, by increasing the space dimension, d, we automatically induce some increase of the possible number of degrees of freedom in the system. Thus, when taking into account the coupling of the late-time growing and relaxation modes, say, in a fairly synchronized viz. power-law way of H-P-G type, one obtains that even though the material’s relaxation goes slower than in the case when such a coupling is proposed in an unsynchronized (Debye-relaxation involving, i.e. rapid) way, one is, however, able to establish or restore an apparent dynamic microstructural order within the system the nonequilibrium (chaotic) measures of which are proposed below. There is, unfortunately, no way of establishing such an order when the coupling fails the power-law type synchronization requirement [3, 4].
Thus, when an ample space amongst the clusters is recovered by the system at its mature growing stage, we consider that the system successfully tries to avoid a chaotic matter organisation in space. Note that temperature may markedly help in surmounting the activation-energy barrier of the agglomeration, especially when it is raised appropriately, whence when not
”damaging’ a possibly smooth evolution of the system. Full success is, however, guaranteed when the limit of d → ∞ is reached. If there is no chance for recovering the ample space, the late-time growing stage is realized in a moderately chaotic way. The mechanical stress relaxation, in turn, enters a readily chaotic regime, since the (nonequilibrium) entropy of the system diverges to plus inﬁnity. The overall scenario resembles, in general, a formation of large (fractal) colloid aggregates that typically occurs with and without temperature and/or space-dimension dependent gravity factor domination, like as if imposing some limits to gelation of colloids [5].
The paper is arranged as follows. In Sect. 2, we deﬁne both the closelypacked as well as loosely-packed agglomerations, calling the latter the aggregation throughout. In Sect. 3, we list some qualitative signatures of chaos in matter-agglomerating systems, and refer brieﬂy to diﬀerent deﬁnitions as well as meanings of chaos. In Sect. 4, we present quantitative measures of chaos signatures in systems of interest, whereas in Sect. 5 we unveil numbertheoretic measures, featuring a chaotic spatio-temporal behaviour of them.
In Sect. 6, according to some suggestions given in [6], on which much of our report is based, in order to see which agglomerations behave orderly or nonchaotically, we explore the limit of d → ∞, and arrive at a certain interesting
(perhaps, surprising) conclusion, favouring aggregation of matter, or some

Agglomeration/Aggregation and Chaotic Behaviour

277

structural loosely-packed, and typically high-temperature, matter rearrangements – in contrast to those emerging under close-packing low-temperature conditions. We close the paper by oﬀering a concluding address in Sect. 7.

2 Agglomeration vs Aggregation of Matter – a Model
Description
Following [7], throughout the present study, we wish to distinguish between the notions of agglomeration and aggregation of matter. By the former we mean an assembly of grains or molecular clusters, kept together by relatively strong forces (e.g. ionic), so that there is no easy possibility of taking the clusters apart, or destroying them. For the latter, because of the appearance of weak bonds between clusters, such as Van der Waals or hydrogen types, the possibility of cluster separation becomes an observed tendency of the matter rearrangement due to their weak bonds. For a schematic explanation of the diﬀerence between both matter arrangements, see Fig. 1.
3

3
1

1

2

2

t1

t1

3

3
2

2

t2

t2

A: Vtotal=const

B: Vtotal≠const

Fig. 1. Typical cluster-merging (three-grain) scenario for closely-packed (left, denoted by A) and loosely-packed (right, denoted by B) agglomerations. Two consecutive time steps t1 and t2 are shown. The former usually goes by a scenario with the preservation of the total agglomerate’s volume (though in a more irregular way, when its logarithmic speed is measured, cf. Sects. 3–6), whereas the latter does not [8]. In the former, the clusters do not perform a translational motion but their boundaries may ﬂuctuate in time and space, even though they are quite strongly conﬁned by their neighbourhood. In the latter, an almost opposite situation in the time-and-space domain is typically observed. Some void is left behind a loosely packed system

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2.1 Basic System of Equations Describing
Model Matter Agglomeration
As in previous work [8, 9] we begin with a local continuity equation
∂
∂ f (v, t) +
J(v, t) = 0 ,
∂t
∂v

(1)

preferentially supplemented by the corresponding initial (of delta-Dirac type as a ﬁrst attempt) and boundary (typically, of absorbing type) conditions
(IBCs).
In a few subsequent studies a thermodynamic-kinetic description [8–10] of model complex matter agglomeration has been worked out. For the current3 in the space of cluster volumes
J(v, t) = − B(v)

∂
∂
Φ f (v, t) − D(v) f (v, t) ,
∂v
∂v

(2)

has been used [9], where f (v, t) is the distribution of clusters of volume v: this means, that f (v, t)dv is the (relative) number of clusters with size in the inﬁnitesimal volume interval [v, v + dv]; t is the time; Φ represents the physical potential, equivalent to the free energy of the system (see [9] for an explanation of the term). It is assumed to be one of the most relevant drivers of the agglomeration process at the mesoscopic level, assuring its nonequilibrium character. In fact, the current (2), in the form presented above, comes from a rigorous derivation, starting from the Gibbs equation for the entropy production [10]. It has quite strong foundations anchored in nonequilibrium thermodynamics [9]. It is worth mentioning that upon inserting (2) into the continuity equation, (1), one gets a second-order partial diﬀerential equation of the Fokker–Planck–Kolmogorov (F-P-K) type [11].
For the dynamics of such a system some routes to chaotic behaviour have been sketched elsewhere by considering the (in)stability of Markov semigroups in [12]. The mobility B(v) is also deﬁned in the conﬁgurational v– space and reads [9]
B(v) =

D α v , kB T

α=

d−1
,
d

(3)

where D is a diﬀusion reference constant. Realize that the principal role of D is to scale the time variable; kB represents the Boltzmann constant.
Note that the mobility B(v) is related to the Onsager coeﬃcient, L(v), that appears in the derivation of the matter ﬂux equation (2) under a set of assumptions, mostly based on the locality of the Kramers-type process [10,
13], etc. L(v), and hence B(v), could be measured by comparing the current and the thermodynamic force [10]. The quantity D(v) = Dv α is to be inferred
3

For a method to derive diﬀusion currents for diﬀerent types of systems one is encouraged to look into [10]

Agglomeration/Aggregation and Chaotic Behaviour

279

from the Green–Kubo (G-K) formula [14], so that there is some quite strong suggestion for deriving B(v) both, experimentally as well as theoretically [9].
There is a debate about a possible violation of the G-K formula [14, 15].
For instance, it is proved that for a gas of charged particles subjected to an external electric ﬁeld, the mean mobility of a charged particle, based on the G-K formula, is reliably well estimated for suitably small values of the external ﬁeld. Moreover, at a microscopic scale one observes a nonlinear (or chaotic) behaviour of the particles, which is, unfortunately, not reﬂected by the macroscopic (mobility) measure. In our case, we assume algebraic correlations in v–space, for a phenomenological formula. The assumption seems to be as natural as possible: D(v) ∝ v α ; that means that both the diﬀusivity D(v) and the mobility B(v) are proportional to the cluster hypersurface, s(d) := Rd−1 . It should be underlined that it is, in our opinion, the common physical case in clustering phenomena, and is working properly at the mesoscopic level considered in our approach.
Notice right here that exactly the same assumption has been used to model in a F-P-K way the formation of surface nanostructure arrays
[16]. Therein, an experimentally-observed passage between direct curvaturedependent ripening of matter nano-islands (our densely-packed agglomeration), and inverse ripening, with an elastic-ﬁeld caused contraction of growing quantum dots [17] (our sparsely-distributed agglomeration of matter), has been presented.
There are, however, matter agglomerations, for a given T , that do not conform usually to
D(v) ∝ B(v) ∼ v α

s(d) .

(4)

To them belong both some physical-metallurgical transformations [18], such as martensitic, and presumably, also certain phase orderings of non-diﬀusive kind, emerging in model biosystems, such as those occurring in lipid biomembranes [19]. Other than algebraic types of correlations in the hyperspace can likely be expected for these. If a power law of the type given by (4) can be kept for further modelling, some additional correlations in time must complete a more comprehensive correlational proposal, cf. [20]. Other types of correlations in the hyperspace, even if they allow to get a general solution to the problem, may not accommodate the boundary conditions [9], so that one would expect either to be left with an unsolved speciﬁc problem or to encounter anomalous or irregular behaviour of the agglomerating system [8, 19, 20]. In such a case another type of ﬁnite, instead of inﬁnite boundary boundary conditions [20], can sometimes give a remedy for the problem [21]. Here, under the term inﬁnite boundary conditions [8] we typically understand the boundary conditions of absorbing (Dirichlet) type f (v = 0, t) = f (v = Vclust , t) = 0

(5)

in which the single cluster volume is taken at inﬁnity, Vclust = ∞, whereas in case of the ﬁnite boundary conditions it assumes a ﬁnite value, 0 < Vclust <

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∞, cf. [21], and a discussion therein. Although the latter unquestionably seems more physical the former is more frequently used to reveal the evolutions in matter-agglomerating systems [22] – this resembles to some extent a situation in statistical-thermodynamical systems undergoing an equilibrium phase transition: As such they are typically considered in the so-called thermodynamic limit (here, with a number of subunits going to ∞) under the mentioned agglomeration-oriented, e.g. condensation conditions, and the analogy would presumably extend over the examined nonequilibrium evolutions too [9, 21].
2.2 Thermodynamic Potentials Driving Matter Agglomerations
In previous work [9] the analytic form of a (so-called) compaction potential was obtained, i.e.
Φ(v) = Φ0 ln (R/R0 ) ,

(6)

where Φ0 , R0 – constants, and R stands for some cluster radius. Because v ≡ v (d) ∼ Rd ,

d = 1, 2, 3, . . . ,

(7)

one gets also Φ(v) ∝ ln (v/v0 ), where v0 is a constant.
The logarithmic potential assures the emergence of rather compact and curvature-involving structures, whence the name of “compaction potential”
[22]. It should be noted that Φ is an entropic potential [9]. Thus, it can be a cause of some desaggregation, or matter-inﬂuenced impingement eﬀects, occurring within the overall aggregation space.
In a previous study on the phase transformation kinetics for loosely packed
“diﬀusive” agglomerates [8] we have written the matter ﬂux of a purely diffusive nature prescribed in conﬁgurational space as follows
J(v, t) = −D(v)

∂ f (v, t) .
∂v

(8)

(The diﬀusion function D(v) = Dv α is proportional to the cluster (grain) surface.) Both closely-packed and loosely-packed agglomerations follow from the general form (2). Indeed, the loosely-packed case is obtained when the ﬁrst
(drift) term in r.h.s. of (2) can be neglected. Formally, B(v) → 0 when
T → ∞. From the physical point of view, it corresponds to suﬃciently high temperatures T ≥ Tpass > 0, where Tpass can be treated as a cross-over temperature4 above which the agglomeration takes place exclusively by yielding loosely-packed microstructures. However, the drift term in r.h.s. of (2) depends both on v and T . It tends uniformly, which means independently of v, to zero at the high temperature limit if
4

Borrowing from the nomenclature of phase transitions and critical phenomena one might sometimes opt for calling it the threshold temperature

Agglomeration/Aggregation and Chaotic Behaviour

B(v)

∂Φ(v)
∂v

C = const .

281

(9)

Then for a given system, temperature Tpass does not depend on v and looks consistently deﬁned.
Some additional argumentation can be provided that such a constant
(limit) C exists and is well deﬁned. Namely, when applying both (3) and (7) one sees with suﬃcient accuracy that
C∝

∆Φ(v)
1
×
.
∆R kB T

(10)

This means that C is essentially determined by a product involving two contributions: a certain curvature-like term, κ = 1/ | ∆R |, and some dimensionless energetic argument, εE = | ∆Φ(v) |/kB T . The above claimed high-temperature limit, with the cross-over temperature Tpass as a reference temperature characteristic of a system of interest, would naturally demand
1 while, because of approaching the mature growing stage any
0 < εE change in the cluster radius must be small, 0 <| ∆R | 1, and therefore, its inverse would tend to some big value, i.e. κ
1. Thus, C will take on a ﬁnite value. It is believed that for certain agglomerations under readily hightemperature conditions it will eventually acquire a small value5 , that means,
0<C
1 naturally holds. It is a case when the potential
Φ(v) ∝ v 1−α

v 1/d .

(11)

In [16] a condition of setting the current equal to zero, J = 0, has been chosen to balance diﬀusive and non-diﬀusive terms in the F-P-K type description, cf. [10], aiming at getting a proper behaviour of the metastable nanostructure [17] arrays. We are of the opinion that such a proposal is legitimate in the relatively low-temperature domain. When the temperature is raised, but agglomeration is still allowed to occur, the proposal may fail.
Thus, the above is a possible solution for the high-temperature limit. A type of localization of the Gaussian distribution, characteristic of the inverse ripening (a metastable state of the nanostructure evolution) can also be obtained within the present modelling, cf. [8]. This is the case of (8) when in a (readily) mature growing stage, since the single volume v of the cluster does not change much. As a matter of fact, there is no small-cluster population available for merging (Fig. 1), i.e. D(v) → const., which nearly corresponds to the high-temperature criteria of (9), or equivalently (10). In so doing, (8)
5

Such a belief comes undoubtedly from the fact that we oﬀer our approach for systems evolving in an overdamped regime, such as those of biopolymeric type. For them the Reynolds number is typically of the order of 10−3 , i.e. very low, so that the mobility per se, even for a single biomolecule but also for a molecular cluster, must clearly be of negligible value [13], regardless of whether we measure it in the v–space or, what is usually done, in a position space

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represents the 1st Fick law in its standard form. Upon inserting it into (1) one immediately arrives at the 2nd Fick law (in the conﬁgurational space) with its standard Gaussian solution, the metastable case being emphasized in [16].
The above potential form (11), designed for loosely-packed agglomeration, seems legitimate here: Note that the ‘force’ Fc−c ∝ ∂Φ(v)/∂v behaves like
Fc−c ∝

1 vα 1 s(d) ,

(12)

because v ∼ Rd . Thus, Fc−c acts as the inverse of the area of the cluster hypersurface, s(d) , which implies that the smaller the area is, the bigger the force acting on the cluster can be, this way impeding the formation of new clusters, which would contribute to an aggregate’s density increase. Qualitatively, a similar dependence is found for the closely-packed matter agglomeration: from (6) one gets, as above, for the ‘force’
Fc−c

1
.
v (d)

(13)

Here, Fc−c acts as the inverse of the hypervolume of the cluster, v (d) := Rd , which makes a clear diﬀerence between closely-packed and loosely-packed agglomerations, presumably leading to a certain relaxation of the surface tension conditions for loosely-packed clusters-containing systems [22].
Referring further to (9) and using the similarity relation, (7), one gets
Φ(R) ∝ Φ(R0 )

R
,
R0

(14)

where
Φ(R0 ) =

kB T
R0 ,
Dα

T ≥ Tpass ,

(15)

and consequently, Φ(R) ∝ R. Moreover,
Dα = D(1 − α) .

(16)

R0 can now be speciﬁed to be the initial cluster radius. Note that Dα is a d–dependent quantity.
2.3 Cluster Volume Fluctuations as Reliable Characteristics of Matter Agglomeration
Aggregations and agglomerations emerge in a ﬂuctuating changing medium.
Therefore, any reasonable quantitative attempt on resolving the ﬂuctuation impact on their speed is worth examining here. In what follows, let us propose an evaluation of the reduced variance

Agglomeration/Aggregation and Chaotic Behaviour

σ 2 (t) =

v 2 (t) − v 1 (t) v 1 (t)

2

2

≡

v 2 (t) v 1 (t)

2

−1,

283

(17)

as a direct measure of the cluster volume ﬂuctuations.
The notation used in (17) refers to the statistical moments
∞

v n (t) =

v n f (v, t) dv

n = 0, 1, 2, . . .

(18)

0

of the stochastic process, where the matter agglomeration is usually described by the local continuity equation, (1).
The explicit solutions, f (v, t)–s, have been presented elsewhere [8, 9, 20], and Refs. therein. The zeroth moment, v 0 (t) , is related to the average number of molecular clusters in the system, and usually shows an algebraic decrease with time [8]. The ﬁrst moment, v 1 (t) , is related to the total volume which is a constant value for closely-packed agglomerations [22] and an increasing function of time for loosely-packed agglomerations [8], cf. Fig. 1.
From the expressions of both moments, it follows that the average cluster radius, Rav (t), behaves as a power law in time, with a growth exponent 1/(d+
1) that apparently contains some signature of random close-packing of matter by having included the super-dimension d + 1 [9,23]. (d + 1 tells us something about the minimum number of non-overlapping neighbours of a given cluster in a d–dimensional space.) These constitute the main characteristics of the model agglomeration/aggregation process in its late-stage (t
1) limit.
The question remains about asymptotic values of the moments < v n (t) > that must be known when applying formula (17). For closely-packed agglomerations, the moments are found to obey a power law [22] v n (t) ∼ t(n−1)/(2−α)

(n = 0, 1, 2),

t

1,

(19)

whereas for matter aggregation one ﬁnds another power law [8] v n (t) ∼ t[(n−1)+α]/(2−α)

(n = 0, 1, 2),

t

1.

(20)

Notice, that for α = 0 (d = 1) both power laws above approach the same form, namely v n (t) ∼ t(n−1)/2 . When utilizing (17) and (19) it appears that for closely-packed agglomerations, σ 2 (t) can be fully identiﬁed with the inverse of v 0 (t) (the average number of clusters), cf. [22] for details, what because of the constancy of v 1 (t) , leads to σ 2 (t) ∝ Vsp (t), where Vsp ≡ v 1 (t) / v 0 (t) , and can be termed the mean speciﬁc volume of
Vsp (t) the tightly-packed agglomerate, being equivalent to the inverse of its mean number density. The speciﬁc volume ﬂuctuations read σ 2 (t) ∝ td/(d+1) , and if d → ∞, σ 2 (t)

Vsp ∝ t.

(21)

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When using (17) and (20), however, it turns out that for loosely-packed agglomerations σ 2 (t) is a quantity equivalent to the average cluster radius
Rav (t), see [8, 9]. They behave in time as σ 2 (t) ∝ t1/(d+1) .

(22)

When d → ∞, σ 2 (t)
Rav (t) → const, which means, that on average the system ceases to grow. Note that the standard diﬀusional regime, is always characterized by the one-half exponent, is achieved exclusively for the aggregations in d = 1 because the only linear characteristic is Rav ≡ Rav (t): Note
3
that Vsp (t) is not a linear characteristic, since Vsp (t) ∝ [Rav (t)] usually holds.
Here the d = 1–case must clearly be disqualiﬁed as standard diﬀusional, cf.
(21).
Commenting on the last relations, (21) and (22), one might furthermore conclude that they reﬂect a well-known Onsager conjecture that the ﬂuctuations in a system undergo the same type of changes as the corresponding macroscopic dynamic variables [4]: Here one may think of the speciﬁc volume of the agglomerate and the grain radius, and their behaviour in the late-time domain, respectively.
2.4 Coupling the Instability (Growing) and Mechanical Stress
Relaxation Modes of Matter Agglomeration
Poisson was likely the ﬁrst who recognized that viscoelastic properties of ﬂuids and solids can reasonably be compared in a suitable, mostly short-time domain, though the speciﬁcation of the domain must be more precise for specifying the systems of interest. Maxwell successfully followed the ideas of his famous French predecessor, arriving at his well-known, in general non–
Markovian, model of relaxation [2]. In what follows we present our Maxwell– model-based ideas on how to distinguish between the two agglomerations under study, and how to switch on a kind of coupling between the (late-stage) growing and relaxational modes in the viscoelastic d–dimensional matrix that we investigate. The existence of the coupling seems to be experimentally justiﬁed, see [5, 15, 16], and involves generically the viscoelastic nature of the mega-cluster late-stage formation [2].
Thus, the afore presented rationale toward quantifying the ﬂuctuations of the system can be strengthened with a supporting phenomenological argumentation. The idea comes from a “coupled” diﬀusion-relaxation picture that appears in such a complex system. In any diﬀusion-migration growing process, the mechanical strain-stress ﬁelds play such a role as well. In our case, such a situation can be safely expected in the temperature domain T ≤ Tpass .
Another type of relaxation of the stress ﬁeld, say σm , is expected to prevail when the closely-packed agglomeration conditions are met. A diﬀerent behaviour may be observed when the closely-packed agglomeration conditions are lost for the ﬁrst time, that is, at T = Tpass , when the loosely-packed context appears. In both temperature regimes, the relaxation of σm (t) over the

Agglomeration/Aggregation and Chaotic Behaviour

285

course of time, is very likely to go in a way essentially described by the current
(2). This is expected to occur [22] presumably under (nearly) homogeneous
1. For an additional motivation of coustrain conditions, εm ≈ const, for t pling matter agglomeration and stress relaxation picture, related to fracture phenomena, see [1, 22].
From [22] it can be learned, that in the absence of non-Arrhenius or fractal type kinetics, seemingly modifying the diﬀusion coeﬃcient D(v) [20], one expects the Maxwell dashpot-and-spring model to reﬂect properly the relaxation behaviour. We wish to set up here a phenomenological picture, showing that both agglomeration and mechanical stress relaxation, where the stress relaxation takes place under slow growth conditions, proper of a mature growing stage in a viscoelastic multiphase medium [13,15], are coupled processes [5, 16]. To work out the problem quantitatively, we will represent one of the two contiguous and matter-exchanging clusters in the agglomerate, say cluster (grain) 1, as an expanding one, equivalent to the spring, growing at the expense of its neighbour, to be named cluster (grain) 2, i.e the dashpot, to which, according to the Maxwell model, the contracting action should be assigned, cf. Fig. 2 for details; see [22].
For the system with non-wide gaps, the Maxwell model conditions are almost satisﬁed, so that the two-cluster action can be extended over all pairs of contiguous clusters until the expanding (growing) eventually survive. In a next step, the same kind of competition appears as in the well argumented
Laplace–Kelvin–Young scenario suitable for cellular systems [9]. This picture holds in the closely-packed context.
In the loosely-packed context (a system with wide gaps), we may have qualitatively almost the same picture [2] but with several diﬀerences which implies that cluster expansion would not be likely so vigourous. Since the corresponding gap is wider, therefore untight, the ﬂuid leakage might be more pronounced. Thus, the ﬂuid response against the piston wall is weaker, and the Maxwell type stress relaxation no longer applies, cf. the caption of
Fig. 2.
The stress relaxation can be described by introducing an exponent χ in the Maxwell-like quasi-fractional model presented here below. This exponent should be, in general, d-dependent, and points to a diﬀerence when comparing with the classical Maxwell model [2]. Here, we oﬀer a coupled matter diﬀusion and stress relaxation picture, but for a random walk performed in the conﬁgurational space [22]. As is known, the Maxwell stress relaxation picture leads to an exponential decay of the stress: σm (t) ∼ exp(−t/τM ) ,

(23)

where τM is a reference time for the concentrated clusters [2] to be eventually inferred from the Einstein–Stokes-like formula [13, 22]. This behaviour holds for T < Tpass . As mentioned above, for T ≥ Tpass we propose σχ dσm
+ m =0, dt τ

(24)

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Fig. 2. Maxwell sequential spring-and-dashpot (quasi-fractional) model with narrow (d–independent) and wide (d–dependent) gaps, shown schematically in two subsequent time instants t1 and t2 , where t2 > t1 > 0, from left to right, respectively. Grain “1” consists of the spring and the piston’s upper wall, to which the second end of the spring is attached, while its ﬁrst end is mounted either on 01 or
02 , from left to right, respectively. Grain “2” consists of the viscous medium inside the cylinder as well as the inner wall of the piston. The cylinder’s walls complete the overall model structure of the viscoelastic grains. The material exchange between
“1” and “2” is assured by the existence of the gaps: narrow CP1−2 gaps in case of closely-packed agglomeration, and some two wider (here, represented by the left(d) hand side gap, LP1 ) in case of the aggregation. Therefore, the piston-and-cylinder system, containing a viscous ﬂuid, here composed of big and small particles, is either more (densely-packed agglomeration) or less (undensely packed formation) leakproof. The overall material exchange is caused by spring expansion along z axis, which results here in a longitudinal expansion of grain “1” at the expense of grain “2”, cf. http://www.j-npcs.org/abstracts/vol2000no4.html. Notice formally that: O1 E = O2 E, and for t1 one has z1 (t1 ) = O1 E, z2 (t1 ) = BE as well as for t2 one gets z1 (t2 ) = O2 I, z2 (t2 ) = IE, which results in grain expansion-contraction behaviour, like O1 B < O2 I and BE > IE, when mutually comparing the distances along the z axis at t1 and t2 , respectively

where the above is usually true when the internal strain ﬁeld, εm is practically constant, see above. When solving (24), one obtains σm (t) ∼ (t/τ )

−1/(χ−1)

,

t

1,

(25)

where χ = 2d + 3; about τ , see [22] or [13]. Notice that for χ = 1 and τ = τM in (24) one gets the solution (23); for χ = 1 (25) is the only solution to the relaxational problem as stated. When comparing (23) and (25) one sees that the relaxational response goes slower for the late-time loosely packed aggregational context than for its densely-packed agglomerational counterpart.

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3 Qualitative Signatures of Chaos in Matter–Agglomerating System
Let us consider a few qualitative signatures of chaos in matter–agglomerating systems from the literature. Such certain signatures for systems of the type studied in the present work are summarized in Table 1.
Table 1. Qualitative signatures of chaos in a model matter–agglomerating system of interest
Item

Signature

Refs.

One:
Two:

Entropic system of molecular-chaotic behaviour
Lack of matter - depletion zones around the charged clusters
Three: Competition-and-loss eﬀect: (un)tight spring-and-dashpot model
Four: IBCs of normal (e.g., Neumann) or abnormal type should be taken appropriately
Five:
Bethe-type frustration in coupled relaxation and late-growing events
Six:
Growth viz. instability: random close-packing with its d + 1–account
Seven: Nonequilibrium entropy measures viz. mean-harmonic speeds Eight: Entropic potential(s) assuring nonequilibrium character of the phenomenon
Nine: G-K type construction of D(v), B(v), and its consequences
Ten:
Diﬀusion-space pre-chaotic (Fibonacci) feature by Dα

[13, 14, 24]
[13, 22, 25]
[2, 23, 26]
[6, 20, 21]
[1, 23, 26]
[9, 23, 24]
[2, 14, 28]
[2, 9, 14]
[2, 14, 27]
[8, 14, 28]

The items stated in Table 1 do not exclude other possible forms to chaos, or its signatures, in matter-agglomerating systems. We do not pretend to describe all of them, or even their majority. For routes to chaos recommended from physical point of view one would usefully consult [27, 28]; which routes, or scenarios of chaos, are recommended by mathematicians, especially when a partial-diﬀerential-equation formalism of F-P-K type is eﬀective, can be found in [6], and in Refs. therein.

4 Some Quantitative Measures of Chaos Signatures in Matter–Agglomerating System
In [22] some entropic-like nonequilibrium measures of growth
(d)
νsp =

ln [σ 2 (t)] ln t for , t 1

d = 1, 2, 3, . . .

(26)

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as well as for the mechanical stress relaxation evolution µ(d) = sp − ln [σm (t)] ln t for , t d = 1, 2, 3, . . .

(27)

1

have been proposed. This seems to be working most appropriately in a growth-and-relaxation synchronization metastable regime
−1/2
σm ∼ Rav ∼ σ −1 ,

(28)

which represents the H-P-G condition [3]
−1/2
σm ∼ Rav ,

(29)

appropriate for the ﬂuctuational late-time regime [13] of interest here. Bear in mind that if certain empirical modiﬁcations of the formula (29) are applied toward obtaining a speciﬁc form, interconnecting (26) with (27), one gets something like
(d)
(30) νsp = qµ(d) , sp (d)

(d)

where typically q > 2. In the classical H-P-G limit νsp = 2µsp holds. However q may also strive for obtaining superplastic eﬀects, i.e. when taking on fractional values, cf. [9], and Refs. therein. This is sometimes termed in physical-metallurgical literature the inverse H-P-G eﬀect.
Because of (22)
(d)
νsp =

1
,
d+1

d = 1, 2, 3, . . . .

(31)

Realize that formulae (28) and (29) might again be interpreted in terms of the Onsager conjecture [2, 4], see above. Since the overall exponent in (25) reads 1
1
=
,
(32) χ−1 2(d + 1)
(d)

which is exactly one half of the growth exponent νsp given in (22), see (31) too, one consequently provides µ(d) = sp 1
,
2(d + 1)

d = 1, 2, 3, . . .

(33)

Let us emphasize here that 1/(χ − 1) stands for the so-called Nutting exponent for relaxation, and can be interpreted in terms of the loss tangent, that means, a well-known dissipation factor in the relaxation phenomena, mostly in dielectric (e.g., macromolecular) environments [22].

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5 Number-Theoretic Measures of Spatial and Temporal
Irregularities in Aggregation-Agglomerating Systems
It is interesting to note here that χ = χ(d) , i.e. χ(d) = 2(d + 1) + 1 .

(34)

A certain generator of the Bethe–lattice elements, is recovered starting from the 3-bond (initial) generator for d = 0, and continuing with d, upon identifying d as the numbers of emerging bonds in a gelling system [5]. This is a very useful tool for the mean-ﬁeld description of gels, and other multi bond-containing systems.
In this way, an odd number Bethe–lattice generator for subsequent d–s can be oﬀered, see Fig. 3.

Fig. 3. An example of a not much developed Bethe lattice, which by itself manifests a frustration because of “having problems” with containing all of its nodes in the available d–dimensional space [26], to some extent so as, for example, the population of Verhulst fellow countrymen does in the available Belgium territory [29]

Another, equally interesting observation can be oﬀered, namely
2
(2) νsp =

1
(1)
νsp

+

1
(3)
νsp

.

(35)

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A. Gadomski and M. Ausloos

This means that for loosely-packed agglomerations the harmonic-mean rule for the growth speed is exactly fulﬁlled. Mutatis mutandis, one can ex(d) pect the same type of rule, (35), for µsp . Let us recall that the ﬂuctuations σ 2 (t) have been proposed as a reliable criterion of diﬀerentiating between aggregation and agglomeration, and that an eﬃciency (harmonic-rule, see (35)) additional criterion, derived from the constructed ﬂuctuational proposal, supports the aggregation in dimension d (d = 1, 2, 3), with an emphasis placed on d = 2, where ‘golden-ratio-like’ or harmonic-mean properties are in favor.
The mean-harmonic speed implies that the center of mass of a moving body, referred consequently to as the molecular cluster, may not span the same distance, say s, back and forth, during a time period. This leads to a quite realistic quantiﬁcation of a mean speed on the distance 2s, and also shows that not an arithmetic mean of the back and forth speeds but a harmonic mean applies here. Such a schedule can likely be extended over the aggregation that essentially relies on random matter attachments and/or detachments of particles [8, 25], in such a way a forward sub-process may essentially go unidimensionally, like in a ballistic motion, whereas its reverse counterpart would explore the whole three-dimensional domain. This situation typically appears in the case of matter desorption, in which detachment occurs part by part from a “reactive” surface spot.
Some other conﬁrmation of (mean) harmonicity, its close relation to the golden rule, and to the Fibonacci sequencing (characterizing well the fractality of “diﬀusive” microstructures), is hidden in the (macro)ion (or, cluster [22]) diﬀusion coeﬃcient Dα , in our model, (16), which is also included in the free energy Φ. The label, or the lower index, is simply α, which for d = 1 results in α = 0 ≡ 0/1, for d = 2 gives α = 1/2, whereas for d = 3 oﬀers α = 2/3. The values of the diﬀusion coeﬃcient (16) are: D0 ≡ D/1,
D1/2 ≡ D/2 as well as to D2/3 ≡ D/3, respectively. They correspond to the ﬁrst ﬁve-number Fibonacci sequence, composed of the numerators and denominators of α-s, like 0,1,1,2,3, and obey an+2 = an+1 + an , for the three subsequent Fibonacci numbers an , an+1 and an+2 . If so, one can provide the following two recursive formulae α(d−1) =

ad−1
,
ad

d=1,

(36)

d = 2, 3 ,

(37)

and α(d−1) =

ad
,
ad+1

where a0 = 0, a1 = 1, a2 = 1, a3 = 2, a4 = 3 are the ﬁrst ﬁve Fibonacci numbers. Since the analogy with gelling systems seems evident [22], this cannot be taken entirely as a surprise. The bonding in gels clearly goes as a branching process, being (as in the case of ultrametric space) quite naturally described geometrically in terms of Fibonacci numbers, thereby involving the notion of fractality [5].

Agglomeration/Aggregation and Chaotic Behaviour

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When ﬁnishing this section, let us note that both the characteristic chaotic measures, cf. (31) for example, have their random close-packing account d + 1 involved. This is a landmark of randomness but readily appears as a spaceﬁlling action of modelled matter reorganisations. Realize that our rationale may apply just in the same vein to cluters-containing assemblies, evolving in a d–dimensional space, – where a cluster is characterized by its fractal dimension 0 < dF < d, cf. [9, 22].

6 Chaos in an Inﬁnite-Dimensional Agglomerating and/or Aggregating System
Consider the case limd→∞ . A corresponding chaotic measure for the late-stage growing event in the agglomeration of matter, very reminiscent of nonequilibrium correlational entropy measure [2], reads
(∞)
νsp = lim

d→∞

ln[σ 2 (t)] ln(t) for

, t (38)

1

whereas its counterpart for the relaxation is given by an analogous formula, namely that
− ln[σm (t)] µ(∞) = lim
,
(39) sp d→∞ ln(t) for t 1 holds. They are consistent formally with the so-called correlational entropy
(Kolmogorov–type) measure, deﬁned in [28] and follow the rationale presented in [6], in which some measures of chaos in dynamical systems described by partial diﬀerential equations have been discussed. For “thermostatic” systems out of equilibrium one has to speak of the so-called generalized fractal dimension formalism, ﬁrst introduced by Grassberger and Proccacia, see [28], and Refs. therein.
The most attractive reason for introducing such measures arises from the
(∞)
(∞) fact that if one evaluates both νsp and µsp , one unambiguously gets for the aggregation
(∞)
(40) νsp = µ(∞) = 0 , sp whereas for the close-packed agglomeration one provides
(∞)
νsp = 1 ,

(41)

µ(∞) = ∞ . sp (42)

and
Thus, for both cases, (41) and (42), one arrives at a chaotic behaviour in the nonequilibrium system [2, 28] of a densely-packed agglomerate. This is not the case of the aggregation for which the common measure of its chaotic character is zero, cf. (40).

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Thus, proceeding consistently with the approach oﬀered in [6] we may conclude that the late-time aggregation process develops in an ordered way.
The case d = 2 appears to be the most eﬃcient since the harmonic-mean rule
(35) is applied; for it the nonequilibrium character of the random process should be emphasized [30]. It is intriguing to realize that the system property called the harmonicity throughout is very much related to its nonequilibrium entropic or chaotic characteristic(s).

7 Concluding Address
Based on the above, we are allowed to state the following:
(i)
(ii)

(iii)

(iv)

(v)

6

in matter-agglomerating systems chaos is revealed as a complex spatiotemporal and temperature-dependent phenomenon; nonequilibrium chaotic measures of any late-stage matter agglomeration modelled can be proposed relying upon the nonequilibrium Kolmogorovtype entropy measure, which makes a reliable (harmonic) quantiﬁcation of the tempo of the process; coupling late-stage matter agglomeration with relaxation of assisting elastic ﬁelds via an Onsager-type [4], or, in the parlance of physical metallurgy, H-P-G conjecture [3], leads to several characteristic subeﬀects (Bethe-lattice generator, ﬁrst-ﬁve Fibonacci-number signatures, random close-packing d + 1–criterion [23], etc.) having their rationale in fundamental properties of the entropic or harmonic-mean character of the phenomenon6 ; as for the formal point of view: The presented mesoscopic system, Sect. 2, serving to describe the matter aggregation can be derived rigorously based on the Gibbs entropy production equation [9, 10, 22], and its chaotic signatures can be inferred as presented in Sects. 3–6, supported somehow by the ideas contained in [6]; at this point, a general task remains to be done as to connect the type of chaos with the entropybased scheme [10, 22] used to derive the equations of F-P-K [11], or diffusion, types [8,9,27], and how far the proposed measures of chaos (also, the ones used in the present review) are reminiscent of those used conventionally in nonlinear science [2, 14, 27, 28, 30]? Perhaps, the Edwards’ entropy measures for slowly moving grains, evolving (bio)polymer- or colloid-type matrices and compacted powders could also contribute to solve the problem [32].

As can be for example observed in clays made of an inorganic material known as laponite [31]

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Acknowledgement
¨
One of us (A.G.) dedicates this study to Prof. Peter Laggner, O.A.W., Graz,
Austria, and Prof. Gerard Czajkowski, U.T.A. Bydgoszcz, Poland, in the year of their 60th anniversary. This work is done under 2P03B 03225 (2003-2006) by A.G. Part of M.A. works is in the framework of an Action de Recherches
Concert´e Program of the University of Li`ge (ARC 02/07-293). Thanks go e e to Prof. Peter Richmond (Dublin) for deciding to support the participation of
A.G. in the meeting “Verhulst 200 on Chaos”, 16–18 September 2004, Royal
Military Academy, Brussels, Belgium from the funds of COST P10. Last but not least we are thankful to Prof. Jerzy Luczka (Katowice) for drawing our attention to Ref. [16] and Prof. Miguel Rub´ (Barcelona) for useful comments ı on the manuscript.

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A Chaos and Fractal Dynamic Approach to the Fracture Mechanics
Lucas M´ximo Alves1 and Rui F.M. Lobo2 a 1

2

GTEME - Grupo de Termodinˆmica, Mecˆnica e Eletrˆnica dos Materiais, a a o lmalves@uepg.br,
Departamento de Engenharia de Materiais, Universidade Estadual de Ponta
Grossa - Caixa Postal 1007, Av. Gal. Carlos Calvalcanti, 4748, Campus
UEPG/Bloco L - Uvaranas, Ponta Grossa, Paran´,CEP. 84030.000 Brasil. a Grupo de Nanotecnologia e Ciˆncia a Nano-Escala, e rfl@fct.unl.pt,
Departamento de F´ ısica, Faculdade de Ciˆncias e Tecnologia da Universidade e Nova de Lisboa, 2829-516 Caparica – Portugal.

Summary. It is shown that the onset of instabilities observed in the fracture of brittle isotropic materials is a consequence of the mathematical structure of chaos that underlies such phenomena. The straight line crack velocity is written in the form of a logistic map explaining the onset of instabilities observed by Fineberg et al. This approach provides a single and concise tool to study this and other nonlinear aspects presented by dynamic crack growth.

1 Introduction
Dynamic fracture has been stimulating a growing interest not only because of its fundamental importance in understanding fracture processes but also because of the challenges to mathematical analysis and experimental techniques.
Fineberg and co-workers performed experiments on fast crack growth [1,2] for diﬀerent brittle materials, such as PMMA, soda-lime glass, etc., revealing many new aspects, deﬁant to the fracture dynamic theory, related to unstable crack growth. They observed the existence of a critical velocity starting from which instability begins and measured the correlation between the ﬂuctuations in crack growth velocity and the ruggedness of the generated surfaces.
They found a time delay (of the order of the stress relaxation time) present during the whole fracture process in the materials used by them. In spite of this experimental observation that can be associated to stress relaxation properties in viscoelastic materials, they were unable to relate it physically to the onset of the instability phenomenon itself. Therefore the need for a correct mathematical description of the instability process in crack growth represent one of the interesting challenges in dynamic fracture.
On the other hand, Kostrov–Nikitin [3] and Christensen [4] modelled the quasi-static crack growth case in viscoelastic materials showing that the stress

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L.M. Alves and R.F.M. Lobo

ﬁeld at the crack tip does not depend on the relaxation properties of the material but that it is determined by the instantaneous elastic modulus at the crack tip. Similarly, analogous considerations have been taken and extended to dynamic case [6, 7, 7]. Analyzing this last situation Freund [8] admits that there is really a paradox because the process zone in front of the crack tip depends strongly on the loading conditions and usually ideal boundary conditions are considered, such as a thin crack tip describing a punctual fracture in a short range eﬀect in the obtaining of asymptotic solutions to the problem. Therefore, such paradox is solved by introducing a separation zone of ﬁnite extend in the form of a cohesive ﬂaw zone of some length, say lo , ahead of the moving crack tip [8, 9] to separate the crack formed from the instantaneous process at the crack tip. Such cohesive zone accompanies the crack growth process at the crack tip, while the crack opens gradually against the resistance of some cohesive stress within this zone. The time, t, required for a crack tip to advance a distance equal to lo , as the crack grows at crack growth velocity, vo , introduces a process time, given by t = lo /vo , that must be compared to a characteristic relaxation time, τ ∼ t, of the material to determine whether the process is “fast” or “slow” [8].
Actually, if it is considered that the relaxation time, τ , is much smaller than the time, t, taken by the process zone to move forward the crack, the zone is stable and the relaxation eﬀects don’t aﬀect the process as it is shown for the quasi-static case (v → 0) [3] by the results of Kostrov–Nikitin and
Christensen [4]. But for fast crack growth the relaxation time τ is very large compared to the time, t, taken by the process zone to move a distance lo .
This happens when the crack growth velocity tends to the Rayleigh waves velocity of the material. In this case the relaxation eﬀects become important and the crack becomes unstable as shown experimentally by Fineberg and co-workers [1, 2] by means of the correlation measured between the ﬂuctuation in crack growth velocity and the ruggedness of the generated surfaces. In these correlation measurements they found a time delay between the oscillations in the crack growth velocity and the surface proﬁle of approximately 3 µs and 1.0 µs for PMMA and soda-lime glass respectively. The critical velocities at which the instability begins are 340 m/s and 1100 m/s, respectively, and both values correspond to 0,34 of the respective Rayleigh surface wave velocity, cR , in these materials, whose values are 975 m/s and
3370 m/s for PMMA and soda-lime glass respectively. This means that at these velocities the characteristics lengths of the process zone is of the order of lo = 3 µs × 340 m/s ∼ 1, 02 mm and lo = 1.0 µs × 1100 m/s ∼ 1.10 mm for the two materials cited above, also agreeing with other experimental results found by Fineberg and co-workers.
In view of these results there are two alternatives to be considered: either the fast crack growth process is described by using the stress ﬁeld modelled with a retardation time that takes into account the eﬀects of relaxation of the material, or the mathematical description of a process zone is used that

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297

keeps all the information of delay produced by the relaxation of the material, leaving unsolved the instability process, and the previous case is ignored. Historically the latest option has been the solution proposed by Irwin and other scientists to describe the complex eﬀect of the plasticity and viscoelasticity at the crack tip [10–12]. It is not a surprise that, when intending to study a Mechanics of the Fracture that can be universally applied to any kind of material and to any mode of loading, it should receive another formulation.
In this chapter, in order to explain Fineberg et al. results [1,2] one makes use of a physical model in which the retardation time is introduced in an explicit way generalizing the classical formulation of Fracture Mechanics. It will also be shown that this mathematical procedure will include the fracture process inside the family of phenomena described by nonlinear dynamic processes with the advantage of using the whole mathematical development so far accomplished for these processes and the classic formulation of Fracture
Mechanics generalized by the explicit inclusion of time retardation.

2 Theoretical Development of a Chaotic Model to Dynamic Fracture
2.1 The Fast Crack Growth in the Fineberg–Gross Experiments
Consider a semi-inﬁnite plane plate under Mode-I loading and plane strain in elastodynamic crack growth conditions, as shown in Fig. 1. The experimental conﬁguration of the body under testing is equivalent to a inﬁnite plate condition to avoid the edges eﬀects on the crack growth process. In this experiment a fast crack growth develops as a result of a high loading rate and a high strain rate in the dynamic fracture process. In agreement with

Fig. 1. Experimental set up of a plane plate under Mode I loading in dynamic fracture process in accord to Fineberg–Gross [1, 2] experiments

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L.M. Alves and R.F.M. Lobo

Fineberg–Gross [1, 2] the aspect of the fracture surfaces is rugged and the crack growth velocity shows a instability from a critical velocity.
In their experiments [1, 2] Fineberg–Gross showed that when the crack speed reaches a critical value a strong temporal correlation between the velocity, vo (t), and the response in the form of the fracture surface at Ao (t + τ ) takes place (having its notation changed, in the present text, to Lo (t) instead of Ao (t) to designate the fracture surface length). The time delay measured between this two magnitudes presents a value τ of about 3 µs for PMMA and 1.0 µs for soda-lime glass, for example, showing that there is a given value for each material.
2.2 The Origin of the Time Delay from Fineberg–Gross Experimental Evidences
Crack growth is a complex process that involves many physical phenomena
(bond breaking, sound and light emission, heat generation, etc.) which participate in the crack growth process. Also, the relative importance of each one depends on the conditions under which crack growth takes place. It is well known that for any given material a certain time delay exists between the loading and the crack growth, at the beginning of the crack growth, in experiments performed under time-dependent loading conditions [8]. Moreover, this time delay tends to disappear with the crack growth [8]. However, since dynamic fracture involves heat generation it is necessary to take into account the fact that a viscoelastic material such as the one used by Fineberg–
Gross [1, 2] shows viscoelastic relaxation phenomena. This is so because the heat developed at the crack tip might have changed the local properties of the material making the hypothesis of viscoelastic relaxation very plausible. In the case of a viscoelastic material, such as PMMA, if a considerable amount of viscoelastic material is formed at the crack tip (size of the order of lo ∼ vo τ ), the time delay appears locally remaining because of the existence of the persistent creep phenomenon that takes place at fast dynamic crack growth, as it is evident by the correlation shown in the Fineberg–Gross experiments [1, 2].
Therefore, this may explain the phenomenology behind the time delay obtained by Fineberg et al. experiments [1, 2]. Taking into account this strong evidence of time delay between the two magnitudes mentioned above the fast dynamic crack growth can be expressed by the elastodynamic energy release rate, GoD , at a given time such as it does not depend on the crack velocity at the same moment [12–15] as shown in Fig. 2 and as it will be shown below.
The time delay obtained by Fineberg and co-workers [1, 2] attributed to the viscoelastic properties of the material at the crack tip, possibly develops in several steps. In the case of the vitreous (glasses) and polymeric materials the viscoelastic properties of the material at the crack tip become evident because at fast crack growth heat develops at crack tip, and due to the poor heat conduction, temperature rises with possible softening of this material,
(in front of the crack tip) as explained above, even at room temperature. In

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299

Fig. 2. Schematic crack tip region with time delay between the input ﬂux energy, φ(t) and output ﬂux energy, ψ(t + τ )

the case of metals, due to higher heat conduction, the viscoelastic properties only appear when the sample is tested at temperatures greater than thirty percent of their absolute melting temperatures [11].
“There are two competing mechanisms involved in the crack growth that characterize the creep deformation. The blunting of the material in front of the crack tip relaxes the crack tip stress ﬁeld and tends to retard crack growth.
The other mechanism results in an accumulation of creep damage in the form of microcracks and voids that enhance crack growth as they coalesce” [11].
2.3 The Foundations of Quasi-Static and Dynamic Fracture Mechanics
For quasi-static crack growth the usual elastic energy release rate, Go , is given by [12] d(F − ULo )
Go ≡
,
(1) dLo where F is the work performed by external forces on the sample, ULo is the change in elastic strain energy caused by the introduction of a crack with length, Lo , into the sample. Lo is the distance between two points of the crack (or projected crack length) and the subscript zero denotes the plane projected magnitudes.
The Griﬃth–Irwin energy balance approach for stable fracture requires that [10, 16]
(2)
Go ≥ Ro , where Ro is the crack resistance per unit thickness deﬁned as
Ro ≡

dUγo
,
dLo

(3)

and Uγo is the product of the speciﬁc elastic surface energy of the material, γo(eﬀ) , by the projected surface area of the crack (two surfaces, length Lo ),

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L.M. Alves and R.F.M. Lobo

i.e., the plane surface area of the crack with actual (rugged) crack length L and unit width. It is important to stress that L refers to the actual (rugged) crack length and Lo to the projected crack length of L, i.e., a distance between two points [17]. Therefore
Uγo = 2Lo γo(eﬀ) ,

(4)

and according to Alves [17] the constant crack growth resistance, Ro = 2γ(eﬀ) needs to be corrected by the ruggedness in the follow way
Ro = 2γeﬀ

dL
,
dLo

(5)

where the mathematical term dL/dLo was called the local ruggedness of the fracture surface [17]. From the quasi-static Griﬃth criteria, the condition to trigger the beginning of crack growth, is given by Go = Ro . Gao [18] introduced the role of the surface roughening and branching instabilities in dynamic fracture proposing a so-called “wavy-crack model” motivated by experimental observations where rapidly moving cracks develop roughened fracture surfaces. The essence of his model consists in separating the microscopic crack-tip motion with local velocity, from the macroscopically observable crack motion with apparent velocity.
Similarly for the stationary case the elastodynamic energy release rate,
GoD , is given by [12]
GoD ≡

d[F − (ULo + To )]
,
dLo

(6)

and To is the kinetic energy of the crack growth which can be written as:
To =

1 2 ρv 2 o

du dx 2

+

du dy 2

dx dy .

(7)

The velocities indexed by o, as vo , refers to the growth rate of the projected crack length, Lo . This equation will be not considered in the calculations that will be followed here; it is just shown up in the text in order to form the body of the classic considerations, that cannot explain the phenomenon in question. A more general expression was included, however, the calculations do not require the explicit use of the expression of the kinetic energy, To , but general principles will just be used.
Mott in 1948 [19] proposed that all excess of the elastodynamic energy release rate, GoD , above the energy necessary to create the fracture surfaces,
2γeﬀ , is transformed into kinetic energy of crack growth. However, it is possible to generalize this proposition to the case of a material which shows a non-constant crack growth resistance, R = 2γeﬀ , that develops a ruggedness such as that of the kind dL/dLo expressed in Alves [17]. Rewriting (6):

A Chaos and Fractal Dynamic Approach to the Fracture Mechanics

GoD =

d(F − ULo ) dTo −
.
dLo dLo 301

(8)

The ﬁrst term on the right side of (8) corresponds to the usual elastic energy release rate, Go given by (1), and so:
GoD = Go −

dTo
.
dLo

(9)

This equation shows the relationship between Go and the elastodynamic energy release rate, GoD . In analogous way to (2), the Irwin–Mott condition of dynamic crack growth is given by
GoD (Lo , vo ) ≥ Γ0 (Lo , vo ) ,

(10)

where Γ0 = Γ0 (Lo , vo ) is the dynamic crack growth resistance, now depending on the crack growth length, Lo , and velocity, vo .
In agreement with Griﬃth fracture criterion for the quasi-static case it is possible to write, in analogous way, the elastodynamic crack growth condition, as follows
(11)
GoD (Lo , vo ) = Γ0 (Lo , vo ) , where Γ0 can be write as Γ0 (Lo , vo ) = Ro − dTo (Lo , vo )/dLo using (5) for crack resistance, Ro .
The expression (6) mentioned here will not be used in the same way, but completes the idea to be explained in the following section by using a variable separation in the non steady-state case, which is a quite knew procedure.
2.4 Advanced Dynamic Fracture Mechanics Considerations
The literature [13] points to the fact that experimental results conﬁrm the continuum theory of dynamic brittle fracture for fast cracks, where expressions similar to 6 are used to explain the phenomena in focus. On the other hand, in classical dynamic fracture mechanics it is usually assumed that rupture is controlled by the expression [8]
GoD

Go 1 −

vo cR = 2γeﬀ .

(12)

This expression equating GoD with vo is only true locally, i.e., in the immediate vicinity of the crack tip. For the conﬁguration of plane stress applied to a medium of inﬁnite extent, for example, a crack will constantly accelerate. 2
For this case, the quasi-static energy release rate, Go = πσ0 Lo /2Eo , in
(12), is proportional to the fault crack length, Lo , so that as the rupture grows, Lo grows, and the velocity, vo , tends to cR in order to maintain 2γeﬀ at a constant value. This formula fails because, accordingly to the well established fracture mechanics theory, as the velocity increases, 2γeﬀ ceases to

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be a constant and depends on rupture speed. At velocities near 0.4 or 0.5 of cR , rupture branches appear and (12) ceases to be applicable [3, 20, 21]. This explanation was well known to Kostrov, Slepyan, Knauss and Ravi–Chandar,
Rice, etc and was rediscovered by physicists in the early 1990s [1,2,13,22,23].
In the dynamical case Slepyan [8,20] proposed that the elastodynamic energy release rate, GoD (Lo , vo ), can be written in terms of the elastic energy release rate, Go (Lo ), for the stable case [16] in the following way:
GoD (Lo , vo ) = Go (Lo )g

vo cR .

(13)

where cR is the Rayleigh velocity.
To write the functional dependence of this elastodynamic energy release rate, GoD (Lo , vo ), Slepyan [20] propose a Maximum Energy Dissipation Principle. In Alves [24] one has also demonstrated the need for correcting this principle in order to include the ruggedness of fracture surface [17]. For the
Fineberg–Gross [1, 2] experimental set up of a semi-inﬁnite plane plate under Mode I loading, the elastodynamic crack growth condition, given by (11) yields: 2γeﬀ (dL/dLo )
= Γ0 (Lo , vo ) ,
(14)
GoD (Lo , vo ) = vo 1 − cR (dL/dLo )
Equation (14) shows a great agreement with Gross experimental results [22] as was ﬁtted by Alves [24].
Using the Maximum Energy Dissipation Principle of Slepyan [20], modiﬁed to include the ruggedness, dL/dLo of fracture surface, and the Irwin–
Mott crack growth dynamic condition given by (10) in (14), including (13), one can assert that: vo cR

Go (Lo )g

≥

2γeﬀ (dL/dLo )
.
vo
1 − cR (dL/dLo )

(15)

Comparing (14) with (13) it can be concluded in agreement with Alves [17] that for the brittle materials case where J ≡ G, that: dL , dLo (16)

1 vo .
1 − cR

(17)

Go (Lo ) ≥ 2γeﬀ and g

vo cR ∼

This is an equation that shows dependence on the particular experimental set up under use. The case studied here is of a plane plate under Mode I loading and boundary conditions of inﬁnite body accordingly to Freund [25].

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303

2.5 The Foundations of Non-Stationary
Dynamic Fracture Mechanics
For, for the non-stationary case the dynamic energetic balance for the bidimensional case can be write written as
˙
˙
P =U +T +φ,

(18)

˙ where φ is the rate of work done by the traction on the surface S, U is the
˙ is the kinetic energy in the region with rate of increase of the strain energy, T
R with area A and F is the energy ﬂux into the crack-tip region.
P =

Ti ui ds ,
˙

(19)

S

U = lim
∗

Γ →0

T = lim
∗

Γ →0

W(

ij )

dA ,

(20)

1 ρui ui dA ,
˙ ˙
2

(21)

R

R

Since the loop Γ ∗ moves with the crack tip, the region R is time-dependent.
In this case the dynamic energy released rate, GoD can be write as
˙
˙ φ = GoD (Lo , vo , t)vo = P − (U + T ) ,

(22)

For the purposes of our calculations we consider that the non-stationary energy released rate GoD (Lo , vo , t) can be described by a function of kind:
GoD (Lo , vo , t) = GoD (Lo , vo )f (t) = Go g

vo cR f (t) ,

(23)

where f (t ≥ τ ) → 1
Considering that the Mott postulate [19] is valid not only at the beginning of the crack growth, at t = 0, but instantaneously at any time, t, during the whole crack growth process, i.e., while, as a crack grows with velocity vo (Lo (t)). Thus the energy ﬂux, φ0 (t), to the crack tip that is characterized by the velocity vo (Lo (t)) is given by: φ0 (t) = GoD [Lo (t), vo (Lo (t)), t] vo (Lo (t)) ,

(24)

where one made use of (9). The coeﬃcient of vo (Lo (t)) is the elastodynamic energy release rate, GoD [Lo (t), vo (Lo (t)), t], given by (23) [8]. Therefore φ0 (t) = Go (Lo (t)) g(

vo (Lo (t))
) f (t) vo (Lo (t)) , cR (25)

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L.M. Alves and R.F.M. Lobo

Based on (13) the energy ﬂux, given in (25), can be written as
GoD [Lo (t), vo (Lo (t)), t] vo (Lo (t)) vo (Lo (t))
= Go (Lo (t)) g f (t) vo (Lo (t)) . cR (26)

For the particular case of a semi-inﬁnite plane plate under Mode I loading the energy ﬂux to the crack tip is derived from (14) as φ0 (t) =

2γeﬀ (dL/dLo ) f (t)vo (t) . vo 1 − cR (dL/dLo )

(27)

2.6 Advanced Considerations Based on Fractal Aspects of Fracture Surface for Dynamic Fracture Mechanics
In a fast crack growth experiment as that performed by Fineberg and Gross the oscillations produced in the crack growth velocity can make that it reaches velocities close to the values of Rayleigh waves speed, (vo → cR ), and when the correspondent energy injected into the crack tip is above this value, it is enough to create new paths for the crack generating branchings [1,2]. Starting from there, if to continue having an increase in the energy injected into the crack tip due to an indeﬁnite loading stress, that is to say, G(t) → ∞, the instability and branching process stays and it starts to happen in diﬀerent scales, i.e., for each new crack made by branching, being obtained a self-aﬃne geometric pattern (invariant by scale transformation). This way it is observed that the fractal nature of instability and of branching it is nothing else than a physical conﬁrmation that the phenomenology of the process described above continues reproducing in scale, indeﬁnitely, while there is an energy excess into the crack tip. It can also be understood, that the chaotic nature of the fracture possesses a kind of “phenomenological memory” that repeats in diﬀerent scales being registered in the fractality of fracture surfaces generated in the dynamic crack growth.
Based in the Fineberg experimental results it can be assumed that the temporal dependence of rugged crack growth of crack length has a structured self-similar [8] or self-aﬃne fractal behaviour [17, 20, 26, 27] in the time, such as, an appropriated function satisfying the relaxation process can be supposed as, (28)
L[Lo (t + τ )] = h[L(Lo (t))] , where τ is a time lag with magnitude of the order of the viscoelastic relaxation time in the sample. This hypothesis means that the crack length generated in the dynamic fracture process elapses on itself.
For the crack growth velocity to be in agreement with (28) another appropriated self-aﬃne equation can be written as

A Chaos and Fractal Dynamic Approach to the Fracture Mechanics

vo [Lo (t + τ )] = vo [Lo (t)]

dh[L(t)] dLo (t)
.
dLo (t) dL(t)

305

(29)

where vo [Lo (t)] = v[L(t)]dLo (t)/dL(t).
This mathematical procedure, naturally, introduces in the equations of dynamic fracture the two necessary conditions to describe the instability process. To know: (i) a decoupling of the dynamical functions between the input and output of system which are the elastodynamic energy release rate,
GoD , and work of fracture, Γ0 , by means of the dynamical variables of length,
Lo , and crack growth velocity, vo , [28] as it is shown in (28) and (29) and (ii) the existence of at least two situations equally probable [20], unifying them into a single condition. This second condition will be focused latter on.
One observes that, while the input energy ﬂux into the crack tip, φ0 (t), vo given by (25) depends on the the crack growth velocity function, g( cR ), the output energy ﬂux, ψ(t), must depend on the local ruggedness created instantaneously in the crack growth process. Therefore, the response in the form of fracture surface, designated by ψ(t + τ ), as being the energy adsorbed to form the rugged crack (shown in Fig. 2), can be written as dUγ [L(t), v(t)] dL(t) dh[L(t)] dL(t) dt dL(t) dUγ [L(t), v(t)] dL(t + τ )
,
= dL(t) dt

ψ(t + τ ) =

(30)

Since ψ(t + τ ) = ψ0 (t + τ ), due to the energetic equivalence among the rugged and the projected crack path, then one can express (30) as ψ0 (t + τ ) = Γ0 [Lo (t), vo (Lo (t))]vo [Lo (t + τ )] ,

(31)

where L(t + τ ) is the ruggedness (i.e. actual) fracture surface length in the time, t + τ , and Lo (t + τ ) is its corresponding plane projected length on the direction of crack growth.
Substituting (14) into (31), one gets ψ0 (t + τ ) =

2γeﬀ (dL/dLo (t))
1−

vo [Lo (t vo [Lo (t)]
(dL/dLo (t)) cR + τ )] ,

(32)

This equation describe the dissipation rate in the formation of the fracture surface formation in the time t + τ .
2.7 Dynamic Fracture Model with a Time Delay
Sharon & Fineberg [13] admit that theory predicting the motion of a crack is governed by the balance between the energy ﬂux into the crack tip, φ0 (t), and the dissipation rate, ψ0 (t), here given by (31) including the time delay between them.

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L.M. Alves and R.F.M. Lobo

φ0 (t) = αψ0 (t) ,

(33)

where 0 < α < 1 is a conversion factor of elastodynamic energy into surface energy (work of fracture).
However it is necessary to take into account the relaxation process at the crack tip. Observe that (26) says that the energy balance between the instantaneous input ﬂux and stationary ﬂux at crack tip, in accordance with
Irwin–Mott’s (10), can be written in a general form as Stationary Flux(time,
t) = Input Flux(time, t)/Coupling function of time, f (t) or φo(stationary) (t) =

φ0 (t)
,
f (t)

(34)

where from (23), (24) and (25) we have: φo(stationary) (t) = GoD [Lo (t), vo (Lo (t))]vo [Lo (t)] ,

(35)

depending on the projected crack length, Lo (t), already formed in each instant, t.
Substituting (35) into (26) one gets:
GoD [Lo (t), vo (Lo (t))]vo [Lo (t)]
= Go (Lo )g

vo (Lo (t)) cR vo [Lo (t)] .

(36)

The particular experimental apparatus in use inﬂuences the functional form of the kinetic energy, To (Lo , vo ) developed by the cracks and this in turn aﬀects the ruggedness developed instantly in the fracture. Therefore it should exist a relationship between the response in the form of a local ruggedness created instantaneously in the crack growth process and the of the crack vo growth velocity function, g( cR ), Thus, assuming a function dh(L(t)) dLo ∼ dLo (t) dL o 1/[g( vo (LR(t)) )] the energy ﬂux, given by (26), can be written as c GoD [Lo (t), vo (Lo (t))] vo [Lo (t)]

dh[L(t)] dLo dLo (t) dL

= αGo [Lo (t)]vo [Lo (t)] .

(37)

Observe from (37) that the Mott postulate, enunciated before, becomes valid instantaneously during the whole crack grow process by means of a new insight in the energy ﬂux balance with time delay. Therefore from the elastodynamic crack growth condition given in (11), and using (29) in (37), this equation can be expressed as
Γ0 [Lo (t), vo (Lo (t))]vo [Lo (t + τ )] = αGo [Lo (t)]vo [Lo (t)] .

(38)

In accordance with the self-aﬃne hypothesis made in (28) and (29) the energy ﬂux balance between the input and output with a time delay, τ , may

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307

be written as α. Relaxed Flux(time past, t) = Output Flux(time present = time past plus relaxation time, t + τ ), or in accordance with Irwin–Mott’s
(10), (38) can be written in a general form as φo(relaxed) (t) =

1 ψ0 (t + τ ) . α (39)

Therefore at time, t, the ﬂux of elastodynamic energy into the specimen to the crack tip, φ0 (t) = GoD [Lo (t), vo (Lo (t)), t] vo [Lo (t)], will create in front of the crack tip, at time t + τ , the conditions for the formation of a process zone (viscoelastic region of size ∼ lo ) that will separate at time, t + τ , forming a rugged crack having length L(t + τ ), necessary for crack growth (e.g. breaking of chemical bonds, formation and nucleation of dislocations, etc.).
This theoretical result resumes all the problematics found by Fineberg and
Gross [1, 2], already explained in the Sects. 2.1 and 2.2 of this chapter.
2.8 Chaotic Nature of Dynamic Fracture
The chaotic nature of fractures can be revealed by diﬀerent forms depending of the particular experimental set up. For the case of a semi-inﬁnite plane plate under Mode - I loading the function g(vo [Lo (t)]/cR ) determines a logistic map well known in the literature concerning chaos theory [29, 30].
Therefore, using (14) in (38) or (41) and (32) in (39) this becomes, αGo [Lo (t)]vo [Lo (t)]
2γeﬀ (dL/dLo (t)) vo [Lo (t + τ )] .
=
o
1 − vo [LR(t)] (dL/dLo (t)) c (40)

Rewriting (40) and multiplying the resulting by dL/dLo term it yields the following expression for the normalized crack growth velocity vo [Lo (t + τ )] corresponding to projected surface vo (Lo (t + τ )) dL cR dLo (t)

t+τ

αGo (Lo (t))
=
dL
2γeﬀ ( dLo (t) )
×

vo (Lo (t)) dL vo (Lo (t)) dL
[1 −
],
cR dLo (t) cR dLo (t)

(41)

which has the form of the equation of the logistic map (Fig. 3) [29–31]. For convenience, instead of the delay in time, τ , (41) can be written, as a sequence of events, i.e., xok+1 = µ0 xok (1 − xok ) ,
(42)
where xok corresponds to the normalized projected surface crack growth velocity and the coeﬃcient, µ0 , to control parameter of (31), i.e.,

308

L.M. Alves and R.F.M. Lobo

Fig. 3. Logistic map of the logistic equation xok+1 = µ0 xok (1 − xok ) related to the input energy ﬂux, φ0 (t), and output ﬂux of dissipated power, P0 (t + τ ) in function of the normalized velocities vo (t)/cR and vo (t+τ )/cR , showing the cycles or periods of iteration

xok ≡ and xok+1 ≡

αGo (Lo )
,
2γeﬀ (dL/dLo )

(43)

vo [Lo (t)] dL
,
cR dLo (t)

µ0 ≡

(44)

vo [Lo (t + τ )] dL
,
cR dLo (t)

(45)

respectively.
In Fig. 4, the iterations of this equation are shown as a function of the control parameter, µ0 , [30] which can be identiﬁed with the crack growth characteristics. Fig. 4. Iterates of the logistic map of (48) as a function of µ0 for 1.0 ≤ µ0 ≤ 4.0. An transient of 200 points has been discarded in each case [30]. The control parameter, µ0 , is given by (43) and vo (t + τ )/cR is the normalized velocity as it was explained in the text

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309

Observe that a normalized crack growth velocity to rugged surface, deﬁned as v(L(t)) = vo [Lo (t)]dL/dLo (t), can also be written in the logistic equation form, in analogous way to plane projected fracture surfaces, dropping all the zero subscripts in the (42), (43) and (44) in a such way that the new rugged crack growth velocity can be written as αG(L) v[L(t)] v[L(t + τ )]
=
cR
2γeﬀ
cR

1−

v[L(t)] cR ,

(46)

where G(L) is the elastic energy release rate to the rugged crack path, giving place to write xk+1 = µxk (1 − xk ). In analogous way it gets xk = v[L(t)]/cR and µ = αG(L)/2γ(eﬀ) . This means that the same conditions can be repeated at each new crack path created, in a indeﬁnite branching process, originating a self-aﬃne spatial fractal pattern formed by all cracks generated during the branched crack growth process.
2.9 The Instability Process Under the Sight of Time Delay
Therefore, the instability dynamic process on the fast crack growth and crack branching in brittle materials like the soda-lime glass and PMMA can be explained in the following way: The existence of a time delay, τ , between the energy ﬂux injected into the crack tip, φo(inst) , (25), and the spent energy ﬂux
(or the dissipated power, ψ0 , (31)) to form the fracture surfaces, (32), (Fig. 3), produces from a critical velocity, (44), a “uncompassing” (Fig. 3) between the crack growth velocity, vo (t), and the rate of formation of the fracture surfaces, vo (t+τ ) = dAo (t+τ )/dt. This “uncompassing” is responsible for an indetermination in the crack growth velocity, (Fig. 4), at crack tip that by its turn gives rise to a dynamic instability in the form of oscillations in the crack growth velocity. This instability produces a rugged fracture surface increasing the consumption of energy. This increase limits the crack growth velocity to a smaller value than the speed of Rayleigh waves in the material, (vo ≤ cR ),
(46), producing a new delay in relation to fracture surfaces formation rate and increasing still more the “uncompassing” between the injected energy ﬂux and the worn-out energy rate to form the fracture surfaces and so forth
(Figs. 3 and 4).

3 Results
Equations (45) and (41) can be explored through the graphs of ψ0 (t + τ ) and φ0 (t) versus vo (t) (Figs. 3 and 4) and of vo (t + τ )/cR versus µ0 (Fig. 4). By means of this diagram one notices that combining the magnitudes, φ0 (t) and ψ0 (t + τ ) it generates an instability whenever a temporal delay exists among them. Actually, (42) is the well-known logistic equation. However, Before analyzing Fineberg et al. experimental results [1] we start from the use of

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L.M. Alves and R.F.M. Lobo

Slepyan’s criterion, given by Maximum Energy Dissipation Principle [20], and modify it in order to include the ruggedness term, dL/dLo , of fracture surface [24]. From this result it was possible to obtain the energy balance between the input and output of the system, where in our notation it is given by (46). Observe that this expression can be written in the following way: αGo [Lo (t)] 1 −
= 2γeﬀ

vo [Lo (t)] dL cR dLo (t)

vo [Lo (t)]

dL vo [Lo (t + τ )] . dLo (t)

(47)

Although its left side reminds the expression of the classical result proposed by Freund [8] where the elastodynamic energy release rate being,
GoD = Go [1 − vo (t)/cR ], for a medium with inﬁnite extension ((8.77) of
Freund [8]). Therefore, a ﬁrst result that must be observed in this paper is that the classical dynamic fracture mathematical formalism is reobtained from (41) when the relaxation process is almost negligible and the persistent creep phenomenon is not considered into a thin crack tip in short range eﬀect described by punctual process zone. In this case it has that vo (t + τ ) = vo (t) and therefore the classical comes out: vo (t)
=
cR

1−

dL
2γeﬀ
Go (Lo ) dLo (t)

dLo (t)
.
dL

(48)

Whenever the elastic energy release rate, Go (Lo ), increases linearly with the crack length, Lo , which in the absence of ruggedness, i.e. dL/dLo = 1, (48) becomes, Loc vo = cR 1 −
.
(49)
Lo
This result shows the stable region of the fracture process formed by “ﬁxed point of order one” of the logistic map (Fig. 4) where the conditions of the crack growth are considered to be slow because the viscoelastic properties of the material do not inﬂuence the process.
On the other side, the ﬁrst graph (Fig. 3) shows the dependence of elastodynamic energy ﬂux, φ0 (t), given by (43) and (44), that ﬂows into the crack tip, and of the dissipated power spent to form the fracture surface, ψ0 (t + τ ), given by (31) and (48), in function of the normalized crack growth velocities vo (t)/cR and vo (t + τ )/cR respectively. This graph shows that the energy ﬂux, φ0 (t), possesses a maximum value at vo /cR = 0.5 and a nonlinear dependence of dissipated power, ψ0 , on the normalized velocity vo (t + τ )/cR as it was proposed by the model (26).
The second graph (Fig. 4) shows the iterations of the logistic map, (41) and (48), of the normalized crack velocity, vo (t + τ )/cR , (44) as a function of the control parameter, µ0 , given by (43). This control parameter, µ0 =
Go
2γeff (dL/dLo ) , for a medium with inﬁnite extension, is linearly proportional

A Chaos and Fractal Dynamic Approach to the Fracture Mechanics

311

to the crack length, Lo (t); then as the crack grows, the control parameter increases. Therefore this graph represents the dependence of the speed with the retardation, vo (t + τ ), in function of the projected crack length, Lo . For values of 1 ≤ µ0 ≤ 3 the crack growth process is dynamically stable since at those speeds the viscoelastic process zone in front of the crack tip is too small to inﬂuence the process. However, the time delay that exists between the energy that ﬂows into the crack tip, φ0 (t), and the dissipated power for crack growth, ψ0 (t+τ ), causes instabilities that show up in the interval 3 ≤ µ0 ≤ 4.
Such instability begins only when the control parameter reaches the value, µ0 = 3. Exactly at this point, µ0 = 3, there is a ﬂip bifurcation [29–31] and it begins the inﬂuence of the time delay (where τ ∼ lo /vo ) into the process zone causing instabilities in the crack growth. As shown in Fig. 4 at µ0 > 3 the ﬁxed points determined by (48) are not stable anymore. Considering that the relaxation with time delay is ﬁxed it is seen that the length of the process zone oscillates in time becoming responsible for the oscillations in the crack growth velocity, vo (t). The velocity function, vo (t+τ ) in (41), which is related to the dissipated power, ψ0 (t + τ ), to µ = 3, has the value equal to 2/3cR .
This value from (48) corresponds to a critical velocity vo(critic) = 1/3cR .

(50)

The main assumption of a time delay is expressed mathematically by formula (48) where the factor µ0 of (43) is taken to be ﬁxed. This means that the crack growth velocity at t + τ is proportional to the energy ﬂux into the growing crack tip at time t. This relationship is very interesting since it leads to the oscillating type of instability for the steady motion (with vo(steady) = cR (1 − 1/µ0 )dLo /dL as the solution (48) if 3 ≤ µ0 ≤ 4. By the way, if µ0 increases (from µoc = 3 to a limit value µo max = 4), the minimum limit of the averaged speed, vo min , is given approximately by vo min
=
cR

1−

1
3

dLo dL = max 2
3

dLo dL ,

(51)

min

just depending on the maximum value of ruggedness, dLmin /dLo , in this point. For a value of ruggedness equal to, dL/dLo = 3/2 the limit of the averaged speed, vo , is given by: vo = 0.44 cR ,

(52)

and the maximum limit of the averaged speed, vo max , is given approximately by dLo vo max
1
3 dLo
= 1−
=
,
(53)
cR
4
dL max
4 dL max depending also on the maximum value of the ruggedness, dLmax /dLo , at this point. Equally for a value to ruggedness value equal to dL/dLo = 3/2 the limit of the averaged speed, vo , is given by:

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L.M. Alves and R.F.M. Lobo

vo = 0.5 cR ,

(54)

from where the normalized crack velocity vok /cR saturates at cR (dLo /dL)max , below of Rayleigh waves velocity, which is approximately the saturation velocity measured by Fineberg et al. [1, 27].
As can be seen in Fig. 3, v0 max and consequently GoD has tendency to approach the maximum of the energy ﬂux. After µ0 ≥ 4 there is no sense to speak about the inﬂuence of µ0 parameter over the crack growth because the
GoD reaches the maximum constant ﬂux and vo reaches a maximum constant velocity and after this the crack growth follows a new stationary state conserving the ruggedness already created to maintain the this maximum values.
It can also be observed that (53) assumes an inﬁnite body sample where the inﬂuence of external stress ﬁeld it is negligible and after the maximum ﬂux condition have been reached the dynamic energy release rate, GoD , does not depend anymore on the sample length.
3.1 Comparison Between Theory and Experiment
Fineberg et al. [1, 2] observed that at the onset of instability the fracture surface changes from a featureless (on a scale larger than 1 µm) to a jagged structure which develops into coherent oscillations and coalesce downstream of the crack growth broadening to extend over the entire width of the sample.
This change in morphology as the crack accelerates can be associated with the geometric increase in the number of ﬂip bifurcations, or, with the number of non-trivial stable ﬁxed points as can be seen in Fig. 4. To each stable ﬁxed point corresponds a given velocity vok+1 /cR . As the crack grows the straight line crack velocity oscillates among the allowed velocities, looking for alternative paths of energy dissipation in excess at the crack tip by means the generation of a rugged surface that mathematically correspond to more energetic path. It is therefore reasonable to associate the issuing increase of ruggedness with the number of possible velocities. At µ0 < 3 there is only one possible velocity (one stable ﬁxed point on the logistic map) and therefore the surface is smooth. As the crack propagates µ0 (given by (43)) increases with the crack length, Lo (t), and the number of allowed straight line crack velocities becomes larger and larger. The straight line crack velocity oscillates among the allowed values (projections of the actual velocity) resulting in morphologies of higher complexity, in agreement with the observations of
Fineberg et al. [1, 2].
Peter Gumbsch [32–34], using molecular dynamics methods in your simulations, reported that the instability in dynamic crack growth sets at
Go /2γo(eﬀ) = 3. This condition for initiate initiating instabilities, considering αdL/dLo 1.0, is equivalent to µ0 = 3.0, as is given by (43). Therefore this result is in reasonable agreement with the result shown by the chaotic model presented in this paper.

A Chaos and Fractal Dynamic Approach to the Fracture Mechanics

313

Taking for the energy conversion factor α = 1.0 one obtains from (48) the value of 0.33 for the normalized crack velocity at which instabilities should occur. However, Fineberg and co-workers observed that beyond a critical value of 0.34 the normalized velocities, vo /cR , start to show instabilities [1,2].
Therefore this theoretical value of 0.33 cR is in excellent agreement with the value of experimental result of 0.34 [1, 2], measured by Fineberg–Gross et al., at the moment the instability starts. Gross [22] also observed that a visible rough structure (branching) appears near from 0.42 cR in the form of a parabolic crack branching.

4 Discussion
The experiments performed by Fineberg and co-workers [1, 2] provide evidence for instability in the brittle fracture of isotropic materials. On the other hand, theories based on conventional concepts such as energy balance [16] and quasi-static conﬁgurational forces at crack tips show no indication of strong oscillatory or branching instabilities [35]. Yoﬀe [36] by analyzing the stresses in the neighborhood of a crack tip growing at high velocity hinted about the emergence of instabilities but the analysis is not a truly dynamic theory of forces and accelerations of fractured surfaces.
The basic property of dynamic fracture mechanics is that the processes near the fault tip occur at near wave velocities, and for this reason the crack tip is independent of the details of loading. This is a famous theorem proved independently by Kostrov and Eshelby in 1964 and 1969, respectively, for the antiplane case and by Kostrov and Nikitin in 1970 for general loading.
Observe that (17) refers to a particular experimental set up. Therefore, in accord to the functional dependence of this equation for the g(vo /cR ) term or depending on of the particular form of the experiments other kinds of logistic maps can be obtained, since that the same procedure of calculations accomplished until now can be done. Equation (46) developed in this paper is equivalent to (12), and the same improvements (ﬁnite size of the sample, inﬂuence of boundary, etc.) proposed to (12) [8] can be incorporated into (43) and (46) without consequences on the results presented in this paper. This is corroborated by experimental evidence [1, 2, 13] showing that the onset of instabilities is independent of the size of the sample and/or of the geometrical set up of the experiment (see also [3]).

5 Summary and Conclusions
This chapter presents arguments in favour of chaotic behaviour of rupture.
The arguments are general and based on energy conservation principle which are totally valid in fracture mechanics. The central hypothesis is the energy ﬂux through the crack tip is converted there into fracture energy with a time

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L.M. Alves and R.F.M. Lobo

delay, τ , due to the development of a viscoelastic process zone in front of the crack tip. It is tacitly assumed in this paper that such a delay exists and it has a well deﬁned time scale, τ , being a characteristic property of the material.
Its magnitude is of the order of the viscoelastic relaxation time of the sample material under local fracture conditions. A key assumption of the theory is that the onset of instability observed in the velocity of dynamic crack growth is due to the time delay, which yields (41) and (46). This time delay factor, τ , in (28) implies the possibility to derive an equation for the crack growth the velocity in the form of a logistic map equation.
Before concluding this chapter it is necessary to notice that the hypothesis of linear energy transfer as given by (46) and (45) is an oversimpliﬁed approach. The energy release rate, Go , is linearly dependent on the crack length whereas the crack resistance, Ro , rises in a non-linear form [10, 12].
Based on (28), (29) and (41), this property will be used in a forthcoming paper in which it is shown that the energy dissipation can also be written in the form of a logistic map having as consequence crack branching and other phenomena so far not explained by the classical fracture theory.
The purpose of this chapter is to show that contrary to what has been thought previously, the most familiar models in fracture mechanics are intrinsically incomplete. Therefore, it was used an as simple as possible case of dynamic crack growth of a semi-inﬁnite body with plane strain condition, and well established concepts and results, to derive an expression for the crack velocity in the well known form of a logistic equation and map. From this map conclusions regarding instabilities of crack growth are drawn and compared with the experimental results obtained by Fineberg and co-workers [1]. This work shows that other logistic maps can be built, accordingly to the particularity of the experiment and accordingly to the expression of its the kinetic energy. This article presents a new picture for dealing with fracture dynamics, making use of logistic maps as a new method for predicting the possible velocities that a crack can reach and from there to try to reproduce its geometric fractal behaviour, ruggedness, etc. Each particular material and each particular experimental testing condition will determine the type of map and the type of crack as well. The logistic map built has an interpretation which enables to understand even more complex situations for the phenomenon under study.
From the above results it is concluded that the instabilities involved in dynamic fracture are consequences of the mathematical structure of chaos that underlies such phenomena. It was possible to write the straight line crack velocity in the form of a logistic map explaining the onset of instabilities observed by Fineberg et al. [1]. This achievement brings into fracture mechanics all the mathematical structure developed for complex systems. This theoretical approach provides a single and concise tool to determine among others properties the conditions under which crack growth becomes dynam-

A Chaos and Fractal Dynamic Approach to the Fracture Mechanics

315

ically unstable and branching takes place as will be shown in a forthcoming paper. The literature usually shade that there are fractals in the quasi-static fracture of surfaces. Then undoubtedly, in fracture dynamics there will be chaotic behaviour in the formation of the same ones. Therefore, if the model proposed in this chapter is not the ﬁnal answer for the subject, at least it is an initial step, it lifts and it opens a new proposal for studying fracture dynamics. Therefore, we want to say that experimental research is needed to illuminate the theoretical evidences more closely.

Acknowledgments
This research work was in part supported ﬁnancially by CNPq, FAPESP,
CAPES and one of the authors, Lucas M´ximo Alves, thanks the Brazilian a program PICDT/CAPES and PROPESQ-UEPG (for concession of a scholarship)and to Profs. Drs. Bernhard Joachim Mokross and Leonid Slepyan for helpful discussions and Prof. Benjamin de Melo Carvalho for concession of a computer obtained through the Project (Ref. Proc. n. 46.8808/00-0) supported by CNPq-BRASIL to accomplish the calculations, typing and printing the ﬁnal version of this work.

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Nonlinear Dynamics and Fractal Avalanches in a Pile of Rice
Rinke J. Wijngaarden1 , Kinga A. L˝rincz2 , and Christof M. Aegerter3 o 1

2

3

Division of Physics and Astronomy, Faculty of Sciences, Vrije Universiteit, De
Boelelaan 1081, 1081HV Amsterdam, The Netherlands rw@nat.vu.nl Division of Physics and Astronomy, Faculty of Sciences, Vrije Universiteit, De
Boelelaan 1081, 1081HV Amsterdam, The Netherlands ka.lorincz@few.vu.nl Division of Physics and Astronomy, Faculty of Sciences, Vrije Universiteit, De
Boelelaan 1081, 1081HV Amsterdam, The Netherlands present address:
Fachbereich Physik, Universit¨t Konstanz, Universit¨tstrasse 10, P.O. Box a a
5560, D-78457 Konstanz, Germany christof.aegerter@uni-konstanz.de 1 Introduction
The result of a measurement x e.g. of the length of Dutch men or the weight of rice grains is usually Gaussian distributed with a well-deﬁned average x
¯
and width σ. Quite to the contrary, around 1944 Gutenberg and Richter [1] discovered that the size of earthquakes follow a power law distribution: an earthquake of energy E has a probability to occur given by
P (E) ∼ E −τ

(1)

where τ is close to 1. Such power law behaviour is quite special since an average and width cannot be deﬁned. A particularly nasty property is that the probability decays only as a power law, while in the Gaussian case it decays faster than exponentially. As a result there is a non-negligible chance for a huge earthquake.
Many systems in nature display dynamics dominated by similar punctuated behaviour, which we call here generically ‘avalanches’. Other examples are: snow-avalanches [2], forest ﬁres, rain fall [3], stock-market indices [4] and the extinction of species in biology [5]. In all these cases, due to a power law distribution function, there is a ﬁnite chance for very big, catastrophic events. In 1988, Bak, Tang and Wiesenfeld [6] introduced the Self-Organized Criticality (SOC) model; it was made accessible for a broad audience by the book
“How Nature Works” [7]. In fact, apart from the power law behaviour, there are now many more criteria [8] for SOC behaviour, which enable a more stringent test.
Although there has been a very signiﬁcant amount of numerical simulations on SOC systems, there are only very few controlled experimental

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R.J. Wijngaarden et al.

investigations. Up to now experimental work was much hampered by the difﬁculty to determine whether a particular system is SOC or not. In fact, for sand piles both SOC and non-SOC behaviour has been reported [9] (based on power law scaling of avalanches only).
We study experimentally a 3-dimensional pile of rice with a 1 × 1 m2 ﬂoor area. Using the more stringent criteria, we discovered [10] that rice behaves as SOC: not only is the avalanche distribution a power law, but also ﬁnite size scaling (FSS) is obeyed: the maximum avalanche size scales in a particular manner with the size of the system, as predicted by SOC-theory.
We determine the exponent (power) of the power law distribution function as well as the fractal dimensions of the avalanche cluster (avalanche exponents).
On the other hand, we also determine the growth and roughness exponents
(surface exponents) for the rough surface that remains after many avalanches.
SOC theory does not ﬁx the exact values of the individual exponents, but does predict scaling relations e.g. between the avalanche and surface exponents.
We ﬁnd that as far as we tested these relations, they are obeyed for our experimental data.
In addition, SOC theory predicts that a system which is not yet ‘critical’ moves in a particular manner toward the SOC state described by the so-called gap-equation. This approach is characterized by another exponent, the value of which, interestingly, is given through another scaling relation containing only values for the SOC state [11]. Also this relation is obeyed for our pile.
After introducing our experimental set-up, we review these ideas and corresponding experimental work in more detail below. We conclude by some
(experimentally tested) ideas on how to prevent big catastrophes.

2 Experiment: A Big Rice Pile
For the experiments we use long grained rice with dimensions of typically
2 × 2 × 7 mm3 , similar to rice A of [12]. Our experimental set-up is shown in
Fig. 1. It consists of a rice pile of 1 × 1 m2 ﬂoor area on the top of which rice rains down from a linear source. Uniformity within 5% in the distribution along this line is guaranteed by the use of a distributor board, see Fig. 2.
By the distributor board the stream of rice is continuously split such that at the bottom we end up with a row of 64 uniformly distributed sub-streams.
A plastic ﬂap at the bottom is used for further spreading and to slow down the rice before it impinges onto the surface of the pile.
To rain down at a uniform rate, the rice is fed to the distribution board from a mechanically stirred funnel, from which the rice emerges at an approximately constant rate of typically 1500 grains per image (taken at 30 s interval), distributed over the whole width of the distributor board.
To monitor the shape of the pile as function of time, a set of coloured
(red, green and blue) lines (see Fig. 3) is projected onto the pile using an overhead projector. Photographs are taken from a diﬀerent viewpoint with

Nonlinear Dynamics and Fractal Avalanches in a Pile of Rice

319

Fig. 1. The rice set-up with 1×1 m2 ﬂoor area. Rice rains down from the distributor board onto the top of the pile. The 3-dimensional shape of the pile is reconstructed from the shape and position of a set of coloured lines projected onto the pile

Fig. 2. The binary distributor board used to create a uniform line source

respect to the projector, such that the shape of the pile can be reconstructed using the geometry of stereoscopy.
Typically an experimental run lasts 4 hr, with a picture being taken every 30 s. The pictures are taken with a digital camera with a resolution of 2048 × 1536 pixels. For each picture the centre of gravity of each line is detected (see Fig. 3, lower panel), from which the 3-dimensional surface is reconstructed. From the 3-dimensional surfaces, the roughening properties (see
Sect. 3 below) and the avalanche properties (see Sect. 4 below) are calculated.

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Fig. 3. (Top panel) detail of the set of coloured lines projected onto the pile.
(Bottom panel) the centre of gravity of the lines as recognized by the software

3 The Rough Surface of the Pile
The surface of the rice pile is not completely smooth, but shows surface ﬂuctuations, which are mainly due to the occurrence of avalanches that remove material from a more elevated part of the pile and deposit it at a lower position. Since the avalanches have all kinds of sizes, they lead to surface ﬂuctuations of all kinds of size.
A common way to analyze such ‘rough’ surfaces is to consider the spatial and temporal dependences of the root-mean-square (RMS) ‘width’ of the
¯
surface. For this we ﬁrst ﬁt a plane h (x, y, t) to the surface h (x, y, t) of the pile. The RMS deviations w (L, t) of the surface with respect to this plane are then easily calculated from

w (L, t) =

1
L2

1
2

L

¯ h (x, y, t) − h (x, y, t)

2

.

(2)

x,y=1

It is well known [13], that initially this width grows as a power law of time w (L, t) ∼ tβ where β is called the growth exponent, while at later times, when lateral correlations span the whole size of the system L, this width grows as a power law of system size w (L, t) ∼ Lα where α is called the roughness exponent. Hence the exponents α and β can be obtained from the slope in a log-log plot of w vs. L and t respectively, see Fig. 4. In fact, a

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321

Fig. 4. (a) Determination of the roughness exponent from the spatiotemporal correlation function C (L, 0) and the distribution function σ (k) as deﬁned in the text.
The line indicates the average slope, which deﬁnes the value for the roughness exponent obtained. (b) Determination of the growth exponent from the spatiotemporal correlation function C (0, t) and the distribution function σ (ω) as deﬁned in the text. The line indicates the average slope, which deﬁnes the value for the growth exponent obtained

more accurate way to determine these exponents was used for this ﬁgure. It is based on the correlation function
C (L, t) =

[h (x + ξ, y + η, t + τ ) h (ξ, η, τ )]

1
2

L,τ

(3)

where the · τ indicates averaging over all τ > 0, and where · L indicates averaging over all points (ξ, η) and all (x, y) at a radius L from the origin.
The behaviour of this correlation function is similar to that of the width i.e.
C (0, t) ∼ tβ and C (L, 0) ∼ Lα . Alternatively [14], one may also start with the radially averaged power spectrum S (k) of the surface deﬁned by
ˆ
S (k) = h (kx , ky )

2

(4)

ˆ where h denotes the 2-dimensional Fourier transform of the surface and k 2 =
2
2 kx + ky . The corresponding spatial distribution function σ (k) behaves in the same manner [14] as the correlation function C (x, t) where k σ 2 (k) =

S (κ) κ dκ .

(5)

0

Similar relations are deﬁned for the temporal distribution function σ (ω).
In Fig. 4 we show this scaling behaviour.
The resulting exponents are α = 0.42(3) for the roughness exponent and β = 0.28(3) for the growth exponent. Before discussing these values, we shall now turn our attention to the avalanche behaviour.

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4 Avalanches on the Rice Pile
A typical avalanche is shown in Fig. 5. The size and shape of the avalanches can be determined from the height diﬀerence of the surface between two consecutive images. In this manner we determine the size of the avalanches
(as a volume, which can be expressed in the number of rice grains participating) and also the shape of the avalanches. In particular, we determine the fractal dimension D of the avalanche cluster and also the fractal dimension dB of the area on which the avalanche happened: this area is a projection of the avalanche shape on the surface of the pile. Both fractal dimensions are determined using box counting [15].

Fig. 5. Typical avalanche on the rice pile. The picture shows the avalanche as the diﬀerence in height between two successive images

A plot of avalanche size as a function of time, presented in Fig. 6, clearly shows the punctuated behaviour.
The avalanches were determined in the central part of the pile with a size of 600 × 600 mm2 . In Fig. 7a we plot the size distribution for the avalanches in this area, but also for subsets of this area.
We observe power law behaviour with a size distribution exponent τ =
1.21 (2) (corresponding to the straight line), but clear deviations from this behaviour occur above a certain size, which is related to the size of the area of observation. Interestingly, SOC-theory makes a precise prediction for this deviation: it occurs because the avalanche ‘feels’ the size of the system L, i.e. because the linear size of the avalanche becomes comparable to L. Clearly

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Fig. 6. Size of the avalanches as a function of time in the steady state, i.e. after the system has been left running for a long time. Results for three experiments are shown. Note the punctuated behaviour

Fig. 7. (a) Size distribution of the avalanches. P (s, L) is the number of avalanches of size s that were observed while monitoring a window of size L × L (b) Finite size scaling of the size distribution of the avalanches. Note the nice data-collapse: the deviation from power law behaviour starts from a size s ∼ LD where D is the fractal dimension of the avalanche volume

this happens for s ≈ LD and hence the deviation should scale with s/LD .
This is veriﬁed in Fig. 7b , where, in addition, the vertical axis was multiplied by sτ to take out the power law. Clearly a very nice data-collapse is observed conﬁrming the ”ﬁnite size scaling”-prediction of SOC theory. This data collapse yields an accurate value for the exponents: D = 1.99 (2) and τ = 1.21 (2).

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5 Relation Between Avalanches and Surface
An avalanche disturbs and changes the surface on which it occurs and intuitively one might expect that there exists a relation between the properties of an avalanche and the surface that remains after the avalanche has taken place. On the other hand, avalanches are quite diﬀerent from granular piles, and hence a comparison is not trivial. Below we discuss two relations between the statistical properties of avalanches and the surface that they leave behind.
Subsequently, we verify whether these are obeyed in our experiments.
The ﬁrst and most simple scaling relation is found by calculating the volume of a (fractal) avalanche. By deﬁnition, this volume is LD since D is the fractal dimension of the avalanche cluster (which can be determined directly in our experiment). On the other hand, the volume should also be equal to the fractal surface times the fractal height. The fractal surface area is the projection of the avalanche cluster on an average (ﬂat) plane trough the pile surface.
The area of this projection is by deﬁnition LdB where dB is the surface fractal dimension of the avalanche cluster, which can be easily obtained in our experiment. The height or thickness of an avalanche is obtained by subtracting the heights (in the direction perpendicular to the average plane, mentioned above) of the piles before and after the avalanche. Since the avalanche modiﬁes the pile only locally, this fractal thickness scales as the surface roughness and is proportional to Lα . Combining these ingredients yields LD ∼ LdB Lα from which we obtain the scaling relation
D = dB + α .

(6)

The second scaling relation follows from the fact that the deviation from power law behaviour seen in Fig. 7 occurs because avalanches above a certain size ‘feel’ that the pile is ﬁnite. We shall now make this more explicit.
As stipulated above, the time evolution of an initially ﬂat surface subject to roughening is such that at short times its root-mean-squared width w increases with time as w ∼ tβ . After some time, however, lateral correlations extend over the whole area of the pile and w does not increase anymore: it is limited to a pile-size dependent value w ∼ Lα , where L is the linear size of the pile and α is the roughness exponent. At the cross-over time t× , both relations hold from which
(7)
t× ∼ Lα/β .
On the other hand, since this cross-over is due to correlations that start to span the whole system, t× is also the moment when the ﬁrst avalanche occurs that spans the whole pile; by deﬁnition the size of this avalanche is s× . Since we seed the pile with a constant rate, t× also is proportional to the amount of mass M we must add to the pile to obtain such a pile-spanning avalanche. However, before we create an avalanche of size s× , many smaller avalanches have occurred, adding to the amount of material that we have to add before a pile-spanning avalanche takes place. Thus the total mass M is

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Fig. 8. The scaling relation D (2 − τ ) = α/β is derived by comparing the time needed for correlations to span the whole pile and the time needed to create avalanches that span the whole pile (see text)

equal to the integral of the size distribution function up to s× , i.e. the shaded area in Fig. 8. This gives s× t× ∼ M =

s×

sP (s) ds ∼

0

0

s s−τ ds ∼ s2−τ .
×

(8)

In the ﬁnite size scaling analysis we found that s ∼ LD , hence we obtain t× ∼ s2−τ ∼ LD(2−τ ) .
×

(9)

Combination with (7) yields the exponent relation
D (2 − τ ) =

α
.
β

(10)

Equations (6) and (10), which were previously derived by Paczuski et al. [8], oﬀer an interesting possibility: one can calculate the roughness and growth exponents, α and β, from the avalanche properties only. In the table below we compare the values [10] from such an analysis with those obtained above from a direct roughness analysis of the surface of the pile: α β from roughness analysis 0.42 (3) 0.28 (3) from avalanche analysis 0.41 (3) 0.26 (2)
Clearly, an excellent agreement is found, which supports the underlying assumption i.e. that SOC theory yields a valid description of the avalanche behaviour in our pile.

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It is interesting to note that the values we ﬁnd for the roughening of our rice pile are very close to those observed for Kardar-Parisi-Zhang [16] (KPZ) systems. For a 2-dimensional interface it was found from simulations that for KPZ systems α = 0.39 and β = 0.24, rather close to the values given in the table above. In addition, if we consider the roughening of contour lines
(lines of equal height) on our pile, we ﬁnd for these 1-dimensional interfaces α = 0.48 (3) and β = 0.33 (3), while it can be shown [13] rigorously that for
1-dimensional KPZ α = 1/2 and β = 1/3. Strangely enough, our system is very unlike a KPZ system: the latter is governed by a Langevin (diﬀerential) equation, which describes the development of the interface in a deterministic and local fashion (although the eﬀects of disorder are very important and must be included in the description). To the contrary, our system seems much more random and non-local due to the occurrence of avalanches. Nevertheless the most likely manner of change of our pile is by increase or decrease of its height, which is most easy in a direction that is parallel to the local normal to the surface. Exactly this is also the main ingredient in the derivation of the KPZ equation: growth always proceeds along the local normal. Possibly, the KPZ model extends to non-Langevin systems obeying this same growth rule. 6 Avalanches as Spatiotemporal Fractals
In this section we discuss the spatiotemporal structure of the avalanches.
It is quite natural under our condition of constant seeding of the pile, to consider all avalanches together as belonging to one superavalanche, which is punctuated in time. In fact, using similar theoretical ideas as above, one can make a testable prediction for the power spectrum of this superavalanche.
To derive a scaling relation for the avalanche power spectrum exponent, let us consider the development of an avalanche, see Fig. 9. Shown are ﬁve successive stages of the development of an avalanche cluster, which is shown full-grown at the top. At the bottom of the ﬁgure, the substrate area of this avalanche is shown. At a certain moment during the avalanche a site is changed (indicated in the middle three pictures by a red colour) or not
(blue). If we consider a single site, then the behaviour as a function of time is punctuated and a time-line through such a site (indicated by the vertical line) is a fractal. We deﬁne the length of this time-line as T θ where T is the total time needed to create the full-grown avalanche and θ is the fractal dimension of the time-line (to be calculated below). If the linear size of the full-grown avalanche cluster is L (note the change in deﬁnition of L), then (according to similar arguments as given above concerning t× ) the time needed to create such correlated cluster is T ∼ Lα/β . Hence the length of the time-line is
T θ ∼ Lθα/β . The full-grown cluster is created during the time T and has a volume that is easily calculated from its substrate area LdB and its height, corresponding to the number of times T θ that activity occurred during its

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Fig. 9. Schematic representation of the spatiotemporal structure of an avalanche.
Shown are ﬁve successive stages of development of an avalanche cluster, shown in full-grown state at the top. At the bottom the substrate area of the full-grown avalanche is shown. As time progresses there is activity (grey) or not (black) at a particular site. A time line running through one site is indicated by the vertical line. For further explanation see text

growth. From this we obtain for the volume of the full-grown avalanche cluster
V = LdB × Lθα/β . On the other hand, this size is also LD , since D is the fractal dimension of the full-grown avalanche. Combining yields the scaling relation 328

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D = dB + θ

α
.
β

(11)

Solving this for θ yields θ = (D − dB )

β
=β
α

(12)

where we have used (6). Hence the fractal dimension of the time line through a single site has the same value as the growth exponent.
We now determine the consequence for the power spectrum. Consider ﬁrst the activity-function A (t) for a certain timeline, such as in Fig. 9. A (t) is one at each moment of activity and zero at all other times. On the average the total number of ones of A during a time t, which we call n (t) is given by t A (τ ) dτ ∼ tβ

n (t) =

(13)

0

since the number of points on a length t of time-line with fractal dimension β is given by tβ .
By deﬁnition, the temporal correlation function for the avalanche is
C (t) = A (τ ) A (τ + t) τ (note that this C (t) for activity during avalanches is diﬀerent from the C (t) calculated above for rough surfaces), where the average is over all starting times τ and all time lines. In this context, C (t) is also called [8] the all-return probability Pall (t) i.e. the probability for a site that is active at τ = 0 to become active again at τ = t. Hence [8] on the average n (t + 1) = n (t) + C (t). Thus
C (t) ∼

dn (t)
∼ tβ−1 . dt (14)

And for the power spectrum [17]:
S (f ) ∼ f −β .

(15)

An experimental power spectrum is shown in Fig. 10. The red line indicates power law behaviour with a negative power of 0.27 (3). According to (15) one expects a value equal to the growth exponent β = 0.28 (3) as determined above. Clearly we ﬁnd a good agreement supporting again the applicability of SOC theory.

7 How to Prevent Avalanches
It is rather worrying that despite vigorous eﬀorts to prevent snow avalanches by controlled explosions, accidents still occur. In fact, the same applies to forest ﬁres (notably in the National Parks in the US), where ‘controlled’ burning sometimes gets out-of-hand and starts large ﬁres. In fact, it is a

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Fig. 10. Power spectrum of the avalanches of Fig. 6. The red line indicates power law behaviour with a negative power of 0.27 (3) , very close to the value of the growth exponent, 0.28 (3) , in good agreement with SOC theory

property of SOC that ‘small’ disturbances may lead to big avalanches, making control very diﬃcult.
In Fig. 11 we replot the data of a comparative study [2] of snow avalanches in two regions of the US, one where controlled explosions were used to try to prevent the occurrence of large avalanches and another, where no explosions were made. Clearly, the power law distributions are very similar, with nearly equal slope. The disturbed region shows even slightly worse behaviour: the sizes are slightly larger and the decay is slightly slower, making large avalanches relatively more likely.
To investigate another method for the prevention of devastating avalanches, we study the transient state. This is the state of the system before it reaches criticality. The rice pile was prepared manually to be at an angle much smaller than the critical angle (we refer to this state hereafter as ‘ﬂat’), after which the measurement was immediately started. A detailed analysis of the multi-scaling properties of the temporal correlation function was made [18], but here we shall concentrate on the avalanche properties.
The punctuated behaviour of the avalanches is shown in Fig. 12, where we see that initially large avalanches do not occur. Indeed averaging the avalanche sizes over 100 consecutive time-steps shows a linear increase with time, see Fig. 13, while the system is moving from the prepared ‘ﬂat’ state towards steady-state SOC behaviour.

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Fig. 11. Replot of the data of Birkeland and Landry [2], showing that the avalanche size distribution for snow avalanches is slightly ‘worse’ for the region where attempts were made to prevent large avalanches by controlled explosions

Fig. 12. Avalanche volume versus time immediately after preparing the pile manually in a rather ﬂat state. Clearly, in the beginning large avalanches do not occur

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Fig. 13. Size of the avalanches of Fig. 12 averaged over 100 time steps. Clearly the average avalanche size increases linearly with time while the system is developing towards the SOC steady state

Interestingly, SOC theory [8] prescribes how the system should approach the SOC state. We call the critical slope of the rice fc . This is the maximum slope that a rice pile can have without starting to slide. The value fc =
0.92 (1) was determined experimentally [11] (i) by slowly tilting a small box with rice until it started to slide and (ii) from the maximum slope observed anywhere in our large pile during many experiments.
We call the maximum slope observed anywhere in the pile at a certain instant in time G. SOC theory describes how G approaches fc after starting from a ‘ﬂat’ state. In particular, it predicts that the average avalanche size ∆V diverges as the SOC state is approached according to
−γ
∆V = (fc − G) . Combining this with the experimental observation that
∆V ∼ t yields fc − G ∼ t−1/γ .

(16)

Indeed, from the experimental data we ﬁnd a reasonable agreement to this behaviour, see Fig. 14, except for the very beginning of the experiment.
In fact, according to SOC theory [8, 11], the observed power 1/γ is given by 1
1 + dB /D − τ
=
.
(17)
γ
2−τ

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Fig. 14. Diﬀerence between actual maximal slope anywhere in the pile G and the critical slope fc of rice as a function of time. The red line indicates power law behaviour with slope 0.8 (1)

This is an interesting relation, because it connects the behaviour in the transient non-SOC state as given by γ to exponents (D, dB and τ ) of the steady SOC state. Experimentally, we ﬁnd 1/γ = 0.8 (1), while using the values obtained above for D, dB and τ, one would expect 1/γ = 0.74 (2), which is a nice agreement.
In addition to the change in average avalanche size, also the avalanche size distribution exponent τ changes during the transient state, see Fig. 15.
Initially τ is larger, indicating a faster decrease of the distribution function with increasing avalanche size and hence a smaller chance for large avalanches.
This makes the transient state safer. An alternative to ﬁring explosives in
(near-)critical snow masses is to disturb the snow at an early stage. Thus the snow may be kept from developing SOC behaviour. In addition, due to the larger τ in the transient, even if triggering provokes an avalanche, the chances that it is very big are signiﬁcantly reduced. Similar arguments hold for controlled burning in the prevention of forest ﬁres.

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Fig. 15. Change of the avalanche size distribution exponent τ as a function of time in the transient state. Clearly at early times the distribution function is steeper, making large avalanches less likely

8 Conclusions
Self-organized criticality is a class of models meant to describe punctuated behaviour in naturally occurring phenomena such as earthquakes, avalanches and extinctions of species in biology. These wildly diﬀerent phenomena have characteristic properties, such as the size of events, that all have power law distribution functions. The exponents of these power laws may vary from system to system, however, SOC theory [8] gives explicit relations between the exponents. In fact, many of the exponents can be expressed in terms of only three of them: D, dB and τ .
We have experimentally investigated the statistical properties of avalanches on a 3-dimensional pile of rice and ﬁnd that avalanche sizes are indeed power law distributed. In addition, we ﬁnd that the starting point for deviations from power law behaviour scales as s× ∼ LD , as predicted by SOC theory. Using this and direct imaging of the avalanches, we have directly measured D, dB and τ . From the properties of the surface of the pile, we independently determined its roughness and growth exponents α and β. We have thus veriﬁed that within experimental accuracy, the scaling relations
D = dB + α and D (2 − τ ) = α/β are obeyed.

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In addition, we have veriﬁed that the value of the exponent of the power spectrum of the avalanches is equal to β as predicted by another scaling relation. It is striking that, while we ﬁnd SOC behaviour in our rice pile and are thus able to verify the exponent relations proposed by Paczuski et al. [8] for our system, for sand the occurrence of SOC is highly debated [9]. It is conceivable that a certain minimum amount of disorder is necessary for SOC behaviour to occur. This idea is supported by the experiments of Altshuler et al. on steel balls [19] and by some of us on superconductors [20], and may also explain why in the Oslo experiments [12], long grained rice (like ours) behaved as SOC while more rounded rice did not.
In experiments that started from a manually prepared ‘ﬂat’ pile, we investigated the approach from this ‘ﬂat’ state towards the SOC state. Again, within experimental accuracy, the scaling relation (17) is corroborated by our experiment. In the approach towards SOC, the avalanches are smaller and their distribution is steeper, making the chance for large avalanches much smaller than in the SOC state. It seems that the best strategy to avoid huge avalanches is to stay away from the SOC state. That is indeed possible by continuously disturbing the pile.

References
1. B. Gutenberg, C.F. Richter: Seismicity of the Earth and Associated Phenomena
(Princeton University Press, Princeton 1949); B. Gutenberg, C.F. Richter: Bull.
Seis. Soc. Am. 34, 185 (1944) and B. Gutenberg, C.F. Richter: Ann. Geoﬁs. 9,
1 (1956)
2. K.W. Birkeland, C.C. Landry: Geophys. Res. Lett. 29, 1554 (2002)
3. C.M. Aegerter: Physica A 319, 1 (2003)
4. X. Gabaix, P. Gopikrishnan, V. Plerou, H.E. Stanley: Nature 423, 267 (2003)
5. P. Bak, K. Sneppen: Phys. Rev. Lett: 71, 4083 (1993)
6. P. Bak, C. Tang, K. Wiesenfeld: Phys. Rev. A 38, 364 (1988)
7. P. Bak: How Nature Works (Copernicus, New York 1966)
8. M. Paczuski, S. Maslov, P. Bak: Phys. Rev. E 53, 414 (1996)
9. H.M. Jaeger, S.R. Nagel, R.P. Behringer: Rev. Mod. Phys. 68, 1259 (1996);
S.R. Nagel: ibid. 64, 321 (1992) and references therein
10. C.M. Aegerter, R.G¨ nther, R.J. Wijngaarden: Phys. Rev. E 67, 051306 (2003) u 11. C.M. Aegerter, K.A. L˝rincz, M.S. Welling, R.J. Wijngaarden: Phys. Rev. Lett. o 92, 058702 (2004)
12. V. Frette, K. Christensen, A. Malthe-Sørenssen, J. Feder, T. Jøssang, P.
Meakin: Nature 379, 49 (1996)
13. A.L. Barabasi and H.E. Stanley: Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995)
14. J. Schmittbuhl, J.-P. Vilotte, and S. Roux: Phys. Rev. E 51 (1995) 131; M.
Siegert: ibid. 53, 3209 (1996); J.M. Lopez, M.A.Rodriguez, R. Cuerno: ibid.
56, 3993 (1997)
15. B.B. Mandelbrot: The Fractal Geometry of Nature (Freeman, New York 1983)

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16. M. Kardar, G. Parisi, Y.C. Zhang: Phys. Rev. Lett. 56, 889 (1986)
17. H.J. Jensen: Self Organized Criticality (Cambridge University Press, Cambridge
1998)
18. C.M. Aegerter, K.A. L˝rincz, R.J. Wijngaarden: Europhys. Lett. 67, 342 (2004) o 19. E. Altshuler, O. Ramos, C. Martinez, L.E. Flores, C. Noda: Phys. Rev. Lett.
86, 5490 (2001)
20. M.S. Welling, C.M. Aegerter, R.J. Wijngaarden: cond-mat/0410369.

A Recent Appreciation of the Singular
Dynamics at the Edge of Chaos
E. Mayoral and A. Robledo
Instituto de F´ ısica, Universidad Nacional Aut´noma de M´xico, Apartado Postal o e
20-364, M´xico 01000 D.F., Mexico e robledo@fisica.unam.mx
Summary. We study the dynamics of iterates at the transition to chaos in the logistic map and ﬁnd that it is constituted by an inﬁnite family of Mori’s q-phase transitions. Starting from Feigenbaum’s σ function for the diameters ratio, we determine the atypical weak sensitivity to initial conditions ξt associated to each q-phase transition and ﬁnd that it obeys the form suggested by the Tsallis statistics. The speciﬁc values of the variable q at which the q-phase transitions take place are identiﬁed with the speciﬁc values for the Tsallis entropic index q in the corresponding ξt . We also describe the bifurcation gap induced by external noise and show that its properties exhibit the characteristic elements of glassy dynamics close to vitriﬁcation in supercooled liquids, e.g. two-step relaxation, aging and a relationship between relaxation time and entropy.

1 Introduction
The logistic equation was introduced in 1845 by the Belgian mathematician and sociologist Pierre-Fran¸ois Verhulst to model the growth of populations c limited by ﬁnite resources [1]. The discrete time variable version of Verhulst’s growth law, the logistic map, has become a foundation stone for the theory of nonlinear dynamics. The logistic map is the archetypal example of how the use as starting point of simple nonlinear discrete maps have often led to signiﬁcant developments in the theory of nonlinear dynamical systems [2].
The discovery of the universal properties associated to the renowned perioddoubling and intermittency routes to chaos displayed by the logistic map, similar to those of conventional critical phenomena in statistical physics, triggered, about three decades ago, an upsurge of activity in the ﬁeld and now both routes, as well as many other remarkable features displayed by the logistic map, are well understood.
Here we might argue that at the present time the logistic map is again becoming a prototypical model system. This time for the assessment of the validity and understanding of the reasons for applicability of the nonextensive generalization of the Boltzmann–Gibbs (BG) statistical mechanics [3, 4].
This is a formalism assumed to be appropriate for circumstances where the system is out of the range of validity of the canonical BG theory. And these circumstances are believed in some cases to be a breakdown in the chain of

340

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increasing randomness from non-ergodicity to completely developed phasespace mixing.
The logistic map contains inﬁnite families of critical attractors at which the ergodic and mixing properties breakdown. These are the tangent bifurcations and the accumulation point(s) of the pitchfork bifurcations, the so-called onset of chaos [2]. At each of the map critical attractors the Lyapunov coeﬃcient λ1 vanishes, and the sensitivity to initial conditions ξt for large iteration time t ceases to obey exponential behaviour, exhibiting instead power-law or faster than exponential behaviour. The pitchfork bifurcations are also critical attractors at which the negative Lyapunov coeﬃcient of periodic orbits goes to zero. There are other attractors at which the Lyapunov coeﬃcient diverges to minus inﬁnity, where there is faster than exponential convergence of orbits.
These are the superstable attractors located between successive pitchfork bifurcations and their accumulation point is also the onset of chaos.
Here we review brieﬂy speciﬁc and rigorous results on the dynamics associated to the critical attractors of the logistic map, or of its generalization to z nonlinearity of order z > 1, fµ (x) = 1 − µ |x| , −1 ≤ x ≤ 1, 0 ≤ µ ≤ 2. (The phase space variable is x, the control parameter is µ and the conventional logistic map corresponds to z = 2). Our results relate to the anomalous sensitivity to initial conditions at the onset of chaos, the associated spectrum of
Tsallis q-Lyapunov coeﬃcients, and the relationship of these with the Mori qphase transitions [5], one of which was originally observed numerically for the
Feigenbaum attractor [5, 6]. In particular, we identify the Mori singularities in the Lyapunov spectra with the appearance of special values for the Tsallis entropic index q. As the properties of the logistic map are very familiar and well understood it is of interest to see how previous knowledge ﬁts in with the new perspective.
Tsallis suggested [7] that for critical attractors ξt (deﬁned as ξt (x0 ) ≡ lim∆x0 →0 (∆xt /∆x0 ) where ∆x0 is the initial separation between two orbits and ∆xt that at time t), has the form ξt (x0 ) = expq [λq (x0 ) t] ≡ [1 − (q − 1)λq (x0 ) t]−1/(q−1) ,

(1)

that yields the customary exponential ξt with λ1 when q → 1. In (1) q is the entropic index and λq is the q-generalized Lyapunov coeﬃcient; expq (x) ≡
[1 − (q − 1)x]−1/(q−1) is the q-exponential function. Tsallis also suggested [7] that the Pesin identity K1 = λ1 (where the rate of entropy production K1 is given by K1 t = S1 (t) − S1 (0), t large, and S1 = − i pi ln pi ) would be generalized to Kq = λq , where the q-generalized rate of entropy production
Kq is deﬁned via Kq t = Sq (t) − Sq (0), t large, and where
Sq ≡

pi lnq i 1 pi W

=

1 − i pq i q−1

(2)

is the Tsallis entropy; lnq y ≡ (y 1−q − 1)/(1 − q) is the inverse of expq (y).

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To check on the above, we have recently analysed [8–11] both the pitchfork and tangent bifurcations and the onset of chaos of the logistic map and found that indeed the Tsallis suggestions hold for these critical attractors, though with some qualiﬁcations. For the case of the tangent bifurcation it is important to neglect the feedback mechanism into the neighbourhood of the tangency to avoid the crossover out of the q-exponential form for ξt . As we explain below, for the onset of chaos there is a multiplicity of q-indices, appearing in pairs qj , and Qj = 2 − qj , j = 0, 1, . . ., and a spectrum of q(k,l)
(k,l)
Lyapunov coeﬃcients λqj and λQj for each qj and Qj , respectively. (The superindex (k, l) refers to starting and ﬁnal trajectory positions). The dynamics of the attractor confers the q-indexes a decreasing order of importance.
Retaining only the dominant indices q0 and Q0 yields a quite reasonable description of the dynamics and considering the next few leading indexes provides a satisfactorily accurate account.
Although in brief, we also describe our ﬁnding [12] that the dynamics at the noise-perturbed edge of chaos in logistic maps is analogous to that observed in supercooled liquids close to vitriﬁcation. The three major features of glassy dynamics in structural glass formers, two-step relaxation, aging, and a relationship between relaxation time and conﬁgurational entropy, are displayed by orbits with vanishing Lyapunov coeﬃcient. The known properties in control-parameter space of the noise-induced bifurcation gap play a central role in determining the characteristics of dynamical relaxation at the chaos threshold.

2 Critical Attractors in the Logistic Map
For our purposes it is convenient to recall some essentials of logistic map properties. The accumulation point of the period doublings and also of the chaotic band splittings is the Feigenbaum attractor that marks the threshold between periodic and chaotic orbits, at µ∞ (z), with µ∞ = 1.40115 . . . when z = 2.
The locations of period doublings, at µ = µn < µ∞ , and band splittings, at µ = µn > µ∞ , obey for large n power laws of the form µn − µ∞ ∼ δ(z)−n and µ∞ − µn ∼ δ(z)−n , with δ = 0.46692 . . . when z = 2, which is one of the two Feigenbaum’s universal constants. For our use below we also recall the sequence of parameter values µn employed to deﬁne the diameters dn of the bifurcation forks that form the period-doubling cascade sequence. At µ = µn the map displays a ‘superstable’ periodic orbit of length 2n that contains the point x = 0. For large n the distances to x = 0, of the iterate positions in
(2n−1 )
(0), have constant ratios such 2n -cycle that are closest to x = 0, dn ≡ fµn dn /dn+1 = −α(z); α = 2.50290 . . . when z = 2, which is the second of the
Feigenbaum’s constants. A set of diameters with scaling properties similar to those of dn can also be deﬁned for the band splitting sequence [2]. Other diameters in the 2n -supercycles are deﬁned as the distance of the mth element

342

E. Mayoral and A. Robledo
(2n−1 )

xm to its nearest neighbour fµn
(m+2n−1 )

dn,m ≡ fµn

(xm ). That is
(m)

(0) − fµn (0), m = 0, 1, 2, . . . ,

(3)

with dn,0 = dn . Feigenbaum [13] constructed the auxiliary function σn (m) =

dn+1,m dn,m (4)

to quantify the rate of change of the diameters and showed that it has ﬁnite
(jump) discontinuities at all rationals, as can be seen by considering the variable y = m/2n+1 with n large (and therefore omitting the subindex n). One obtains [2,13] σ(0) = −1/α, but σ(0+ ) = 1/αz , and through the antiperiodic property σ(y + 1/2) = −σ(y), also σ(1/2) = 1/α, but σ(1/2 + 0+ ) = −1/αz .
Other discontinuities in σ(y) appear at y = 1/4, 1/8, etc. As these decrease rapidly in most cases it is only necessary to consider the ﬁrst few.
An important factor of our work is that the sensitivity to initial conditions ξt (x0 ) can be evaluated for trajectories within the Feigenbaum attractor via consideration of the discontinuities of σn (m). If the initial separation is chosen to be a diameter ∆x0 = dn,m and the ﬁnal time t is chosen to have the form t = 2n − 1, then ∆xt = dn,m+2n −1 , and ξt (x0 ) can be written [14] as ξt (m)

σn (m − 1) σn (m)

n

, t = 2n − 1, n large .

(5)

For the clarity of the presentation, we shall only use absolute values of positions so that the dynamics of iterates do not carry information on the self-similar properties of “left” and “right” symbolic dynamic sequences [2].
This choice does not aﬀect the results on the sensitivity to initial conditions.

3 Mori’s q-Phase Transitions in the Logistic Map
During the late 1980’s Mori and coworkers developed a comprehensive thermodynamic formalism to characterize drastic changes at bifurcations and at other singular phenomena in low dimensional maps [5]. The formalism was also adapted to the study of critical chaotic attractors and was illustrated by considering the speciﬁc case of the onset of chaos in the logistic map [5,6,15].
For critical attractors the scheme involves the evaluation of ﬂuctuations of the generalized ﬁnite-time Lyapunov coeﬃcient λ(t, x0 ) =

1 ln t

t−1

ln i=0 dfµ∞ (xi )
,
dxi

t

1.

(6)

Notice the replacement of the customary t by ln t above, as the ordinary
Lyapunov coeﬃcient

A Recent Appreciation of the Singular Dynamics at the Edge of Chaos

1 t→∞ t

t−1

λ1 (x0 ) = lim

ln i=0 dfµ∞ (xi ) dxi 343

(7)

vanishes at µ∞ , t → ∞.
The probability density distribution for the values of λ, P (λ, t), is written in the form
1,
(8)
P (λ, t) = t−ψ(λ) P (0, t), t where ψ(λ) is a concave spectrum of the ﬂuctuations of λ with minimum ψ(0) = 0 and is obtained as the Legendre transform of the ‘free energy’ function φ(q), deﬁned as
1
ln Z(t, q) , t→∞ ln t

φ(q) ≡ − lim

(9)

where Z(t, q) is the dynamic partition function
Z(t, q) ≡

dλ P (λ, t) t−(q−1)λ .

(10)

The ‘coarse-grained’ function of generalized Lyapunov coeﬃcients λ(q) is given by λ(q) ≡ dφ(q)/dq and the variance v(q) of P (λ, t) by v(q) ≡ dλ(q)/dq
[5, 6]. Notice the special weight t−(q−1)λ in the partition function Z(t, q) as this shapes the quantities derived from it. These functions are the dynamic counterparts of the Renyi dimensions Dq and the spectrum f (α) that characterize the geometric structure of the attractor.
A “q-phase” transition is indicated by a section of linear slope mc = 1−qc in the spectrum (free energy) ψ(λ), a discontinuity at qc in the Lyapunov function (order parameter) λ(q), and a divergence at qc in the variance (susceptibility) v(q). For the onset of chaos at µ∞ (z = 2) a single q-phase transition was numerically determined [5, 6, 15] and found to exhibit a value close
−0.7; arguments were provided for this value to be to mc = −(1 − qc ) mc = −(1 − qc ) = − ln 2/ ln α = −0.7555 . . . Our analysis described below shows that the older results give a broad picture of the dynamics at the
Feigenbaum attractor and that actually an inﬁnite family of q-phase transitions of decreasing weights take place at µ∞ .

4 Tsallis Dynamics at the Edge of Chaos
By taking as initial condition x0 = 0 we found that the resulting orbit consists of trajectories made of intertwined power laws that asymptotically reproduce the entire period-doubling cascade that occurs for µ < µ∞ [9, 11]. This orbit captures the properties of the so-called ‘superstable’ orbits at µn < µ∞ , n = 1, 2, . . . [2] (see Fig. 1), and can be used as reference to read all other orbits within the attractor. At µ∞ the Lyapunov coeﬃcient λ1 vanishes and

344

E. Mayoral and A. Robledo

10

0

1

9 13 17

5
2

11 15 19
10 14
18

7

3
6
4

12

−1

10

20

8
16

xτ 10

−2

−3

10

−4

10

0

10

1

10

τ

10

2

10

3

Fig. 1. Absolute values of positions in logarithmic scales of iterations τ for a trajectory at µ∞ with initial condition x0 = 0. The numbers correspond to iteration times. The power-law decay of the time subsequences can be clearly appreciated
(k,l)

in its place there appears a spectrum of q-Lyapunov coeﬃcients λq . This spectrum was originally studied in [6] and [15] and our recent interest has been to study its properties in more detail to examine their relationship with the Tsallis statistics. Recent analytical results about the q-Lyapunov coeﬃcients and the q-generalized Pesin identity are given in [9] and [11].
Now, consider a ﬁrst approximation to the function σn (m) for n large, applicable to general non-linearity z > 1. This is to assume that half of the diameters scale as α (as in the most sparse region of the attractor) while the other half scale as α0 = αz (as in the most crowded region of the attractor).
This approximation captures the eﬀect on ξt of the most dominant trajectories within the attractor. With these two scaling factors σn (m) becomes the periodic step function
⎧
0 < m ≤ 2n−1 ,
⎪ α0 = αz ,
⎪
⎨ n−1 1
< m ≤ 2 · 2n−1 , α, 2
=
(11) z n−1
< m ≤ 3 · 2n−1 , σn (m) ⎪ −α0 = −α , 2 · 2
⎪
⎩
−α ,
3 · 2n−1 < m ≤ 4 · 2n−1 , . . .
Use of (11) into (5) for the sensitivity ξt (m) yields the result

A Recent Appreciation of the Singular Dynamics at the Edge of Chaos

ξt (m) =

α−(z−1)n , m = (2k + 1)2n−1 , α(z−1)n , m = (2k + 2)2n−1 ,

345

(12)

where k = 0, 1, . . . and where the ﬁnal observation time is of the form t = l2n − 1, l = 1, 2, . . .. With the introduction of the total time variable τ ≡ m + 1 + t, (12) can be rewritten in terms of the q-exponential functions
(k,l)

ξτ (m) =

[1 + (1 − q0 )λq0 τ ]1/(1−q0 ) , m = (2k + 2)2n−1 , n−1 (k,l)
,
[1 + (1 − Q0 )λQ0 τ ]1/(1−Q0 ) , m = (2k + 1)2

where q0 = 1 −

ln 2 ln 2
=1−
, ln α0 /α
(z − 1) ln α

(14)

(z − 1) ln α
,
(k + 2l + 1) ln 2

(15)

(k,l) λ q0 =

Q0 = 1 +
(k,l)

(13)

ln 2 ln 2
=1+
, ln α0 /α
(z − 1) ln α

(16)

2(z − 1) ln α
.
(2k + 4l + 1) ln 2

(17)

λQ0 = −

Notice that Q0 = 2 − q0 . For z = 2 one obtains Q0 1.7555 and q0 0.2445, this latter value agrees with that obtained in several earlier studies [7,9,11,16].
The two scales considered describe correctly only trajectories that start at the most sparse region of the multifractal (x 0) and terminate at its most crowded region (x
1), or the inverse. (This is why we obtain the two conjugate values q and Q = 2 − q, as the inverse of the q-exponential function satisﬁes expq (y) = 1/ exp2−q (−y)). The vertical lines in Fig. 2a represent the
(k,l)

(k,l)

ranges of values obtained when z = 2 for λq0 and λQ0 . See [14] for more details. We consider the next discontinuities of importance in σn (m). Independently of the number of discontinuities taken into account one obtains qexponential forms for ξt . The value of σ(y = m/2n+1 ) at y = 1/4 measures
1/α1 , with α1 5.4588 for z = 2, and it is associated to one ‘midway’ region between the most crowded and most sparse regions of the attractor (the other
‘midway’ region being associated to σ(3/4)). With three scaling factors, α, α0 and α1 , we have three values for the q index, q0 , q1 and q2 (together with the conjugate values Q0 = 2 − q0 , Q1 = 2 − q1 and Q2 = 2 − q2 for the inverse trajectories). To each value of q there is a set of q-Lyapunov coeﬃcients running from a maximum λqi ,max to zero (or a minimum λQi ,min to zero).
The results when z = 2 for the ranges of values obtained for the q-Lyapunov coeﬃcients are shown as the vertical lines in Figs. 2(a), 3(a) and 4(a). Similar results are obtained for the case of four discontinuities in σn (m), etc.

346

E. Mayoral and A. Robledo

λ (q )

(a)

1.323
0

−2.646 q 0= 0.2445

Q0=1.7555 ψ(λ) (b)

φ (q)

q

(c) m c= −(1−q 0 )

Mc= λ q0 ,max = 1.323
0

M c= λ Q0,min
= −2.646 q0 =0.2445

Q0 =1.7555

q

m c= ( 1−q 0 ) λ Q ,min
0

0

λ q0,max

λ

Fig. 2. q-phase transitions with index values q0 and Q0 = 2 − q0 obtained from the main discontinuity in σn (m). See text for details

5 A Family of q-Phase Transitions at the Edge of Chaos
As a function of the variable −∞ < q < ∞ the q-Lyapunov coeﬃcients obtained in the previous section are functions with two steps with the jumps located at q = qi = 1 − ln 2/ ln αi (z)/α(z) and q = Qi = 2 − qi . Immediate contact can be established with the formalism developed by Mori and coworkers and the q phase transition obtained in [6]. Each step function for λ(q) can be integrated to obtain the spectrum φ(q) (λ(q) ≡ dφ/dq) and from this its Legendre transform ψ(λ) (≡ φ − (1 − q)λ). We illustrate this ﬁrst with σn (m) obtained with two scale factors, as in (11). We show numerical values for the case z = 2. From (14) to (17) we obtain
⎧
⎨ λq0 ,max , q ≤ q0 = 1 − ln 2/(z − 1) ln α 0.2445 , q0 < q < Q0 ,
(18)
λ(q) = 0,
⎩
λQ0 ,min , q ≥ Q0 = 2 − q0 1.7555 , where λq0 ,max = ln α/ ln 2 1.323 and λQ0 ,min = −2 ln α/ ln 2 −2.646.
The free energy functions φ(q) and ψ(λ) that correspond to (18) are given by ⎧
⎨ λq0 ,max (q − q0 ), q ≤ q0 , q0 < q < Q0 , φ(q) = 0,
⎩
λQ0 ,min (q − Q0 ), q ≥ Q0 ,

A Recent Appreciation of the Singular Dynamics at the Edge of Chaos

λ (q )

347

(a)

2.24996

0
−1.12498
q1= 0.1111

(b)

φ (q)
0

Q1=1.8889

q

(c)

ψ(λ) m c =−(1− q1 )

Mc= λ q1 ,max = 2.24996
M c= λ Q1,min
= −1.12498 q1 =0.1111

Q1 =1.8889

q

m c= ( 1−q 1 ) λ Q ,min

0

1

λ q ,max
1

λ

Fig. 3. q-phase transitions with index values q1 and Q1 = 2 − q1 obtained from the main two discontinuities in σn (m). See text for details

and ψ(λ) =

(1 − Q0 )λ, λQ0 ,min < λ < 0 ,
(1 − q0 )λ, 0 < λ < λq0 ,max .

We show these functions in Fig. 2. The constant slopes of ψ(λ) represent the q-phase transitions associated to trajectories linking two regions of the attractor, that in this case are its most crowded and most sparse, and their values 1 − q0 and q0 − 1 correspond to those obtained for the q-exponential ξt (13). The slope q0 − 1 −0.7555 coincides with that originally detected by Mori and colleagues [5].
When we consider also the next discontinuity of importance in σn (m), at σ(1/4) = 1/α1 , we obtain a set of two q-phase transitions for each of the three values of the q index, q0 , q1 and q2 . We show in Figs. 2, 3 and 4 the functions λ(q), φ(q), and ψ(λ) obtained for these three cases. The parameter values for the q-phase transitions at 1 − q0 and q0 − 1 appear again, but now we have also two other sets at 1 − q1 and q1 − 1, and at 1 − q2 and q2 − 1, that correspond, respectively, to orbits that link a ‘midway’ region of the attractor with the most sparse region, and with the most crowded region of the attractor.

348

E. Mayoral and A. Robledo

λ(q)

(a)

0.1986
0
−0.3972 q2= −4.0348 Q2=6.034

(b)

φ (q)
M c= λ q

2 ,max

0

q

(c)

ψ(λ) m c −(1 −q2 )
=

= 0.1986
Mc = λ Q2 ,min
= −0.3972

q 2 =−4.0348 Q2 =6.034

q

mc =(1 −q2 ) λ Q , min
2

0

λ q ,max
2

λ

Fig. 4. q-phase transitions with index values q2 and Q2 = 2 − q2 obtained from the main two discontinuities in σn (m). See text for details

6 Noisy Dynamics at the Edge of Chaos
Consider now the logistic map z = 2 in the presence of additive noise xt+1 = fµ (xt ) = 1 − µx2 + χt ε, −1 ≤ xt ≤ 1, 0 ≤ µ ≤ 2 , t (19)

where χt is the random variable with average χt χt = δtt , and ε measures the noise intensity [2,17]. Except for a set of zero measure, all the trajectories with µ∞ (ε = 0) and initial condition −1 ≤ x0 ≤ 1 fall into the attractor with fractal dimension df = 0.5338 . . .. These trajectories represent nonergodic states, since as t → ∞ only a Cantor set of positions is accessible out of the total phase space. For ε > 0 the noise ﬂuctuations erase the ﬁne features of the periodic attractors as these widen into bands similar to those in the chaotic attractors, yet there remains a deﬁnite transition to chaos at µ∞ (ε) where the Lyapunov exponent λ1 changes sign. The period doubling of bands ends at a ﬁnite value 2N (ε) as the edge of chaos transition is approached and then decreases at the other side of the transition. This eﬀect displays scaling features and is referred to as the bifurcation gap [2, 17]. When ε > 0 the trajectories visit sequentially a set of 2n disjoint bands or segments leading to a cycle, but the behaviour inside each band is completely chaotic. These trajectories represent ergodic states as the accessible positions have a fractal

A Recent Appreciation of the Singular Dynamics at the Edge of Chaos

349

dimension equal to the dimension of phase space. Thus the removal of the noise ε → 0 at µ∞ leads to an ergodic to nonergodic transition in the map.
In the absence of noise (ε = 0) the diameter positions x2n = dn = α−n visited at times τ = 2n by the trajectory starting at x0 = 1 is given by the
Feigenbaum ﬁxed-point map solution g(x), xτ = g

(τ )

(x0 ) = τ −1/(1−q0 ) g(τ 1/(1−q0 ) x0 ) ,

(20)

that in turn is obtained from then → ∞ convergence of the 2n th map composition to (−α)−n g(αn x) with α = 21/(1−q0 ) . When x0 = 0 one obtains in general [9] xτ = g (2l+1) (0)g (2

n−1

)

(0) = g (2l+1) (0) α−n , τ = (2l + 1)2n , l, n = 0, 1, . . .

(21)
When the noise is turned on (ε always small) the 2n th map composition converges instead to
(−α)−n [g(αn x) + χεκn GΛ (αn x)] ,

(22)

where κ a constant whose numerically determined [18, 19] value κ 6.619 is
√
well approximated by ν = 2 2α(1 + 1/α2 )−1/2 , the ratio of the intensity of successive subharmonics in the map power spectrum [2, 19]. The connection between κ and the ε-independent ν stems from the necessary coincidence of two ratios, that of noise levels causing band-merging transitions for successive 2n and 2n+1 periods and that of spectral peaks at the corresponding parameter values µn and µn+1 [2,19]. Following the same procedure as above we see that the orbits xτ at µ∞ (ε) satisfy, in place of (20), the relation xτ = τ −1/(1−q0 ) g(τ 1/(1−q0 ) x) + χετ 1/(1−r) GΛ (τ 1/(1−q0 ) x) ,

(23)

where GΛ (x) is the ﬁrst-order perturbation eigenfunction, and where r =
1 − ln 2/ ln κ 0.6332. So that use of x0 = 0 yields xτ = τ −1/(1−q0 ) 1 + χετ 1/(1−r)

(24)

xt = exp2−q0 (−λq0 t) [1 + χε expr (λr t)]

(25)

or where t = τ − 1, λq0 = ln α/ ln 2 (λq0 ,max of the previous section) and λr = ln κ/ ln 2.
At each noise level ε there is a ‘crossover’ or ‘relaxation’ time tx = τx − 1 when the ﬂuctuations start destroying the detailed structure imprinted by the attractor on the orbits with x0 = 0. This time is given by τx = εr−1 , the time when the ﬂuctuation term in the perturbation expression for xτ becomes ε-independent, i.e.

350

E. Mayoral and A. Robledo
−1/(1−q0 ) xτx = τx
|1 + χ| .

(26)

Thus, there are two regimes for time evolution at µ∞ (ε). When τ < τx the ﬂuctuations are smaller than the distances between adjacent subsequence positions of the noiseless orbit at µ∞ (0), and the iterate positions in the presence of noise fall within small non overlapping bands each around the ε = 0 position for that τ . In this regime the dynamics follows in eﬀect the same subsequence pattern as in the noiseless case. When τ ∼ τx the width of the ﬂuctuation-generated band visited at time τx = 2N matches the distance between two consecutive diameters, dN − dN +1 where N ∼ − ln ε/ ln κ, and this signals a cut-oﬀ in the advance through the position subsequences. At longer times τ > τx the orbits are unable to stick to the ﬁne period-doubling structure of the attractor. In this 2nd regime the iterate follows an increasingly chaotic trajectory as bands merge progressively. This is the dynamical image – observed along the time evolution for the orbits of a single state µ∞ (ε) – of the static bifurcation gap ﬁrst described in the map space of position x and control parameter µ [17, 18].

7 Analogy with Glassy Dynamics
We recall the main dynamical properties displayed by supercooled liquids on approach to glass formation. One is the growth of a plateau and for that reason a two-step process of relaxation, as presented by the time evolution of correlations e.g. the intermediate scattering function Fk [20]. This consists of a primary power-law decay in time t (so-called ‘β’ relaxation) that leads into the plateau, the duration tx of which diverges also as a power law of the diﬀerence T − Tg as the temperature T decreases to a critical value Tg . After tx there is a secondary power-law decay (so-called ‘α’ relaxation) that leads to a conventional equilibrium state [20]. A second characteristic dynamical property of glasses is the loss of time translation invariance, a characteristic known as aging [21]. The time decay of relaxation functions and correlations display a scaling dependence on the ratio t/tw where tw is a waiting time.
A third distinctive property is that the experimentally observed relaxation behaviour of supercooled liquids is described, via regular heat capacity assumptions [20], by the so-called Adam–Gibbs equation, tx = A exp(B/T Sc ) ,

(27)

where the relaxation time tx can be identiﬁed with the viscosity, and the conﬁgurational entropy Sc is related to the number of minima of the ﬂuid’s potential energy surface (and A and B are constants).
Returning to the map, phase space is sampled at noise level ε by orbits that visit points within the set of 2N bands of widths ∼ ε, and this takes place in time in the same way that period doubling and band merging proceeds in the presence of a bifurcation gap when the control parameter is run

A Recent Appreciation of the Singular Dynamics at the Edge of Chaos

351

through the interval 0 ≤ µ ≤ 2. That is, the trajectories starting at x0 = 0 duplicate the number of visited bands at times τ = 2n , n = 1, . . . , N , the bifurcation gap is reached at τx = 2N , after which the orbits fall within bands that merge by pairs at times τ = 2N +n , n = 1, . . . , N . The sensitivity to initial conditions grows as ξt = expq0 (λq0 t) (q0 = 1 − ln 2/ ln α < 1) for t < tx , but for t > tx the ﬂuctuations dominate and ξt grows exponentially as the trajectory has become chaotic and so one anticipates an exponential ξt (or q = 1). We interpret this behaviour to be the dynamical system analogue of the ‘α’ relaxation in supercooled ﬂuids. The plateau duration tx → ∞ as ε → 0. Additionally, trajectories with initial conditions x0 not belonging to the attractor exhibit an initial relaxation stretch towards the plateau as the orbit falls into the attractor. This appears as the analog of the ‘β’ relaxation in supercooled liquids. See [12].
Next, we determine the entropy of the orbits starting at x0 = 0 as they enter the bifurcation gap at tx (ε) when the maximum number 2N of bands allowed by the ﬂuctuations is reached. The entropy Sc (µ∞ , 2N ) associated to the state at µ∞ (ε) has the form
Sc (µ∞ , 2N ) = 2N εs ,

(28)

since each of the 2N bands contributes with an entropy εs where
1

s=−

p(χ) ln p(χ)dχ ,

(29)

−1

and where p(χ) is the distribution for the noise random variable. In terms of tx , given that 2N = 1 + tx and ε = (1 + tx )−1/(1−r) , one has
Sc (µ∞ , tx )/s = (1 + tx )−r/(1−r)

(30)

tx = (s/Sc )(1−r)/r .

(31)

or, conversely,
Since tx ε , r − 1 −0.3668 and (1 − r)/r 0.5792 then tx → ∞ and
Sc → 0 as ε → 0, i.e. the relaxation time diverges as the ‘landscape’ entropy vanishes. We interpret this relationship between tx and the entropy Sc to be the dynamical system analog of the Adam–Gibbs formula for a supercooled
−1
liquid. See [12]. Notice that (31) is a power law in Sc while for structural
−1
glasses it is an exponential in Sc [20].
Last, we examine the aging scaling property of the trajectories xt at µ∞ (ε). The case ε = 0 is more readily appraised because this property is actually built into the position subsequences xτ = g (τ ) (0) , τ = (2l + 1)2n , l, n = 0, 1, . . .. These subsequences are relevant for the description of trajectories that are ‘detained’ at a given attractor position for a waiting period of time tw and then ‘released’ to the normal iterative procedure. We chose the holding positions to be any of those along the top band shown in Fig. 1 for a waiting time tw = 2l + 1, l = 0, 1, . . .. Notice that, as shown in Fig. 1, r−1 352

E. Mayoral and A. Robledo

for the x0 = 0 orbit these positions are visited at odd iteration times. The lower-bound positions for these trajectories are given by those of the subsequences at times (2l + 1)2n (see Fig. 1). Writing τ as τ = tw + t we have that t/tw = 2n − 1 and
(32)
xt+tw = g (tw ) (0)g (t/tw ) (0) or xt+tw = g (tw ) (0) expq0 (−λq0 t/tw ) .

(33)

This property is gradually modiﬁed when noise is turned on. The presence of a bifurcation gap limits its range of validity to total times tw + t < tx (ε) and so progressively disappears as ε is increased. See [12].

8 Concluding Remarks
We have re-examined the dynamical properties at the onset of chaos in the logistic map and obtained further understanding about their nature. We exhibited links between original developments, such as Feigenbaum’s σ function,
Mori’s q-phase transitions and the noise-induced bifurcation gap, with more recent advances, such as q-exponential sensitivity to initial conditions [9], q-generalized Pesin identity [11], and dynamics of glass formation [12]. The dynamics at the edge of chaos is anomalous because it is an incipient chaotic attractor with vanishing ordinary Lyapunov coeﬃcient λ1 . Chaotic orbits possess a time irreversible property that stems from mixing in phase space and loss of memory, but orbits within critical attractors are non-mixing and have no loss of memory. As a classic illustration of the latter case the attractor at the onset of chaos presents dynamical properties with self-similar structure that result in a set of power laws for the sensitivity to initial conditions. We determined exact analytical expressions for ξt .
Our most striking ﬁnding is that the dynamics at the onset of chaos is constituted by an inﬁnite family of Mori’s q-phase transitions, each associated to orbits that have common starting and ﬁnishing positions located at speciﬁc regions of the attractor. Each of these transitions is related to a discontinuity in the σ function of ‘diameter ratios’, and this in turn implies a q-exponential ξt and a spectrum of q-Lyapunov coeﬃcients for each set of orbits. The transitions come in pairs with speciﬁc conjugate indexes q and
Q = 2 − q, as these correspond to switching starting and ﬁnishing orbital positions. Since the amplitude of the discontinuities in σ diminishes rapidly, in practical terms there is only need of evaluation for the ﬁrst few of them.
The dominant discontinuity is associated to the most crowded and sparse regions of the attractor and this alone provides a very reasonable description, as found in earlier studies [7,9,11,16]. Thus, the special values for the Tsallis entropic index q in ξt are equal to the special values of the variable q in the formalism of Mori and colleagues at which the q-phase transitions take place.

A Recent Appreciation of the Singular Dynamics at the Edge of Chaos

353

As described, the dynamics of noise-perturbed logistic maps at the chaos threshold presents the characteristic features of glassy dynamics observed in supercooled liquids. In particular our results are [12]: (i) The two-step relaxation that takes place when ε → 0 is obtained in terms of the bifurcation gap properties, speciﬁcally, the plateau duration tx is given by a power law in the noise amplitude ε. (ii) The map analogue of the Adam–Gibbs law is given also as a power-law relation between tx (ε) and the entropy Sc (ε) associated to the noise widening of chaotic bands. (iii) The trajectories at µ∞ (ε → 0) are shown to obey a scaling property, characteristic of aging in glassy dynamics, of the form xt+tw = h(tw )h( t/tw ) where tw is a waiting time.
The limit of vanishing noise amplitude ε → 0 (the counterpart of the limit
T − Tg → 0 in the supercooled liquid) brings about loss of ergodicity. This nonergodic state with λ1 = 0 corresponds to the limiting state, ε → 0, tx →
∞, for a family of small ε states with glassy properties, which are expressed for t < tx via the q-exponentials of the Tsallis formalism. It has been suggested on several occasions [22, 23] that the setting in which nonextensive statistics appears to come out is linked to the prevalence of nonuniform convergence, such as that involving the thermodynamic N → ∞ and the inﬁnitely large time t → ∞ limits. Here a similar situation happens, that is, if ε → 0 is taken before t → ∞ a nonergodic orbit restrained to the Feigenbaum attractor and with fully-developed glassy properties is obtained, whereas if t → ∞ is taken before ε → 0 a chaotic orbit with q = 1 would be observed.

Acknowledgement
AR is grateful for the support provided by the Verhulst 200 organizers. We also acknowledge support from CONACyT and DGAPA-UNAM, Mexican agencies. References
1. P.-F. Verhulst: Recherches math´matiques sur la loi d’accroissement de la pope ulation, Nouv. Mem. de l’Acad. Roy. des Sciences et Belles-Lettres de Bruxelles
XVIII.8, 1-38, 1845
2. See, for example, H.G. Schuster: Deterministic Chaos. An Introduction, 2nd
Revised Edition (VCH Publishers, Weinheim 1988)
3. C. Tsallis: J. Stat. Phys. 52, 479 (1988)
4. For recent reviews see, Nonextensive Entropy – Interdisciplinary Applications, ed by M. Gell-Mann, C. Tsallis (Oxford University Press, New York 2004). See http://tsallis.cat.cbpf.br/biblio.htm for full bibliography
5. H. Mori, H. Hata, T. Horita, T. Kobayashi: Prog. Theor. Phys. Suppl. 99, 1
(1989)
6. T. Horita, H. Hata, H. Mori, K. Tomita: Prog. Theor. Phys. 82, 897 (1989)

354

E. Mayoral and A. Robledo

7. C. Tsallis, A.R. Plastino, W.-M. Zheng: Chaos, Solitons and Fractals 8, 885
(1997)
8. F. Baldovin, A. Robledo: Europhys. Lett. 60, 518 (2002)
9. F. Baldovin, A. Robledo: Phys. Rev. E 66, 045104(R) (2002)
10. A. Robledo: Physica D 193, 153 (2004)
11. F. Baldovin, A. Robledo: Phys. Rev. E 69, 045202(R) (2004)
12. A. Robledo: Phys. Letters A 328, 467 (2004)
13. M.J. Feigenbaum: Commun. Math. Phys. 77, 65 (1980); Physica 7D, 16 (1983)
14. E. Mayoral-Villa, A. Robledo: submitted
15. G. Anania, A. Politi: Europhys. Lett. 7, 119 (1988)
16. M.L. Lyra, C. Tsallis: Phys. Rev. Lett. 80, 53 (1998)
17. J.P. Crutchﬁeld, J.D. Farmer, B.A. Huberman: Phys. Rep. 92, 45 (1982)
18. J. Crutchﬁeld, M. Nauenberg, J. Rudnick: Phys. Rev. Lett. 46, 933 (1981)
19. B. Shraiman, C.E. Wayne, P.C. Martin: Phys. Rev. Lett. 46, 935 (1981)
20. For a recent review see, P.G. De Benedetti, F.H. Stillinger, Nature 410, 267
(2001)
21. See, for example, J.P. Bouchaud, L.F. Cugliandolo, J. Kurchan, M. Mezard. In
Spin Glasses and Random Fields, ed by A.P. Young (World Scientiﬁc, Singapore
1998)
22. C. Tsallis. In Nonextensive Statistical Mechanics and Its Applications, ed by S.
Abe, Y. Okamoto, Lecture Notes in Physics 560, 3 (Springer, Berlin 2001)
23. C. Tsallis, A. Rapisarda, V. Latora, F. Baldovin. In Dynamics and Thermodynamics of Systems with Long-Range Interactions, ed by S. Ruﬀo, E. Arimondo,
M. Wilkens, Lecture Notes in Physics 602, 140 (Springer, Berlin 2002)

Quantum Chaos Versus Classical Chaos: Why is Quantum Chaos Weaker?
H. Kr¨ger, J.-F. Laprise, G. Melkonyan, and R. Zomorrodi o D´partement de Physique, Universit´ Laval, Qu´bec, Qu´bec G1K 7P4, Canada e e e e hkroger@phy.ulaval.ca 1 Introduction
Chaotic systems occur in nature everywhere. Examples are: the turbulent ﬂow of gases and liquids, e.g. when pouring milk into a cup of coﬀee, in the bath tub, in hurricanes, or in the formation of galaxies. Chaos occurs in biology in the dynamics of populations and species. It occurs in the rate of heart beats, in attacks of epilepsy, and in the brain. It occurs also in chemical reactions. Most of the dynamics occurring in nature is chaotic. The occurrence of non-chaotic systems in nature is more of an exception. The mathematical description of chaos has reached a mature state over the last decade. The tools being used are classical phase space, Lyapunov exponents,
Poincar´ sections, Kolmogorov–Sinai entropy and others. e There is also chaos in the quantum systems [1–3]. People have investigated in quantum physics the analogues of classically chaotic systems. For example, a stadium-shaped billiard is classically chaotic. The corresponding quantum system is a system of ultra-cold atoms bouncing against walls of stadium shape, being created by interaction of the atoms with laser beams.
Quantum chaos has been found to play an important role in dynamical tunnelling [4, 5]. The ﬁngerprints of chaos in quantum systems and the mathematical tools of its description are quite diﬀerent from those used in classical chaos. The necessity for a diﬀerent treatment is due to the nature of quantum mechanics: There is no proper phase space in quantum systems – Heisenberg’s uncertainty principle forbids that a point in phase space (uncertainty zero in position and momentum) exists. Heisenberg’s uncertainty relation is a direct consequence of quantum mechanical ﬂuctuations. A common approach to describe quantum chaos is random matrix theory and the use of energy level spacing distributions [3]. A fundamental conjecture by Bohigas, Giannoni and
Schmit [6] postulates that the energy level spacing distributions possesses a dominant part – which depends on the particular system – and a subleading universal part – independent of the particular system. The universal part gives a level spacing distribution of Wignerian type, if the corresponding classical system is fully chaotic. Such level spacing distribution can be generated also by random matrices of a certain symmetry class. The conjecture has not been rigorously proven yet, but has been veriﬁed and found to be valid in almost all cases.

356

H. Kr¨ger et al. o This approach is successful and very popular. However, it has some shortcomings: First, the conjecture by Bohigas et al. holds strictly only in the case of a fully chaotic system, while in nature most systems are only partly chaotic,
i.e. so-called mixed systems where its classical counterpart has coexistence of regular and chaotic phase space. In such cases the quantum system yields a level spacing distribution, which is neither Wignerian nor Poissonian (where the latter corresponds to a completely regular system). There is no mathematical prediction of the functional form of such distribution. However, a number of interpolations between the Poisson and Wigner distribution have been proposed [7–11]. Second, we may ask: How about the comparison of the classical with the quantum system? And what is the quantitative degree of chaos? How can we answer this when the instruments used to measure chaos are quite diﬀerent for both systems?
Starting from this perspective and having in mind the goal to compare classical with quantum chaos, one may try the following strategy: Find a uniform description of chaotic phenomena, valid for both, classical and quantum systems. In more detail: Starting from nonlinear dynamics and phase space in classical systems, one may look for a suitable quantum analogue phase space.
Starting from random matrix theory and energy level spacing distributions of quantum systems, one may seek a random matrix description and a level spacing distribution of suitable dynamical objects in classical physics. In the following we shall discuss some progress recently made in this direction. Using some of those results, we shall compare for a particular system the chaotic behaviour of the quantum system with the classical system. The numerical analysis shows that the quantum system is globally less chaotic than the classical system. We believe that such ﬁnding is not limited to the particular system. In particular, we want to understand the underlying reason for such behaviour. 2 Cases Where Quantum Chaos Was Found to Be Weaker
First, Casetti et al. [12] considered the N -component Φ4 theory in the presence of an external ﬁeld and in the limit of large N . They used mean ﬁeld theory and observed a strong suppression of chaos in the quantum system, due to quantum corrections causing the system to move away from a hyperbolic ﬁxed point responsible for classical chaos. Second, Matinyan and
M¨ller [13] studied another ﬁeld theoretic model which is classically chaotic, u namely massless scalar electrodynamics. They investigated the corresponding quantum ﬁeld theory using eﬀective ﬁeld theory and loop expansion. They noticed that quantum corrections increase the threshold for chaos due to a modiﬁcation of the ground state of the system. A third example is the kicked rotor which is a classically chaotic system in 1-D. Schwengelbeck and

Quantum Chaos Versus Classical Chaos

357

Fig. 1. Volume of chaotic phase space over total phase space versus energy. Comparison of results from classical action (dotted line) with quantum action (full line)

Faisal [14] considered the corresponding quantum system using Bohm’s interpretation of quantum mechanics to introduce trajectories and a quantum equivalent phase space. They found that the Kolmogorov–Sinai entropy goes to zero in the quantum system, i.e. it is non-chaotic. This approach has also been applied to study chaos in anisotropic harmonic oscillators [15], coupled anharmonic oscillators [16] and the hydrogen atom in an external electromagnetic ﬁeld [17]. Finally, Caron et al. [18] considered the anharmonic oscillator in 2-D, which is classically a mixed chaotic system. Using the concept of the quantum action functional, a quantum analogue phase space has been constructed. As a result, the phase space portrait of the quantum system was found to be slightly but globally less chaotic for all energies (see Fig. 1).
Because the quantum action has been constructed from the classical action by taking into account quantum ﬂuctuations [19–25], hence the softening of chaos in the quantum system must be due to quantum ﬂuctuations. We suspect that such softening eﬀect may not solely show up in chaos. Indeed,
2

1 in the context of looking at a double-well potential V (x) = λ x2 − 8λ tunnelling, we observe that quantum eﬀects cause the quantum potential to be much “weaker” than the classical potential (see Fig. 2), i.e. the potential wells are less pronounced and the potential barrier is much lower for the quantum potential (note that the quantum potential has a triple-well shape).
The shape of the potential for tunnelling translates into the shape of the instantons. Thus it comes as no surprise that the instanton of the quantum

358

H. Kr¨ger et al. o Fig. 2. Comparison of classical potential and quantum potential (multiplied by a scale factor 1.7331) for quartic coupling λ = 1/32. Also shown is the ground state wave function and two lowest energy levels

action (actually a double-instanton) is softer than the classical instanton (see
Fig. 3).

3 Uniform Description of Chaos
Let us recall that the conjecture by Bohigas et al. is about random matrix theory and energy level spacing distributions to describe chaos in quantum systems, while chaos in classical systems is conventionally described in terms of phase space. That is, the tools to describe chaos in classical and quantum physics are diﬀerent. For the purpose to better understand the physical content of this conjecture or eventually to ﬁnd a mathematical proof, it is highly desirable to use the same language, respectively, tools in quantum physics as in classical physics. This can be viewed in two ways: Either one adopts the point of view that chaos should be analysed in terms of phase space. Then a uniform description is achieved by use of classical phase space in classical physics and the quantum analogue phase space in quantum systems. How about the point of view that chaos should be analysed in terms of random matrix theory and level spacing distributions? In the following we propose how to construct such a level spacing distribution, which will play the same role in classical physics as the energy level spacing distribution in the quantum system.

Quantum Chaos Versus Classical Chaos

359

Fig. 3. Comparison of classical instanton with instanton from the quantum action

One may immediately object that the energy level spacing distribution in quantum physics is due to the fact that one considers a system of bound states
(with a “conﬁning” potential), which due to the rules of quantum mechanics gives a discrete spectrum. The discreteness of the spectrum is a quantum eﬀect, i.e., a physical eﬀect. Having in mind to construct a counterpart in classical physics, one may object that classical physics is continuous. There is no discreteness inherent in classical physics. If any discreteness occurs it will be due to some (mathematical) approximation. Hence, how can a level spacing distribution derived from a classical function be discrete and physical? First we want to propose a function and show how to construct a meaningful level spacing distribution. Afterwards, we shall try to answer the last question.
We would like to emphasize two properties: First, the level spacing distribution of the quantum system – in the case of a classically fully chaotic system – corresponds to a Wigner distribution. The type of Wigner distribution is determined by the symmetry of the Hamiltonian. E.g., there is the
Gaussian orthogonal ensemble (GOE), the unitary ensemble and the symplectic ensemble. Thus, we need a function which carries the same symmetries. A function convenient for this purpose is the classical action. Another property to be emphasized is locality. Let us recall that the level spacing distribution,
e.g. the GOE distribution, is invariant under orthogonal transformations. Because an orthogonal transformation maps any orthogonal basis of states onto another orthogonal basis, the level spacing distribution is essentially independent of the particular choice of the basis. That means, we may choose a basis that is quasi-local, i.e., built from square integrable functions which are identically zero everywhere except in a small interval where they are

360

H. Kr¨ger et al. o non zero and constant. Those box functions are almost local. The advantage of locality is the fact that symmetries of the Hamiltonian often have to do with transformations in position space, and locality facilitates the analysis of such symmetries. Why do we need to bother about symmetries? Because the energy level spacing distribution is meaningful only (and gives the Wigner distribution for a chaotic system, respectively a Poissonian distribution for an integrable system) if the energy levels have all the same quantum numbers
(except for the quantum number of energy). For example, for the spectrum of the hydrogen atom, one should take the bound states all with the same angular momentum quantum numbers, e.g., l = m = 0.
3.1 Action Matrix
We consider a system deﬁned by a Lagrange function L(q, q, t). Let S denote
˙
the corresponding action,
T

dt L(q, q, t) .
˙

S[q(t)] =

(1)

0

Let qtraj denote the trajectory, i.e. the solution of the Euler–Lagrange equation of motion. Such trajectory is a function, which makes the action functional stationary. Each trajectory is speciﬁed by indicating the initial and ﬁnal boundary points, i.e., q(t = 0) = qin and q(t = T ) = qﬁ . We assume that those boundary points are taken from some ﬁnite set of nodes, qin , qﬁ ∈ {q1 , . . . , qN }. Thus with each pair of boundary points, (qk , ql ) we traj associate a trajectory qkl (t). Then we introduce an action matrix Σ, where the matrix element Σkl corresponds to the value of the action S evaluated traj along the trajectory qkl (t), traj Σkl = S[qkl (t)] .

(2)

All matrix elements Σkl are real. They are also symmetric, Σkl = Σlk . Thus the action matrix Σ is a Hermitian N × N matrix. Consequently, Σ has a discrete spectrum of action eigenvalues, which are all real, σ(Σ) = {σ1 , . . . , σN } .

(3)

3.2 Symmetry
For the purpose to compute a level spacing distribution from the action eigenvalues, one must ﬁrst address the issue of symmetry. Let us consider for example the harmonic oscillator in 2-D. We may take the coordinates qk to be located on the nodes of a regular grid (reaching from −Λ to +Λ on the x and y axis), with a spacing ∆x = ∆y = a = const. For example, the action of the harmonic oscillator, with the classical trajectory going from xa to xb in time T is given by

Quantum Chaos Versus Classical Chaos

Σa,b =

mω
(x2 + x2 ) cos(ωT ) − 2xa · xb . a b
2 sin(ωT )

361

(4)

This function is invariant under rotations. When choosing the coordinates qk to be located on the regular grid, the continuous symmetry of rotation will become a discrete symmetry of ﬁnite rotations (a group). We have to ﬁnd the irreducible representations of such group and sort the action eigenvalues according to those irreducible representations (this can be done by inspecting the properties of the corresponding eigenvector). Then one has to select a particular representation and retain a subset of eigenvalues in that representation. The action level spacing distribution can then be obtained from the action eigenvalues in such subset.
This procedure is feasible. However, it has two disadvantages. First, ﬁnding the irreducible representations and classifying the eigenvectors accordingly is laborious. More importantly, the fact that one has to work with a subset of eigenvalues only and not the whole ensemble of eigenvalues means a drastic reduction of the size of the statistical ensemble. In other words, the statistics of the resulting level spacing distribution will deteriorate.
For those reasons it would be highly desirable to avoid the above strategy.
This is indeed possible by using the following trick. One can camouﬂage the symmetry by choosing the coordinates oﬀ the nodes of the regular grid. That means, for example to deﬁne new coordinates as follows, deform = qk + qk k

,

(5)

where k denotes a randomly chosen small deformation (in angle and length).
In this way the nodes are irregularly distributed. Consequently, the discrete symmetry of the action matrix Σkl disappears when replacing it by the “deformed” action matrix, deform = action evaluated along the classical trajectory
Σkl
deform deform from node qk to node ql
.

(6)

In doing so we avoid a laborious symmetry analysis and secondly shall have a better statistics!
3.3 Action Level Spacing Distribution
Now we want to construct a level spacing distribution of action levels. We proceed in analogy to random matrix theory and the method of constructing an energy level spacing distribution. For an overview on how to compute level spacing distributions see [26]. One has to separate the dominant system dependent part from the subleading universal part which describes the properly normalized ﬂuctuations. Because we are here interested only in the ﬂuctuation part, we suppress the leading part. One should note that this means to discard all physical information which depends on the particular system.

362

H. Kr¨ger et al. o For example thermodynamical functions can not be computed from the subleading ﬂuctuating part. The strategy to obtain the subleading part is called unfolding. One constructs a ﬁt to the original spectrum and multiplies the spectrum such that on average the mean spacing distribution becomes unity.
Also the integrated level spacing distribution will be normalized to unity.
We have applied this to simple integrable systems in 1-D. For integrable systems one would expect a Poissonian distribution for the action level spacing distribution. Preliminary results are compatible with a Poissonian distribution. The following remarks are in order. The ﬁrst numerical results for integrable systems have to be repeated with precision and analysed carefully.
Second, one wants to see what happens in chaotic systems. Possibly such strategy applied to a fully chaotic system will result in a Wignerian action level spacing distribution. Finding an answer will be computationally much more involved, simply because the action functions for the integrable systems considered above are analytically known, while for a chaotic system (i.e. nonintegrable) this needs to be calculated numerically. Moreover, the numerical precision required needs to be suﬃcient to resolve small ﬂuctuations. In the statistical sense, one is interested in a sample of large size. But that means that after unfolding the ﬂuctuations will become small and hence require a high numerical precision for its resolution. Presently, numerical studies of such question are under way.
Let us get back to the question posed at the beginning of Sect. 3: How can a level spacing distribution derived from a classical function be discrete and physical? In our opinion, the answer lies in the fact that the level spacing distribution is universal, that is, it does not depend, for example, on the parameters of the discrete grid. Diﬀerent grids give the same result. This has been veriﬁed numerically. It also should not depend on the deformation k
(as long it is not too close to zero and the discrete symmetry is restored).
Also this has been veriﬁed numerically and found to be satisﬁed. Thus one could in principle go with the volume of the lattice V = (2Λ)D to inﬁnity and with the lattice spacing ∆x = ∆y to zero. The result should not change, but one would have reached the continuum limit. The discreteness would then disappear. 4 Renormalisation Flow of Parameters of the Quantum Action
In Sect. 2 we have seen examples for the observation that quantum chaos seems to be weaker than chaos in the corresponding classical system. Of course it would be interesting to explore a much wider class of systems in order to see if such observation holds more generally. Here we want to pick one of the above examples, namely the chaotic anharmonic oscillator in 2-D and try to understand why quantum chaos is weaker than classical chaos.
The classical action is given

Quantum Chaos Versus Classical Chaos
T

S=

dt
0

363

1
(x2 + y 2 ) − V (x, y) ,
˙
˙
2m

mω 2 2
(x + y 2 ) + λx2 y 2 = v0 + v2 (x2 + y 2 ) + v22 x2 y 2 .
V =
2

(7)

For λ = 0 the system is reduced to the standard harmonic oscillator, which is integrable. The chaoticity is introduced and controlled by the parameter
λ. Thus small λ causes mild chaos, while large λ makes the system strongly chaotic. The quantum action has been postulated to be of the functional form like the classical action, i.e. the kinetic term of the quantum action may diﬀer in the value of the mass, and the potential of the quantum action should also be local, depend only on coordinates, but may have a diﬀerent functional form. Here let us consider an Ansatz of the following form
T

˜
S=

dt
0

1
˜
(x2 + y 2 ) − V (x, y) ,
˙
˙
2m
˜

˜
V = v0 + v2 (x2 + y 2 ) + v22 x2 y 2 + higher order polynomials .
˜
˜
˜

(8)

As a quantitative measure of the strength of chaos we take the strength of the parameters of the action, in particular, the parameter λ. The parameters of the quantum action can be interpreted as a “renormalisation eﬀect” of the parameters of the classical action. The calculation of those parameters has to be done numerically, following the deﬁnition of the quantum action to be a functional which ﬁts the transition amplitudes [19]. However, in a certain limit, the quantum action is known to be an exact representation of the transition amplitudes and moreover the action is related via diﬀerential equations to the classical action [22]. This limiting case is using imaginary time and let time go to inﬁnity (Feynman–Kac limit). Because we want to obtain an analytical result, we shall use perturbation theory. This means that we consider the regime of small λ = v22 .
In order to simplify the matter, let us start by considering the system in
1-D. Thus we have the potential
V (x) =

1 mω 2 x2 + λx4 ≡ v2 x2 + v4 x4 .
2

(9)

We assume λ to be small, λΛsc v2

1,

(10)

where Λsc introduces a physical length scale, e.g. the analogue of the Bohr radius. According to [23] the following relation between the classical and the quantum potential holds,
2m(V (x) − Egr ) = 2m(V (x) − v0 ) −
˜ ˜
˜

d
˜ ˜ dx 2m(V (x)

2

− v0 )
˜

2m(V (x) − v0 )
˜ ˜
˜

sgn(x) .

(11)

364

H. Kr¨ger et al. o We deﬁne the functions
W (x) = 2m(V (x) − Egr ) ,
U (x) = 2m(V (x) − v0 ) ,
˜ ˜
˜

(12)
(13)

where Egr denotes the ground state energy of the lowest eigenstate of the quantum mechanical system (obtained from the Schr¨dinger equation). Due o to (9), the function W (x) must have the form
W (x) = w0 + w2 x2 + w4 x4 .

(14)

This function is symmetric with respect to parity. Then the function U (x) representing the quantum potential, will be parity symmetric too. We make an Ansatz of the form
U (x) = u0 + u2 x2 + u4 x4 + u6 x6 + · · ·

(15)

The assumption that the expansion parameter λ is small is now expressed by
(0)

w4 ≡ w4

,

1

(16)

Now using (14) and (15) in combination with (11), we obtain w0 + w2 x2 + w4 x4 = u2 x2 + u4 x4 + u6 x6 + · · ·
2u2 + 4u4 x2 + 6u6 x4 + · · ·
√
−
2 u2 + u4 x2 + u6 x4 + · · ·

for x > 0 .

(17)

The smallness of w4 implies that the terms of fourth order and higher in x occurring in the function U (x) are small compared to the second order, i.e. u4 x2 + u6 x4 + · · ·

u2 .

(18)

Now doing a Taylor expansion in the small terms u4 x2 + u6 x4 + · · · allows to express the r.h.s. of (17) as a polynomial in x. Then comparing terms in x order by order, we ﬁnd the following relations w0 = − √ 3 4u2 ,
2
4 u2 w2 = u2 − √ 3 6u2 u4 ,
4 u2 w4 = u4 − √ 3 (10u2 u6 − 4u2 ) .
4
4 u2

(19)

Now we try to ﬁnd the parameters u2 and u4 as solution of those equations.
The ﬁrst equation gives
(20)
u2 = (w0 / )2 .

Quantum Chaos Versus Classical Chaos

365

On the other hand we have, due to (13), w0 = −2mEgr .

(21)

Due to the anharmonic perturbation the ground state energy is diﬀerent from the ground state energy of the harmonic oscillator. However, because of the smallness of the perturbation, we can express the ground state energy Egr using perturbation theory as a power series in ,
Egr = E (0) + E (1) +

2

E (2) + · · · ,

(22)

osc where E (0) = Egr . Thus from (21) and (22) we obtain

√
2mE (1)
+ O( 2 ) , u2 = mω +

(23)

or u 2 = m2 ω 2 +
= w2 [1 +

4m2 E (1)

+ O( 2 )

2E (1)
+ O( 2 )] .
E (0)

(24)

Next let us consider (19b). We obtain u4 = −

2√ u2 (w2 − u2 ) .
3

(25)

Recalling w2 = m2 ω 2 and (24), we ﬁnd u4 = −

2 m3 ω 3 E (1)
+ O( 2 ) .
3
ω

(26)

In (24), (26) we have expressed the parameters of the quantum action in terms of the parameters of the classical action. However, it remains to compute the energy E (1) . Again we use stationary perturbation theory. The Hamiltonian is given by, taking into account (14), (16),
H = H (0) + H (1)
1
p2
+ mω 2 x2
H (0) =
2m 2
(0)
w
H (1) = 4 x4 .
2m
To ﬁrst order of perturbation theory in

(27)

the energy E (1) is given by
(0)

osc
E (1) = ψgr |

w4 4 osc x |ψgr .
2m

(28)

366

H. Kr¨ger et al. o which yields the result
(0)

3w4 2
.
8m3 ω 2
Substituting this result into (24), (26) we ﬁnally obtain
E (1) =

u2 = w2 1 + ( and 3
)w4 + O( 2 ) ,
2m3 ω 3

1 u4 = − w4 + O( 2 ) .
4

(29)

(30)

(31)

5 Interpretation
Let us see what happens when we keep the classical parameters ﬁxed, except
(0)
for w4 , i.e., we keep w4 ﬁxed and vary . Note that w2 > 0 and w4 > 0. We
(0)
also have w4 > 0 and > 0. Now we want to study what happens when
→ 0. Equation (30) yields u2 > w2 u2 −→ →0 w2 .

(32)

Likewise, (31) yields u4 < w4 u4 −→ →0 w4 −→

→0

0.

(33)

In other words, in the limit → 0 the classical potential approaches the potential of the harmonic oscillator. The potential of the quantum action asymptotically approaches the classical potential, hence also the harmonic oscillator potential. That is, the renormalisation group ﬂow of the parameters u2 ( ), u4 ( ) goes to a Gaussian ﬁxed point. Second, for any value of the value the quadratic term of the potential is larger for the quantum potential than for the classical potential. Third, for any value of the value the quartic term of the potential is smaller for the quantum potential than for the classical potential. Recall that the quadratic term is the term, which, if it would stand alone, would make the system integrable. On the other hand, the quartic term is the term which drives the system away from integrability (and introduces chaos in 2-D). Thus we ﬁnd that quantum ﬂuctuations, which are the cause for the diﬀerences ∆2 = w2 − u2 and ∆4 = w4 − u4 to be nonzero, have the tendency to drive the quantum system closer to the regime of integrability.
Now the above perturbative calculations were performed in 1-D. Chaos in time-independent Hamilton systems exists only for D ≥ 2. A similar, but more tedious calculation can be performed in 2-D. It conﬁrms the above

Quantum Chaos Versus Classical Chaos

367

result that quantum ﬂuctuations drive the system closer to the regime of integrability and away from the regime of chaos.
Of course such perturbative calculations are meaningful only in a neighbourhood of the Gaussian ﬁxed point. It would be desirable to extend the calculations to a larger regime. However going to higher order of perturbation theory would make those calculation much more tedious. Nevertheless the perturbative result gives some insight into the dynamical consequences of quantum ﬂuctuations.

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F. Haake: Quantum Signatures of Chaos (Springer, Berlin 2001)

On the Prediction of Chaos in the Restricted Three-Body Problem
Houman Safaai1 and Mohammad Hasan Ghaﬀari Saadat2
1

2

Department of Mechanical Engineering, Amirkabir University of Technology,
Tehran, Iran,
Houman1359@yahoo.com
Department of Mechanical Engineering, Amirkabir University of Technology,
Tehran, Iran,
Ghaffari@aut.ac.ir

In this chapter it is shown that the elementary tools of Riemannian diﬀerential geometry can be successfully used to explain the origin of Hamiltonian chaos beyond the usual picture of homoclinic intersections. The stability of dynamics is related to the curvature of the conﬁguration space manifold, this is possible because the natural motion of standard Hamiltonian systems
(i.e. with a quadratic kinetic energy term) can be seen as geodesics of the conﬁguration space manifold equipped with the standard Jacobi metric. The stability properties of such manifold can be investigated through the Jacobi equation for geodesic spread and a geometric indicator of chaos similar to the largest Lyapunov exponent can be used. The case of a general two degree of freedom system is considered. The method is applied to the restricted three-body problem and the results of the Jacobi equation is compared with the detailed qualitative information provided by the approximate Lyapunov exponents. Complete agreement is found. The conﬁguration space manifold associated with the Hamiltonian studied here is everywhere of positive curvature. The ﬂuctuations of curvature of manifold along the geodesics yield parametric instability of the trajectories and thus chaos. Chaotic ﬂows of physics have nothing to do with Anosov ﬂows deﬁned on negative curvature manifolds. 1 Introduction
The diﬀerential geometric approach of Hamiltonian chaos has recently been proposed and successfully applied to the study of chaos in Hamiltonian systems [1–3]. This method provides a geometric explanation of Hamiltonian chaos and an eﬀective method to quantify it [4]. The starting point of this method is that trajectories of a standard Newtonian system can be viewed as the geodesics of a Riemannian manifold endowed with a suitable metric. The instability properties of geodesics, and therefore of corresponding dynamics are shown to be related to the curvature properties of the underlying manifold through the Jacobi equation for the evolution of geodesic separation.

370

H. Safaai and M.H.G. Saadat

By combining analytic computations with numerical simulations, we have a powerful tool for the study of chaos in Hamiltonian systems.
In the present work, we verify that the geometry of the conﬁguration space appears to enfold all the information of a nonlinear Hamiltonian system for what concerns regular and chaotic dynamics.
We show that the Riemannian approach is eﬀective for systems with two degrees of freedom. For such a system, we can solve the Jacobi equation and we show that the solution of Jacobi equation brings all those detailed information about order and chaos that can be obtained by analyzing Lyapunov exponent at diﬀerent energies.
In Sect. 2, we recall the basic deﬁnitions and concepts of the geometric formulation of Newtonian mechanics, including the stability criteria and the
Jacobi equation for a generic two-dimensional system. In Sect. 3, we apply the theory to the restricted three-body problem. We show that geometry contains all the information about order and chaos and is in complete agreement with the results of the largest Lyapunov exponent.

2 The Relationship Between Mechanics and Riemannian Geometry
We aim at considering Newtonian mechanics from a geometrical point of view. More precisely, the trajectories of a Hamiltonian ﬂow are regarded as geodesics of a Riemannian manifold, equipped with a suitable metric, so that weak or strong chaotic instability can be related with geometrical properties of the underlying manifold.
The natural geometrical setting of Hamiltonian dynamics is within the framework of symplectic or, more generally, Poisson geometry. However, since we are mostly interested in the study of Hamiltonian systems that have a standard kinetic part, also Riemannian geometry can be used. The advantage of using a Riemannian manifold relies on the possibility of measuring the distances between two points, thus the separation between two diﬀerent trajectories. It is well known how Newtonian mechanics can be rephrased in the Riemannian geometrical language.
Diﬀerent choices are possible for the ambient space (conﬁguration space, conﬁguration space-time, phase space); therefore, diﬀerent metrics can be used. Besides, for more general problems, when the Riemannian description is not possible, still a geometrical approach in the same spirit can be developed with the aid of Finsler spaces.
In the following, as the ﬁrst step to use the geometry, we use the conﬁguration space manifold equipped with a Riemannian metric.
The relationship between dynamics and Riemannian geometry is established through variational formulation of both geodesics and Newtonian dynamics. The geodesics of a Riemannian manifold are deﬁned as the extrema of the arc length functional:

On the Prediction of Chaos in the Restricted Three-Body Problem

371

B

δ

ds = 0 .

(1)

A

On the other hand, the trajectories, in conﬁguration space of a mechanical system according to the Maupertuis least action principle are given by the extrema of the action integral [6]: δ 2T dt = 0 ,

(2)

γ(t)

where T is the kinetic energy and γ(t) are all the isoenergetic curves joining two points q0 and q1 .
Comparing (1) and (2) and setting ds = 2T dt, it is seen that the trajectories of a mechanical system can be viewed as geodesics of the conﬁguration space manifold M equipped with a suitable metric which can be obtained through: gij = 2aij (E − V (x)) .

(3)

In local coordinates, the geodesic equations of a Riemannian manifold are given by [7]: j k d2 xi i dx dx
=0,
+ Γjk ds2 ds ds

(4)

i where s is the proper time and Γjk are the Christoﬀel coeﬃcients and xi ’s are the conﬁguration space coordinates.
Equation (4) are also the equations of motion that can be obtained from the Lagrange equations. In local coordinates the components of the Riemann tensor are given by: l l l m l m l
Rkij = ∂i Γkj − ∂j Γki + Γkj Γmi − Γki Γmj .

(5)

i
The trace Rkl = Rkil is the Ricci tensor and the scalar R = g kl Rkl is the scalar curvature of M , which is called Ricci scalar.
It has been shown that from the curvature properties of a Riemannian manifold some relevant consequences about the stability properties of its geodesics can be derived [1, 2, 7, 8]. This is the central point that this work is focussed on.
Now consider some region of the manifold through whose points geodesics can be drawn in various directions. Concentrate on one geodesic and its neighboring geodesics. Let s designate the parameter that locates points on the geodesic and τ is a continuous parameter that labels diﬀerent geodesics.
In order to measure the degree of sensitivity to initial conditions, we consider a congruence of geodesics γ(τ, s) and a separation vector J , which is Lie dragged by the congruence (as is shown in Fig. 1) [1]. Now choose any given geodesic γ(τ0 , s) as the “reference geodesic” and deﬁne the parameterization

372

H. Safaai and M.H.G. Saadat

Fig. 1. bundle of geodesics and Jacobi vector, base vectors also are represented

so that γ(τ0 , s) = γ(τ = 0, s). then the magnitude of the separation vector
J (s) is deﬁned by:
J(s) =

∂γ(τ, s)
∂τ

.

(6)

τ =0

Using ﬁnite diﬀerence in τ, s, as is shown in Fig. 2, we can see that J(s) can be interpreted as the distance between two nearby geodesics. Two geodesic coordinates e1 , e2 which we will use as the base vectors of the decomposition of J are shown in Fig. 1.

Fig. 2. The Jacobi can be used to measure the separation between nearby geodesics

By taking the second variation of J we can obtain the Jacobi equation
[1, 7]:
∇ ∇
J (s) + R(J , v)v = 0 ,
(7)
ds ds where v = γ and ∇/ds = ∇γ is the covariant derivative along the geodesic.
˙
˙
The Jacobi equation (7) gives the dynamics of the geodesics with respect to the “reference geodesic”. Divergence of J explain the instability of the geodesic and so the relevant Hamiltonian system [1, 8]. The instability of the system can be studied by means of this equation. It shows the relationship between geodesic instability and chaos.

On the Prediction of Chaos in the Restricted Three-Body Problem

373

For a general two degree of freedom system, we can assume the following decomposition for J :
2

J (s) =

ξi (s)e(i) (s) ,

(8)

i=1

where e(1) , e(2) is an orthonormal system of parallely transported vectors.
In this reference we have [1, 8]:
2

∇s ∇s J = i=1 d2 ξi e(i) (s) . ds2 (9)

Moreover,
R(v, J )v =

R(v, J )v, e(i) e(i)

(10)

j

=

Rv, e(i) )v, e(i) ξi e(i) . j So (7) can ﬁnally be written as: d2 ξi
+
ds2

Qij ξj = 0

i, j = 1, 2

(11)

i

with Qij = R(v, e(i) )v, e(i) .
We can choose our basis e(1) , e(2) such that e(1) ⊥v, e(2) v so that gij ei ej = δαβ [8]; then since v is parallely transported by deﬁnition of the
(α) (β) geodesic, e(2) is parallely transported too, and if e(1) ⊥e(1) then e(2) is also parallely transported. With respect to a local(static) reference frame they can be written as: e(1) = e(2) =

dx2 dx1
,
, ds ds dx1 dx2
,
. ds ds

−

(12a)
(12b)

They are the same as the base vectors of Fig. 1.
Now (11) becomes: d2 ξ 1
+ Q11 ξ 1 + Q12 ξ 2 = 0 , ds2 d2 ξ 2
+ Q22 ξ 2 + Q21 ξ 1 = 0 . ds2 (13a)
(13b)

The components of the Riemann tensor are derived from (5). In two dimensions it can be shown that the only nonvanishing component of R is R1212 and for the Jacobi metric it is equal to [1, 8]:

374

H. Safaai and M.H.G. Saadat

R1212 =

∆V (x) (∇V (x))2
+
,
2
2W

(14)

where W = E − V (x) and ∆, ∇ are the Euclidean Laplacian and gradient, respectively. We used MAPLE to calculate the Riemann tensor components analytically. Combining (14) and (12) we obtain Q12 = Q21 = Q22 = 0 and
Q11 =

1 ∆V (x) (∇V (x))2
+
W2
2
2W

.

(15)

So (13a) and (13b) become: d2 ξ 1
1
+ Rξ 1 = 0 , ds2 2 d2 ξ 2
=0,
ds2

(16a)
(16b)

where R denotes the scalar curvature of conﬁguration space manifold.
It is obvious that a negative curvature yields an unstable solution of (16a),
(16b). This is the case of abstract geodesic ﬂows of ergodic theory, which are deﬁned on hyperbolic manifolds (more precisely, they are deﬁned on the unitary tangent bundle of manifolds of everywhere negative curvature). However, for many systems like the restricted three-body problem, the curvature is mainly positive and the way to make the solution unstable is the rapid change of the curvature along the geodesic, which brings about parametric resonance and chaos [9]. Hyperbolicity is not the only way to make unstable solutions of (16a). If R is not constant, the loss of stability of the geodesics can also be introduced by parametric instability. Let us brieﬂy recall what parametric instability is. If the parameters of a dynamical system vary periodically in time, then a stable solution can be made unstable even if it is stable for each value of the parameters; this is the case of a harmonic oscillator whose frequency is periodically modulated in time with a suitable period [9, 10]. It has been shown in several papers that this is the dominant mechanism of instability in several physical geodesic ﬂows on high dimensional manifolds. In Sect. 3, we show that this is also the case for a typical two degrees of freedom system.
Passing from the proper time s to the physical time t in (16a), (16b) we ﬁnd: 1 dW dξ 1 d2 ξ 1
+ 2R1212 ξ 1 = 0 ,
−
2 dt W dt dt d2 ξ 2
=0.
dt2

(17a)
(17b)

From (17a), (17b) we see that the only component of the geodesic separation vector which conveys information about the stability behaviour of nearby geodesics is the transverse component ξ 1 .

On the Prediction of Chaos in the Restricted Three-Body Problem

375

Equations (17a), (17b) describe without approximation the stability properties of the dynamics of two-degree of freedom Hamiltonian systems.
To solve the Jacobi equation, we can use a new function:
Y (t) = ξ 1 (t) exp −

1
2

dt

˙
W
W

ξ 1 (t)
≡ √
.
W

(18)

Inserting (18) into (17a) we obtain the Hill equation: d2 Y
+ Q(t)Y = 0 , dt2 (19)

where,
˙
(∇V )2
1 W
Q(t) = ∆V +
−
W
4 W
˙
1 W
= 2R1212 −
4 W

2

+

2

+

˙
1 d W
2 dt W

˙
1 d W
2 dt W

(20)

.

√
√
From (18) we have ξ 1 (t) = W Y (t), the prefactor W has only bounded oscillations and ξ 1 has the same stable or unstable behaviour as Y (t).
The stability (19) which is derived from the Jacobi equation is our main tool to observe the chaos and instability in a two-dimensional dynamical system (restricted three-body problem).
Whenever Q(t) in (19) has a nonvanishing stochastic component the solution Y (t) has an exponentially growing envelope whose growth rate provides a measure of the degree of chaoticity. This rate of growth is similar to Lyapunov exponent and can be used as an indicator of chaos that is derived from geometry. It is deﬁned as [13]:
Λ = lim ln t→∞ ˙
Y 2 (t) + Y 2 (t)
.
˙
Y 2 (0) + Y 2 (0)

(21)

This quantity is a measure of the divergence of the nearby geodesics. It is similar to the largest Lyapunov exponent, but to produce this quantity, it is not necessary to linearize the equations of motion and it can be derived from an exact (19). We computed this quantity by solving (19) and in the next section we report it for each case and compare it with the classical Lyapunov exponent. At this point one can ask what is the diﬀerence with respect to the common deﬁnition of chaos, because here apparently the same deﬁnition is given.
First, it is necessary to distinguish between the usual explanation of the origin of chaos and the operational method to detect it numerically (Lyapunov exponents). Dating back to Poincar´, Melnikov, and others, the origin of e 376

H. Safaai and M.H.G. Saadat

chaos is attributed to homoclinic intersections of perturbed separatrices near hyperbolic points. This is a perturbative picture. It requires the use of action– angle variables and applies to quasi-integrable Hamiltonians. Moreover, for typical Hamiltonians of physical interest, the explicit change of coordinates to action–angle ones, in general, is a lengthy work that quickly becomes intractable with the increase of the dimension of the system. In practice and particularly at large dimensions, homoclinic intersections mainly gives a qualitative picture of the source of chaos. The main idea of this work is to show the tight relationship between local instability of the trajectories and local geometry (curvature) of the underlying manifold. The link between local instability and chaos is made by the compactness of the manifold where the trajectories live. The geometrical approach makes use of the natural coordinates (positions and velocities) and applies at any energy and at any strength of the nonintegrable part of the Hamiltonian, and allows a uniﬁed treatment of both the explanation of the origin of chaos and the method to measure its intensity. 3 Numerical Computation for Restricted
Three-Body Problem
In order to understand the capabilities of the Riemannian approach to retrieve the information about order and chaos, we apply the previous analysis to a two-degree of freedom system. Therefore, by studying the stability of the solutions of (19) along the trajectories originating from regular and chaotic regions, it is possible to obtain qualitative information about order and chaos.
In addition, we evaluate the both largest Lyapunov exponent and geometric indicator of chaos (21) approximately and make quantitative comparison between geometric analysis and the traditional one.
One of the simplest systems that can exhibit a chaotic behaviour is the restricted three-body problem, a schematic of which is presented in Fig. 3.

Fig. 3. Restricted three-body system

On the Prediction of Chaos in the Restricted Three-Body Problem

377

In this system we have a large body with mass M , a small body with mass mj and a very small body with mass me which is much smaller than
M and mj . We assume that me is so small that it cannot inﬂuence mj and so mj have a regular planar motion around M . The motion of the smallest body get inﬂuence from both M and mj and can have a regular or chaotic motion. From now on we choose physical units in which we have G = 4π 2 and
M = 1 for convenience. In these units, all the masses are actually expressed as fractions of M .
The Hamiltonian of the system in these units can be written in the form:
1
1 me x2 + me y 2 +
˙
˙
2
2
1
1
˙
˙
T = me x2 + me y 2 ,
2
2
1
4π 2 me
2 2
V = mj rj ωj −
2
r

H=

4π 2 mj
1
4π 2 me
4π 2 me mj
2 2 mj (rj ωj ) −
−
−
,
2 r rj r12 (22)
−

2

2

4π mj
4π me mj
−
. rj r12

Equations of motion for me can be derived with the Lagrange-Euler method: r = −4π 2
¨

r − rj r + mj
3
3 r |r − r j |

.

(23)

We consider a circular motion around M for mj , so the position of mj can be written in cartesian coordinates as: xj = rj cos(ωj t) , yj = rj sin(ωj t) ,

(24)

where r(x, y) is the position vector of me and r j (xj , yj ) is the position vector of mj which have a circular motion around M with angular speed ωj . The biggest mass M is always in the origin. In (23) we do not see M , because we ﬁx it as a constant. Now we can write the equations of motion for me , explicitly: ⎧
⎪
x−rj cos(ωj t) x ⎪ x = −4π 2
⎨¨
3 + mj
3
(x2 +y 2 ) 2
[(x−rj cos(ωj t))2 +(y−rj sin(ωj t))2 ] 2
(25)
⎪ y−rj sin(ωj t) y ⎪ y = −4π 2
⎩¨
3 + mj
3
2
2
2
2
(x +y ) 2

[(x−rj cos(ωj t)) +(y−rj sin(ωj t)) ] 2

From (3), we can obtain the metric of the ambient manifold: g11 = 2me (E − V (r, r j )) , g12 = g21 = 0 , g22 = 4me (E − V (r, r j )) .
Or explicitly,

(26)

378

H. Safaai and M.H.G. Saadat

⎛
⎧
⎪
⎪
⎪
⎪ g = 2m ⎜ E −
⎪ 11 e⎝ ⎪
⎪
⎪
⎪
⎪
⎨
g = g21 = 0 ,
⎛
⎪ 12
⎪
⎪
⎪
⎪
⎪
⎪ g = 4m ⎜ E −
⎪ 22 e⎝ ⎪
⎪
⎩

1
2

1
2

2

2

mj rj ωj +

2 2 mj rj ωj

+

4π 2 me
(x2 + y 2 )

4π 2 me
(x2 + y 2 )

+

4π 2 mj

+

4π 2 mj

rj

rj

+

+

4π 2 mj me
(x − rj cos(ωj t))2 + (y − rj sin(ωj t))2
4π 2 mj me
(x − rj cos(ωj t))2 + (y − rj sin(ωj t))2

⎞
⎟
⎠,
⎞
⎟
⎠.
(27)

The equations of motion have been integrated numerically by means of a Hamming modiﬁed predictor-corrector algorithm of fourth order, which is ideal for dynamical models. In order to perform an accurate numerical integration of the equations of motion, the time step is typically varied and adjusted in the interval 10−3 − 10−4 . With this time step range, the error
∆E/E did not exceed 10−5 − 10−6 . Together with the equations of motion and by means of the same algorithm, we have also integrated the Jacobi equation in its equivalent form (19) and after computing, we have computed the geometric indicator of chaos (Λ) with the use of (21). Using the same algorithm, we obtained the approximate largest Lyapunov exponent (λ) with the method of [12]. Besides its simplicity, this algorithm has the advantage of being symplectic, which means that its eﬀect upon the equations of motion is equivalent to a canonical transformation of variables. This ensures a faithful numerical representation of a Hamiltonian ﬂow [1].
We ﬁx the following parameters through the computations:
⎧
⎪ rj = 5.203 AU ,
⎪
⎨
2π
ωj = 11.86 year−1 ,
(28)
⎪ me = 2.96 × 10−6 ,
⎪
⎩ mj0 = 317.9 me .
These are exactly the parameters of the (Sun, Earth, Jupiter) system in the units where Msun = 1. Here, the unit of distance is the Astronomical Unit and the unit of time is the year, also mj0 is a constant which mj is considered as some multiple of it.
The only variable is mj . We considered three diﬀerent values for mj and solved the equations of motion in each case. we considered mj = 60 mj0 , mj =
300 mj0 , mj = 450 mj0 , with mj0 deﬁned in (28). For each mj we chose the
˙
˙ same initial conditions x0 = 1, y0 = 0, x0 = 0, y0 = 1 for me . We have kept
˙
Y (0) = 10−9 and Y (0) = 0 as initial conditions of the Jacobi equation in all cases. Y (t) is proportional to the separation of the two nearby geodesics (trajectories). The divergence of Y (t), in spite of Y (0) being small, reveals sensitive dependence on initial conditions which is a signature of chaotic motion.
The solution of (19) for mj = 60 mj0 is plotted in logarithmic scale in
Fig. 4(a). As it is seen, the envelope of Y (t) appears to be bounded or linearly growing for this initial condition. After solving Y (t) we can compute the geometric indicator of chaos Λ. We see that it is a negative number so the geodesic ﬂow is stable. In Fig. 4(b) the Lyapunov exponent is obtained by using a simple method which is explained in [12]. We can see that its

On the Prediction of Chaos in the Restricted Three-Body Problem

379

Lyapunov exponent is negative and it agrees with which we obtained from the geometric method (Fig. 4(a)) and the geometric indicator of chaos. In
Fig. 4(c) the nonzero component of the Riemann curvature tensor of the manifold is plotted and we see that it is always positive and ﬂuctuating. the orbit of the third mass is plotted In Fig. 4(d). We can see that because of the inﬂuence of mj , its orbit deviates from a regular elliptic orbit.

(a) log Y (t) − t, Λ = −0.0152

(c) R1212 − t

(b) log β(t) − N t, λ = −0.0023

(d) Orbit of me

Fig. 4. mj = 60 mj0

The solution of (19) for mj = 300 mj0 is plotted in logarithmic scale in Fig. 5(a). As it is seen, the envelope of Y (t) appears to be bounded or linearly growing for this case. The geometric indicator of chaos is computed and it is also a negative number which shows the validity of this argument. In
Fig. 5(b) the Lyapunov exponent is computed. We can see that its Lyapunov exponent is negative and it agrees with which we obtained from the geometric

380

H. Safaai and M.H.G. Saadat

(a) log Y (t) − t, Λ = −0.089

(c) R1212 − t

(b) log β(t) − N , λ = −0.017

(d) Orbit of me

Fig. 5. mj = 300 mj0

method. Both the geometric indicator of chaos and Lyapunov exponent are greater than those of the previous case and it shows that by increasing mj we can make this system chaotic. In Fig. 5(c) the nonzero component of the
Riemannian curvature tensor of the manifold is plotted and it is obvious that it is always positive. The orbit of the me is plotted In Fig. 5(d). From this ﬁgure, one can ﬁnd that this orbit is more irregular than the orbit of the previous case and the eccentricity of the orbit changes more than that of the previous case.
As the third example, we did the same steps for the third case with mj = 450 mj0 which we shall see that it is a chaotic case. We solved (19) and plotted Y (t) in logarithmic scale in Fig. 6(a). In this case, we see that this function is increasing rapidly and the geometric indicator of chaos is a positive number. The magnitude of Y (t) after a short time increases 106 times and so

On the Prediction of Chaos in the Restricted Three-Body Problem

(a) log Y (t) − t, Λ = 0 186

(b) log β(t) − N , λ = 0 0578

(c) R1212 − t

381

(d) Orbit of me

Fig. 6. mj = 450 mj0

the geodesics which originate from a compact region will diverge and go far away. Also we computed the Lyapunov exponent in Fig. 6(b). The Lyapunov exponent also is a positive number and it shows that we have chaotic motion.
This is in agreement with which we obtained by the geometric method and by solving the Jacobi equation. The nonzero component of the Riemann curvature tensor is plotted in Fig. 6(c) and we see that it is always positive and ﬂuctuating. So chaos is generated with the mechanism of parametric resonance. In Fig. 6(d) the orbit of me is plotted and we can see that it is more irregular than in the previous cases.
We have just seen that the comparison between the detailed information given by the geometry (from Y (t) and Λ) reveals a complete qualitative and quantitative agreement with results obtained from the classical method
(Lyapunov exponents). Here we can speak about the origin of chaos.

382

H. Safaai and M.H.G. Saadat

Evaluation of R1212 by using (14), and the scalar curvature R =
2R1212 /W 2 , shows that both are always positive for the restricted threebody problem, no matter whether it is computed along a regular or chaotic orbit. Therefore, in the geodesic ﬂow deﬁned by the restricted three-body problem, chaos is due to the parametric resonance of the nearby geodesics induced by ﬂuctuations of the positive curvature. By comparing Figs. 4(c),
5(c) and 5(c) it is obvious that in all three cases the curvature is always positive and rapidly ﬂuctuating, but the diﬀerence between these three cases is that by increasing mj the amplitude of the ﬂuctuations increases. For example the magnitude of the ﬂuctuations in the regular case mj = 60 mj0 is of the order 10−10 but this magnitude for the chaotic case mj = 450 mj0 is of the order 10−3 . It increases by the order of 107 . So we can conclude that for occurring parametric resonance and so chaos in a system it is necessary that not only the curvature ﬂuctuates rapidly but also its amplitude be large enough. A deep understanding of the parametric resonance can help us to understand more about the occurrence of chaos in Hamiltonian systems. It may be possible to study the chaotic criteria of a system with only studying the behaviour of the curvature of its equivalent Riemannian manifold.
Finally we can summarize the results in a table:
Table 1. Λ and λ in three cases mj = 60 mj0 geometric indicator of chaos (Λ): largest Lyapunov exponent (λ):

−0.0152
−0.0023

mj = 300 mj0
−0.089
−0.017

mj = 450 mj0
0.1860
0.0578

4 Conclusion
In this chapter, we have shown that the geometric description of Hamiltonian chaos based on Riemannian geometry is in agreement with the standard approaches (Lyapunov exponents). This has been shown for a two degree of freedom system (restricted three-body problem) and the results of the stability (19), which has been derived from the Jacobi equation are computed and the behaviour of the system (chaotic or regular) speciﬁed and compared with the phenomenological standard methods. Also we used a geometric indicator of chaos (Λ) and it was shown that it can be used for predicting the behaviour of system like the Lyapunov exponents but it is derived from an exact equation (not a linearized one as in deriving Lyapunov exponents). A complete agreement is found. This method can be applied for studying chaos in any Hamiltonian system in any energy and in any range of the nonlinearity parameters and so it is very useful to apply this method to real systems like engineering problems.

On the Prediction of Chaos in the Restricted Three-Body Problem

383

The results reported in this paper are not simply alternatives to the conventional tools to describe chaos, but have a deeper meaning [13]. They provide an explanation of the origin of Hamiltonian chaos that is an alternative to the conventional interpretation, which is based on homoclinic intersections.

References
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2.
3.
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M. Pettini: Phys. Rev. E 47, 828 (1993)
L. Casetti, M. Pettini: Phys. Rev. E 48, 4320 (1993)
M. Cerruti-sola, M. Pettini: Phys. Rev. E 51, 53 (1994)
R. Abraham, J. E. Marsden: Foundations of Mechanics (Addison-Wesley, Redwood city CA 1998)
J. Jos´, E. Saletan: Classical Dynamics: A contemporary approach (Cambridge e university press, USA 1998)
M. Nakahara: Geometry, Topology and Physics (Adam Hilger, Paris 1990)
M. Pettini, R. Valdettaro: Chaos 5, 646 (1995)
M. Cerruti-sola, M. Pettini: Phys. Rev. E 53, 179 (1996)
R. Lima, M. Pettini: Phys. Rev. A 41, 726 (1990)
A.H. Nayfeh, D.T. Mook: Nonlinear oscillations (Wiley, New York 1979)
H.T. Davis: Introduction to Nonlinear Diﬀerential and Integral equations
(Dover Publications Inc., New York 1962) pp 50–60
L. Hand, J. Finch: Analytical Mechanics (Cambridge university press, USA
1998)
L. Casetti, C. Clementi, M. Pettini: Phys. Rev. E 54, 5969 (1996)

Order and Chaos in Some Hamiltonian
Systems of Interest in Plasma Physics
D. Constantinescu1 and B. Weyssow2
1

2

Department of Applied Mathematics, Association EURATOM-MEC, University of Craiova, 13 A.I. Cuza Street, Craiova 200585, Romania dconsta@central.ucv.ro Department of Statistical Physics and Plasmas, Association
EURATOM-Belgian State, ULB CP 231, Bvd du triomphe, 1050 Bruxelles,
Belgium
bweyssow@ulb.ac.be

1 Introduction
The name Tokamak, an acronym for the Russian expression TOrichnaia
KAmera MAgnitnaia Katushka [1], is given to toroidal chambers in which plasma particles are conﬁned by a magnetic ﬁeld which has two basic orthogonal components, one acting in the direction of the major curvature of the torus, the other in the direction of the minor curvature of the torus.
The magnetic ﬁeld line equations deﬁne a Hamiltonian system [3, 4]. In the ideal i.e. integrable case the helical magnetic ﬁeld lines lay on constant pressure surfaces, called magnetic surfaces, having the topology of nested tori surrounding the magnetic axis. Deviations to the ideal case are always present in experiments either due to internal factors such as instabilities and ﬂuctuations or due to external causes such as magnetic imperfections arising from the poloidal distribution of the toroidal coils. The perturbed systems, generically non integrable [3] can be studied using the Poincar´ map associe ated to a given poloidal cross-section. This map is an area-preserving map which has to be compatible with the toroidal geometry. A typical phase portrait of the Poincar´ map exhibits a complex structure of regular and chaotic e zones which form a fat fractal [2]. The ﬁeld line transport properties are determined by the competition between order and chaos which is observable by varying the parameters of the system.
The localization of the regions of phase-space covered by regular orbits and the description of the mechanisms for their destruction is important from both theoretical and practical points of view. From a practical point of view, the study of magnetic internal transport barriers relates to the more general study of particle internal transport barriers (ITB) which is of great importance for nuclear fusion because the particle radial transport is strongly reduced in their presence and hence the particle conﬁnement time is increased.
For magnetic ITB a reduction of particle transport is expected due to a reduction of the magnetic ﬁeld line radial wandering (charge particle guiding centres follow magnetic ﬁeld lines). From a mathematical point of view, a

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magnetic internal transport barrier is an invariant set, mainly formed by regular orbits, separating a core chaotic zone situated near the magnetic axis of the tokamak, from a peripheral chaotic zone situated near the tokamak wall. This magnetic ITB cannot be traversed by magnetic ﬁeld lines, i.e. the magnetic transport through the ITB is suppressed and all magnetic ﬁeld lines passing through the central stochastic zone are conﬁned.
The aim of the chapter is to present recent results on the conﬁguration of the transport barriers in some realistic models of magnetic ﬁeld line dynamics in a tokamak geometry. This study leads to a new understanding of the dynamic of this kind of magnetic systems based on very ingenious correlations with the theory of numbers and on hidden geometrical aspects. The chapter is organized as follows. Section 2 is devoted to the presentation of the mathematical models. Some deﬁnitions and basic results from the theory of dynamical systems are presented in Sect. 3. The transport barriers as well as their localization in phase-space are described in Sect. 4. Section 5 contains a description of some scenarios of magnetic ﬁeld line reconnection and of magnetic island merging. Conclusions are drawn in Sect. 6.

2 The Mathematical Models
A set of toroidal coordinates (r, θ, ζ), where ζ is the toroidal angle around the symmetry axis of the torus and (r, θ) are the polar coordinates in a (circular) poloidal cross-section at distance R0 to the symmetry axis and having maximal radius a, is usually used for the description of the magnetic line conﬁguration. However, since canonicity of the coordinates is needed for the derivation of the discrete dynamical systems describing the behaviour of the magnetic ﬁeld lines one uses the toroidal ﬂux ψ = r2 /2 instead of the radial coordinate r [3]. A Poincar´ section is deﬁned from the intersection points e of a magnetic ﬁeld line starting at position (θ0 , ψ0 ) with the poloidal section S : ζ = cst. The intersection point after n toroidal turns is denoted by
(θn , ψn ). The third coordinate ζ does not appear here as it is a constant parameter depending only on the position of the poloidal section. The application
TK : S 1 × R+ → S 1 × R+ , TK (θn , ψn ) = (θn+1 , ψn+1 )

(1)

has to be an area-preserving map compatible with the toroidal geometry (if ψ0 = 0 then ψn = 0 for all n ∈ N and if ψ0 > 0 then ψn > 0 for all n ∈ N). A map satisfying the imposed constraints is derived from the mixed generating function FK : S 1 × R+ → S 1 × R+ (see [3], [5]):
FK (θn , ψn+1 ) = ψn+1 θn + α0 (ψn+1 ) + K P (θn , ψn+1 ) .
The corresponding discrete system is

(2)

Order and Chaos in Some Hamiltonian Systems

∂P (θn , ψn+1 )
∂θn
∂P (θn , ψn+1 )
= θn + W (ψn+1 ) + K
∂ψn+1

387

ψn+1 = ψn − K θn+1 (mod 1)

(3)

where W (ψ) = dα0 (ψ)/dψ. The analytical expression of TK is obtained by determining θn+1 and ψn+1 from (3) as a function of θn and ψn . The application W (ψ) is often called the winding function, q ≡ 1/W the safety factor or more simply the q-proﬁle and s = d ln q/d ln ψ the shear proﬁle. The latter quantity is positive when q is an increasing function and it is negative when q is a decreasing function of the toroidal ﬂux (or radius). By choosing adequately the forms of α0 (ψ) and P (θ, ψ) in (2) some well-known models used in plasma physics are recovered. For instance, with α0 (ψ) = ψ 2 /2 and P (θ, ψ) = −(2π)−1 cos(2πθ) one obtains the Chirikov–Taylor system generated by the celebrated “standard map” [6]. With α0 (ψ) = ψ 2 /2 and
P (θ, ψ) = −(2π)−1 ψ cos (2πθ) the “Wobig map” is obtained [7] while with α0 (ψ) = a (ψ − ψ 3 /3) and P (θ, ψ) = −(2π)−1 cos(2πθ) one obtains the nontwist standard map [23,24]. None of these maps is applicable to toroidal fusion devices because either the positivity of the toroidal ﬂux is not guaranteed or the proﬁle for the winding function is unphysical. They should therefore be considered as academic models.
More realistic models of magnetic ﬁeld line dynamics also satisfying (2) are known as the Tokamap [3] and the Rev-Tokamap [10]. They involve a monotonous and a non-monotonous winding function, respectively. The perturbation P = P (θ, ψ) is chosen in order to be consistent with the toroidal geometry. The tokamap is obtained by choosing in (3)
P (θ, ψ) = −
WT (ψ) =

ψ
1
cos(2πθ)
2 1+ψ
(2π)

1
(2 − ψ) (2 − 2ψ + ψ 2 ) ,
4

(4)

where an index T stands for Tokamap. A magneto-hydrodynamic derivation of the realistic (decreasing) winding function W (or safety factor q = 1/W ) is given in [8]. The tokamap is deduced also as a particular case of a map for guiding centre particles [9]. The study of reversed shear conﬁgurations can be performed using a diﬀerent winding function [10]
WRT (ψ) = w 1 − A (C ψ − 1)2

(5)

where the lowerscript RT stands for reversed tokamap (or rev-tokamap). The parameters are w − w0 w − w1
A=
, C =1+
,
(6) w w − w0 where w0 = W (0) and w1 = W (1).
This map is RTK : S 1 × R+ → S 1 × R+ , RTK (θ, ψ) = (θ, ψ) with

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ψ=

1
2

ψRT +

2 ψRT + 4ψ

¯ θ = θ + w [1 − A (C ψ − 1)2 ] −

1
K
cos(2πθ)
4π (1 + ψ)2

(mod 1)

(7)

K where ψRT = ψ − 1 − 2π sin(2πθ). The winding function W has a local maximum when ψ = 1/C. In order to further exemplify some results to be shown in this chapter two other maps are proposed. One is the bounded tokamap obtained from (11) by choosing WBT (ψ) = WT (ψ), as in the usual tokamap, and

P (θ, ψ) = −

1 ψ (1 − ψ) cos(2πθ) (2π)2 1 + ψ

The analytical expression of the bounded tokamap is BTK :

ψ=

ψBT +

2 ψBT + 4ψ (1 −

2 (1 −

K
2π

K
2π

sin(2πθ))

sin(2πθ))
2

θ = θ + WBT (ψ) −

ψ + 2ψ − 1
K
cos(2πθ)
2
4π
(ψ + 1)2

(mod 1)

(8)

K where ψBT = ψ − 1 − 2π sin(2πθ). The main diﬀerence between the tokamap and the bounded tokamap is that BTK has two invariant geometrical circles; one ψ = 0 near the magnetic axis of the tokamak and the other ψ = 1 that deﬁnes the tokamak wall. The latter transport barrier is induced by the perturbation, not by the winding function W .
The second proposed map, the degenerate tokamap, is obtained from (11) by choosing

ψ
1
cos(2πθ)
(2π)2 1 + ψ
1
WDT (ψ) = (2 − ψ) (2 − 2ψ + ψ 2 ) + ψ 2
4
P (θ, ψ) = −

2−

7 ψ 4

.

(9)

The analytic expression of this map is DTK : S 1 × R → S 1 × R: ψ= 1
2

ψDT +

2 ψDT + 4ψ

θ = θ + WDT (ψ) −

1
K
cos(2πθ)
4π (1 + ψ)2

(mod 1)

(10)

K with ψBT = ψ − 1 − 2π sin(2πθ). The perturbation is the same as for the tokamap but the winding function is modiﬁed. It is still a monotone decreasing function but such that WDT (1/2) = WDT (1/2) = 0 (a prime denotes a

Order and Chaos in Some Hamiltonian Systems

389

∂/∂ψ derivative) in contrast to the tokamap which has WT (ψ) < 0 for all
(θ, ψ) ∈ S 1 × [0, 1]. The conditions WDT (0) = WT (0) and WDT (0) = WT (0) ensure that the tokamap and the degenerate tokamap have similar behaviour near the polar axis ψ = 0. All four maps described in this chapter are submitted to the condition K < 2π which must hold since the analytical properties of the maps have to be at least of class C 1 .

3 Deﬁnitions and Basic Results
The prototype of area-preserving maps which will be studied in the sequel not is f : S 1 × R → S 1 × R, f (θ, ψ) = (θ, ψ) deﬁned by the system of implicit not equations (thus the notation = ): ψ = ψ − K g (θ) h(ψ) θ = θ + W (ψ) + K g(θ) h (ψ)

(mod 1)

(11)

where g : S 1 → R, h : R → R are bounded functions of class C 2 (two times derivable) and g is a 1-periodic function in θ.
– An orbit of (θ0 , ψ0 ) is the set {(θn , ψn ), n ∈ N} such that (θn+1 , ψn+1 ) = f (θn , ψn ). not – A lift of f , i.e. F : R × R → R × R, F (θ, ψ) = (x, ψ), can be obtained from (11) by removing the mod 1 in the second equation.
– The winding (or rotation) number of an orbit is deﬁned as ω = lim

i→∞

xi i (12)

if the limit exists.
– The orbit of (θ0 , ψ0 ) is periodic of period n if f n (θ0 , ψ0 ) = (θ0 , ψ0 ). The rotation number of a periodic orbit is a rational. The stability of the periodic point (θ0 , ψ0 ) is characterized by the eigenvalues of the Jacobian matrix
J(θ0 ,ψ0 ) f n which are called the multipliers of (θ0 , ψ0 ) and will be denoted by λ1 and λ2 . Because f is an area-preserving map (det(J(θ,ψ) f ) = 1 for all (θ, ψ) ∈ S 1 × R) their product λ1 λ2 is 1. Elliptic periodic points have complex multipliers (and λ2 = λ1 ), while hyperbolic periodic points have real multipliers λ1 = 1/λ2 = ±1 and parabolic periodic points have λ1 = λ2 = ±1.
– A set homeomorphic to a circle that is mapped onto itself by f is called a rotational invariant circle. All orbits contained in an invariant circle densely ﬁlls it and they have the same winding number, which is generically irrational. An orbit for which there is no winding number is chaotic.
– A transport barrier is a f -invariant subset of S 1 × R separating two other f -invariant subsets of S 1 × R. The transport barriers cannot be crossed by the orbits starting from its neighbourhood.

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– The map f is a positive twist map (respectively negative) if
∂θ
(θ, ψ) > 0
∂ψ

(respectively

∂θ
(θ, ψ) < 0)
∂ψ

for all (θ, ψ) ∈ S 1 × R.
– A map for which (∂θ/∂ψ)(θ, ψ) = 0 for some (θ, ψ) ∈ S 1 × R is called a non-twist map.
The study of twist area preserving maps initiates with the works of H.
Poincar´ (see [17] pp 38). Their dynamics is described by the Poincar´– e e
Birkhoﬀ theorem and the KAM (Kolmogorov–Arnold–Moser) theory.
The unperturbed map corresponds to K = 0. It is an integrable map in the sense that the orbit of (θ0 , ψ0 ) is included in the (geometrical) circle ψ = ψ0 and it is periodic if W (ψ0 ) ∈ Q, or quasiperiodic (dense in the circle)
/
/ if W (ψ0 ) ∈ Q. Every circle ψ = ψ0 with ψ0 ∈ Q is a transport barrier.
In the case of the perturbed map (K > 0) some orbits are dense in rotational invariant circles which are transport barriers. The elliptic periodic points are surrounded by stability islands. The hyperbolic periodic points having same rotation number are connected by their stable and unstable manifolds. If these manifolds do not intersect transversely the dynamics near them is regular (the orbits of the neighboured points are dense in rotational circles or in curves surrounding elliptic periodic points ). If the manifolds intersect transversely, a chaotic layer is formed in a zone containing their
(topological) closure. The phase portrait of the tokamap (which is a negative twist map [8]) is presented in Fig. 1 where the coordinates θ and ψ are represented on the horizontal and vertical axis , respectively. Near the vertical line θ = 0.5 the elliptic periodic points with the rotation number 1/1, 2/3,
1/2, 1/3, 1/4, and 1/5 can be identiﬁed. Some chaotic layers surrounding the hyperbolic periodic points are observed in the upper part of the ﬁgure.
These chaotic layers are separated by invariant circles. A large chaotic zone is observed in the central part of the phase space. In this region some manifolds of hyperbolic points having the rotation number 1/3, 1/2 respectively 2/3 overlap. Since the tokamap is almost integrable near the polar axis ψ = 0 a regular zone is also observed in the lower part of the phase space.
For K
1 the maps given by (11) are close (at least in a C 0 topology) to the integrable map corresponding to K = 0, hence they are almost integrable. In this case the main part of the orbits is still regular and the eventually formed chaotic layers are very thin (being separated by very frequent rotational circles).
The Poincar´–Birkhoﬀ theorem [11] ensures that to each rational number e n/m in an appropriate interval corresponds a pair of periodic orbits of rotation number n/m in the perturbed map: an elliptic one and an hyperbolic one. The points of these orbits interlace and they are included in the so-called n/m-type Poincar´–Birkhoﬀ island chains. e Order and Chaos in Some Hamiltonian Systems

391

Fig. 1. Phase portrait of tokamap (a twist map) corresponding to K = 4.5 and w=1 The KAM theory provides the conditions for the existence of invariant circles in a slightly perturbed integrable twist map: the invariant circles having diophantine rotation number are more robust than the others [12].
For larger values of K this theorem may not be applicable, even if numerical evidences indicate that some resistant barriers have diophantine rotation number [8,13]. Some conditions for the existence of invariant circles were proposed in the eighties by J.M. Greene [14], S. Aubry [15] and J.N. Mather [16].
In this case the enlargement of the chaotic zone can be explained by the converse KAM theory (cone crossing criterion, folding properties, overlapping criterion etc.) [17].
The non-twist maps started to be explored only in the last decade, although the richness of new phenomena exhibited by such maps (the existence of twin Poincar´–Birkhoﬀ chains with the same rotation number, reconnece tion, meandering tori) were pointed out early in the eighties [18]. The phase portrait of rev-tokamap (a non-twist map) is presented in Fig. 2. The twin
Poincar´–Birkhoﬀ island chains having the rotation number 1/2 can be obe served. The lower one is contained in a regular region, but the upper one is surrounded by a chaotic layer. The hyperbolic points of twin Poincar´ e 392

D. Constantinescu and B. Weyssow

Fig. 2. The phase portrait of the rev-tokamap (a non-twist map) with parameters
K = 1.95, w = 0.67, w0 = 0.3333, w1 = 0.1667

chains with the rotation number 2/3 are connected both by homoclinic and heteroclinic connection. Between them a meandering circle is observed.
The non-twist maps are used to describe the behaviour of magnetic ﬁeld lines in reversed tokamaks [3], chaotic advection in ﬂuids [19,20], chaotic ionization in atomic physics [21], oscillations in astrophysics [22]. It is also to be noted that non-twist maps appear naturally in iterations of twist maps because the composition of twist maps is generally not a twist map. There exist only few theoretical results concerning the dynamical behaviour of these maps. However, even if the Aubry–Mather variational theory for twist maps does not generalize to non-twist maps, an analogue of the KAM theory has been developed [23–25]. The regions of instability for non-twist areapreserving maps are studied in [26]. The non-twist maps with reversing symmetry group studied in [27] and an analytical method for determining the reconnection threshold was proposed in [28].
The tokamap and the bounded tokamap and negative twist maps. They deviate the lines θ = cst to the left.
The rev-tokamap and the degenerate tokamap are non-twist maps because the equation (∂θ/∂ψ)(θ, ψ) = 0 has solutions in S 1 × R+ . Their properties

Order and Chaos in Some Hamiltonian Systems

393

1

0.8

tokamap

0.6 psi bounded−tokamap

degenerate−tokamap

0.4

0.2

rev−tokamap
0

0

0.1

0.2

0.3

0.4

0.5 theta 0.6

0.7

0.8

0.9

1

Fig. 3. The eﬀect of tokamap, bounded tokamap, degenerate tokamap (all corresponding to K = 5.9, w = 1) and rev-tokamap (K = 5.9, w = 0.67, w0 = 0.3333, w1 = 0.1667) on the vertical line θ = 0.1

are diﬀerent because the winding function of the rev-tokamap is a quadratic function having a maximum, while the winding function of the degenerate tokamap is a decreasing function (having an inﬂexion point ψ = 1/2).
Figure 3 shows the line θ = 0.1 and its images produced by the tokamap, the bounded tokamap, the degenerate tokamap and the rev-tokamap. The image is tilted to the left by the tokamap and the bounded tokamap showing they are negative twist maps. For the two remaining maps, the rev-tokamap and the degenerate tokamap, the image is no more a graph of a function depending on ψ. This results from non-twist property of the maps.
In order to describe the transport barriers in non-twist maps we introduce the following deﬁnitions. not Deﬁnition 1. Let f : S 1 × R → S 1 × R, f (θ, ψ) = θ, ψ be a non-twist map deﬁned by (11).
– The regular set of f is Creg : W ψ = 0.
– The shearless orbit has a rotation number which is a local extremum of all rotation numbers of orbits in the map. The closure of the shearless orbit, denoted by Csh , is called the shearless curve.

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– The critical twist set of f is the curve Cnt : (∂θ∂ψ)(θ, ψ) = 0.
– The non-twist annulus is the closure of all orbits starting from Cnt .
The unperturbed system (K = 0 in (11)) deﬁned with a quadratic winding function satisﬁes the obvious equalities (∂θ/∂ψ)(θ, ψ) = W (ψ) = W ψ =
0 which imply that the curves Cnt , Creg and Csh coincide i.e. Cnt = Creg =
Csh : ψ = ψmax where ψmax is the value of ψ for which W is maximum.
If the winding function has no extremum (as in the case of the degenerate tokamap) the shearless curve does not exist but the regular and the critical twist curves are well deﬁned.
In the perturbed case (K = 0) these curves are diﬀerent and a natural question arises on how to relate the dynamical properties of the system with each of them. These curves have the following equations:
Cnt : W
Creg : W

¯ ψ + kg (θ) h ψ =0,

¯ ψ =0,

(13)
(14)

The analytical form of the shearless curve (if it exists) cannot usually be given because it involves the numerical computation of the (perturbed) winding number. The shearless curve is f -invariant but the regular set and the critical twist set are not f -invariant. For this reason the non-twist annulus (which is f -invariant) has to be considered.
If the equation (∂θ/∂ψ)(θ, ψ) = 0 has a unique solution for every θ ∈ S 1 , which is a typical situation, the critical twist set of f is a rotational circle.
The non-twist annulus collects all the points having a non-twist dynamic.
The phase space splits in three invariant zones on which the map f acts separately. In Fig. 4 the non-twist annulus and the chaotic invariant zones of the rev-tokamap (a non-twist map [10]) are represented. The map BTK acting below or above the non-twist annulus has opposite twist properties due to the opposite monotonicity of W . The map DTK of the degenerate tokamap case has negative twist properties both below and above the non-twist annulus because WDT is a monotonous function.
The existence of the shearless curve is a non-twist essential phenomenon hence the shearless curve is contained in the non-twist annulus.

4 Transport Barriers
In the twist systems the transport barriers which are usually analysed are invariant circles. However, large transport barriers can be obtained by using appropriate perturbations which create some zones where the twist map is almost integrable. This is what is happening with the bounded tokamap (8).

Order and Chaos in Some Hamiltonian Systems

395

Fig. 4. The non-twist annulus and the invariant chaotic regions in the rev-tokamap model (k = 2.5, w = 0.67, w0 = 0.3333, w1 = 0.1667)

Proposition 1. In an annulus surrounding the circle ψ = 1 the map (8) is almost integrable.
Proof : In an annulus surrounding the invariant circle ψ = 1 the map (8) is not close to the map IK : S 1 × R → S 1 × R, IK (θ, ψ) = (θ , ψ ) deﬁned by ψ =ψ θ =

θ + W (1) +

K cos(2πθ) 8π 2

(mod 1)

(15)

because d2 ((θ , ψ ), (θ, ψ)) =

K ψ (1 − ψ) sin(2πθ) 2π 1 + ψ
2

2

K ψ + 2ψ − 3
+ W (ψ) − W (1) +
16π 2 (ψ + 1)2

2

is a small quantity for ψ ≈ 1 (which means ψ = 1 because ψ = 1 if and only if ψ ≈ 1). (Q.E.D.)

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Fig. 5. The phase portrait of the bounded tokamap (K = 5.9, w = 1)

It results from this that a transport barrier having positive area appears in a region surrounding the circle ψ = 1. In Fig. 5 some regular orbits can be observed near the circle ψ = 1. The chaotic layers formed around the
Poincar´–Birkhoﬀ chains having the rotation numbers 1/2, 3/7, 2/5, 1/3, e 2/7, 1/4, 1/5 etc. are getting closer and closer to ψ = 0 but are separated by invariant circles. We note that for K = 5.9 (the value considered in Fig. 5) there are no transport barriers in the tokamap model.
Proposition 2. The region where the non-twist map f given by (11) is closest (in C 0 topology) to a rigid rotation is an annulus intersecting the curve
Creg .
Proof : Consider (θ0 , ψ0 ) ∈ S 1 × R+ and the map fc deﬁned by the implicit equations ψ =ψ θ = θ + W ψ0 .
It results that d2 ((θ, ψ), (θ , ψ )) = K g (θ) h(ψ)

2

+ W (ψ) − W (ψ 0 ) + K g(θ) h ψ

2

.

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Now, because W (ψ) − W (ψ 0 ) = W (ψ 0 )(ψ − ψ 0 ) + 1 W (ψ 0 )(ψ − ψ 0 )2 + · · ·
2
is smaller near Creg (here W (ψ 0 ) = 0) it results that f and fc restricted to the disk surrounding (θ0 , ψ0 ) are closer in C 0 topology if (θ0 , ψ0 ) ∈ Creg than in all other cases. (Q.E.D)
It is important to mention that the winding function W and the functions g and h which deﬁne the perturbation are independent objects. It results from the integrability of fc that the region where f has a chance to be almost integrable (hence to have regular dynamics) is an annulus intersecting Creg .
Such an annulus does not exist for large perturbations (i.e. Kg (θ0 )h(ψ 0 ) or Kg (θ0 ) h ψ 0 are large for all (θ0 , ψ0 ) ∈ Creg ) because f and fc are not close enough to ensure the regularity of f . It can be very large for small perturbations, usually when K
1.
Figure 6 shows the phase portrait of the degenerate tokamap (a non-twist map) for K = 5.9. The central region of the phase space presents a clear regular zone. It is the annulus where the map is almost a pure rotation.
In order to relate the non-twist properties of a map to the existence of an annulus where the map is almost integrable we consider the rev-tokamap system. Fig. 6. The phase portrait of the degenerate tokamap (K = 5.9)

398

D. Constantinescu and B. Weyssow

Using the explicit form of WRT one obtains Creg : ψ = 1/C, i.e.
Creg : ψ =

K
1
+ sin(2πθ) C
2π (1 + C)

The implicit (13) cannot be solved analytically for the rev-tokamap hence Cnt is computed numerically (using the Broyden method with the error 10−15 ).
Computer experiments point out that Creg and Cnt are very close even for
¯
large values of K ∈ [0, 2π). This eﬀect is due to the fact that Kg (θ) h ψ <
3 −1
[2π (1 + 1/C) ] is usually small. We can conclude that in the non-twist annulus the rev-tokamap is almost integrable.
The analytical result concerning the regular annulus is in agreement with the experimental observations in tokamaks which point out the existence of a regular zone surrounding the so-called “zero shear region” i.e. the shearless curve. In order to give a diﬀerent perspective on the objects Creg and Cnt , we consider their intersections with the line θ = 0.5. These points are represented in Fig. 7 for a ﬁxed winding function characterized by the parameters w =
0.67, w0 = 0.3333 and w1 = 0.1667 and for various values of K in the range
[0.6, 4.5]. For small values of K the non-twist annulus is thin (it reduces to the curve ψ = 1/C for K = 0). Its size increases as K increases hence the nontwist dynamical property becomes more important. The shearless curve and the regular curve are located between the non-twist annulus. The steep jump observed in the position of the shearless curve for K ∈ [1.8, 1.9] is explained by the collision-annihilation of some periodic points having a rotation number slightly less than 0.67.
The non-twist annulus is, as shown above, a useful analytic tool, but the computer experiments have also revealed a transport barrier containing the non-twist annulus for small values of K. For K = 0 the map is integrable and the non-twist annulus reduces to a curve. For small perturbations (K < 2.5) the regular zone extends down to the circle ψ = 0 and no chaotic zone can be easily identiﬁed in this region. By further increasing K some invariant circles are destroyed, a chaotic zone is formed around the hyperbolic points having the rotation number 1/2 and both sides of the transport barrier are clearly identiﬁed. The transport barrier gets narrower while the non-twist annulus gets larger as K is increased. This phenomenon is shown in Fig. 7.
Very sharp changes in the position of the upper bound of the ITB is observed when K ∈ (1.2, 1.3), K ∈ (1.7, 1.8), K ∈ (2.7, 2.8) and K ∈ (3.4, 3.5).
These abrupt variations are explained by the chaotisation of some Poincar´– e Birkhoﬀ chains in the negative twist region. The Poincar´–Birkhoﬀ chain of e 1/3-type enters in the globally stochastic zone for a value of K in the range
K ∈ (1.2, 1.3). The same phenomenon occurs for the periodic orbit of 1/2type for K in the range K ∈ (1.7, 1.8), for the periodic orbit of 4/7-type for
K in the range K ∈ (2.7, 2.8) and ﬁnally for the periodic orbit of 3/5-type for K in the range K ∈ (3.3, 3.4). The width of the islands in a Poincar´– e Birkhoﬀ chain generally decreases when the period increases; for this reason

Order and Chaos in Some Hamiltonian Systems

399

1
0.9
the upper boundary of ITB

0.8
0.7

psi

0.6
0.5
0.4
0.3

the boundaries of tne nontwist annulus the shearless curve 0.2 the regular curve 0.1
0

1

1.5

2

2.5

3

3.5

k

Fig. 7. The position of ITB, of the boundaries of non-twist annulus, of the shearless curve and of the regular curve on the line θ = 0.5 for the rev-tokamap with the same winding function (w = 0.67, w0 = 0.3333, w1 = 0.1667) and various values of
K ∈ [0.6, 3.9]

the eﬀect of the chaotisation of 1/2-type periodic orbits (K ∈ (1.7, 1.8)) on the position of the ITB upper bound is more important than for the other cases. Many other periodic orbits also enter into the globally stochastic zone, but these cannot be observed in Fig. 7 because their periods are large and the corresponding island chains are very thin.
The rev-tokamap has positive twist property below the non-twist annulus and negative twist property above it. As long as an invariant circle is contained in these regions their destruction can be explained by arguments used in the theory of twist maps, the KAM theory or the cone crossing criterion for example. More interesting is the case of invariant circles contained in the non-twist annulus.

400

D. Constantinescu and B. Weyssow

5 Reconnection and Transport Barriers
Reconnection is a global bifurcation in the dynamics of a non-twist area preserving map. It is a change in the topology of the invariant manifolds of the hyperbolic points of two Poincar´–Birkhoﬀ chains having the same e rotation number m/n.
We shall study the reconnection process in the rev-tokamap model. In the degenerate tokamap model this phenomenon does not occur because there are not twin Poincar´–Birkhoﬀ chains (the winding function is monotonous). e Reconnection appears in systems with ﬁxed winding function W whose maximum value decreases to a value slightly larger than m/n as the stochasticity parameter K is increased or for ﬁxed K as the maximum value of W is decreased. The general scenario for reconnection is as follows: before reconnection the Poincar´–Birkhoﬀ chains having same rotation number are separated by e invariant sets (usually invariant circles), at the reconnection threshold the hyperbolic periodic points of the two chains are connected by their manifolds and after reconnection two dimerized island chains are formed. Each of them contains a set of hyperbolic periodic points connected either by heteroclinic connections or by homoclinic connections surrounding elliptic periodic points.
These dimerized islands are separated by meanders.
In the generic position before reconnection the periodic points situated approximately at the same θ have opposite stability (one is elliptic, the other is hyperbolic). In this case, the twin Poincar´–Birkhoﬀ chains approach each e other until the hyperbolic points get connected by a heteroclinic connection.
This is the typical behaviour in the rev-tokamap model.
Starting from a nongeneric position before reconnection, the periodic points situated approximately at the same θ have the same stability. Before reconnection a saddle-centre bifurcation occurs in a Poincar´–Birkhoﬀ e chain i.e. a new pair of periodic points having the same rotation number as the preexistent ones is created. By modifying the parameter the new points annihilate the old elliptic orbit. Finally only one elliptic orbit remains in a generic position with the other Poincar´–Birkhoﬀ chain. From this moment e the generic route to reconnection is followed.
We shall describe the reconnection of the 2/3 Poincar´–Birkhoﬀ chains for e K = 2 and K = 4 for decreasing values of w:
– The value K = 2 is small enough to ensure that the reconnection occurs in the regular zone:
– For w > wr ≈ 0.6702 the island chains are in the regular zone in a generic position and they are separated by invariant circles;
– For w = wr ≈ 0.6702 the reconnection occurs;
– For w < wr ≈ 0.6702 the dimerized islands are separated by meanders.

Order and Chaos in Some Hamiltonian Systems

401

Fig. 8. Twin Poincar´–Birkhoﬀ chains from the rev-tokamap model (K = 4, w = e 0.71, w0 = 0.3333, w1 = 0.1667)

– For K = 4 the perturbation is stronger. The reconnection occurs in a chaotic layer produced by the manifolds of the hyperbolic points (these manifolds intersect transversely):
– For w = 0.7 the island chains are outside the regular annulus in a nongeneric position (see Fig. 8).
– For w = 0.685 a saddle-centre bifurcation occurred in the upper island chain. Because at the bifurcation threshold the periodic points which bifurcate is parabolic (its multipliers are λ1 = λ2 = 1) the map is almost integrable around it, hence the upper chain enters the regular zone (Fig. 9)
– For w = 0.68213 the reconnection occurs (Fig. 10) in a chaotic layer.
By decreasing w the periodic points of the upper island chain situated approximately on the same vertical approach each other, collide and annihilate. The other island chain enters the regular zone and remains there until the collision–annihilation of its points occurs.
The main conclusion of this section is that the reconnection takes place inside the non-twist annulus (because it is speciﬁc to the non-twist dynamics)

402

D. Constantinescu and B. Weyssow

Fig. 9. The saddle-centre bifurcation occurred in the upper Poincar´–Birkhoﬀ chain e (rev-tokamap K = 4, w = 0.685, w0 = 0.3333, w1 = 0.1667)

in a zone which can be regular or chaotic, depending on the stochasticity parameter. In our last example, the shearless curve disappears at the end of the reconnection process (because it would be located between the dimerized island chains in a chaotic region) but a regular zone i.e. a transport barrier can however be identiﬁed (Fig. 9). Therefore the shearless curve of the revtokamap is not the most resistant invariant circle. This result, in contradiction with what is observed for the non-twist standard map [19], is explained by the fact that the latter map has time reversal and symmetry properties which are not possessed by the former. For this reason the rev-tokamap exhibits more general dynamical properties.

6 Conclusions
Four Poincar´ maps giving the intersection points of a magnetic ﬁeld line with e a given poloidal cross section are presented: the tokamap representing usual tokamak conditions, the rev-tokamap for a tokamak running with reversed

Order and Chaos in Some Hamiltonian Systems

403

Fig. 10. The rev-tokamap phase portrait at the reconnection threshold (K = 4, w = 0.68213, w0 = 0.3333, w1 = 0.1667)

shear conditions, the bounded map with the outmost tokamak magnetic surface forced to be a smooth surface and the degenerate tokamap. Each of the four maps develops its own magnetic structure which are explained using some of the new results on non-twist maps reported here.
To be speciﬁc, the maps may be characterized in terms of the ITB width, the formation of the ITB and the process of magnetic reconnection:
(a) Previous studies of the ITB width considered as important three particular curves: the regular curve, the non-twist curve and the shearless curve which coincide in the unperturbed case for non-twist maps. The detailed study of the four maps for diﬀerent parameter values has deﬁnitively showed that the regular curve indeed relates to the regular dynamics but that the shearless curve does not.
(b) The mechanisms leading to the formation of ITB’s with positive area has been clariﬁed in two situations which had not been considered previously: twist systems, like for instance the bounded tokamap, support the formation of large ITB because of their proximity to an integrable system and non-twist systems with monotonous unperturbed winding function, like the degenerate tokamap (see Sect. 4).

404

D. Constantinescu and B. Weyssow

(c) The principal diﬀerence between twist and non-twist maps lies in the possibility in the former to proceed with a reconnection mechanism. This process of reconnection is described in details in two limits, for small perturbations (in a regular zone) and for large perturbations (in the chaotic zone). The main consequence of a reconnection in a chaotic layer is that in the rev-tokamap system the shearless curve is not always the most resistant invariant circle (this result diﬀers from the usual behaviour of models that have a symmetry group).
Maps for magnetic ﬁeld line behaviour for tokamak geometries have already given a number of unexpected results that could facilitate the analysis of experimental results, even considering particles instead of magnetic ﬁeld lines in ITB’s. Further progresses are however still awaited in the area of guiding centre maps where some diﬃculties arise due to more stringent topological constraints.

Acknowledgement
The authors would like to thanks the members of the International working group “Fusion BFR”, for their great interest in the present work and in particular R. Balescu, J. Misguich and E. Petrisor for many detailed discussions on non-twist maps.

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B. Weyssow, J.H. Misguich: Europhys. Conf. Abstracts J 23 795 (1999)
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(May 1982), Physica 7D (1983), pp 240–258
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Numerical Simulation 8, 355 (2003)
D. del-Castillo-Negrete, J.M. Greene, P.J. Morrison: Physica D 91, 1 (1996)
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A. Delhams, R. de la Llave: Siam J. Math. Anal. 31, 1235 (2000)
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E. Petrisor: Chaos, Solitons and Fractals 14, 117 (2002)
E. Petrisor, J.H. Misguich, D. Constantinescu: Chaos, Solitons and Fractals 18,
1085 (2003).

Index

α relaxation 350, 351 β relaxation 350, 351 d–dimensional 283, 284, 289, 291 q-phase transition 339, 340, 342, 343,
346–348, 352
ACP model 224, 225, 235 activation 170 activation-energy 276 agglomeration 275–286, 290–292 aggregation 241, 275–277, 280,
282–284, 286, 289–292
Anderson model 99, 101 anharmonic oscillators 357 anisotropic harmonic oscillators 357
Anosov ﬂows 369 anticipative control 53, 74
Apollonian network 77, 81, 89–95 aquatic environments 117 autocatalytic growth 118
Bak–Sneppen model 159, 224 band merging 350 band-merging 349
Belgian mathematicians 22 bi-dimensional 303 bifurcation 400–402 bifurcation diagram 34, 54, 67, 69–74 bifurcation threshold 401 biological growth 121 biological growth rate 121, 122 boundary condition 78, 120, 279, 296,
302
bounded tokamap 388, 392–394, 396,
403
bubble 40, 46, 48, 239–242, 244, 245,
251, 255, 257, 275

canonical chaos map 54, 65–67, 74 catastrophe 40 catastrophic senescence 199 causality network 49 chaos map 53, 54, 56–60, 62–64, 66, 67 chaos theory 3, 10, 22, 29, 40, 49, 53,
307
chaotic anharmonic oscillator 362
Christoﬀel coeﬃcients 371 classical chaos 355, 356, 362 cluster merging 277 coherence 77, 80, 86, 91, 93–95 coherent state 77, 80–88, 93, 95 colloid 275, 276, 292 complex Lyapunov exponent 226 conﬁguration space 369–371, 374 coupled anharmonic oscillators 357 crack growth 295–301, 303–305,
307–312, 314 crack growth length 301 crack growth process 296–298, 303,
305, 306, 309, 311 crack growth resistance 300, 301 crack growth velocity 295, 296, 298,
304–307, 309–311 crash 40, 41, 46–48, 233, 235, 238–257 crash signature 245, 257 crash time 226, 235, 237, 238, 240, 241,
244, 245, 249, 255, 256 crashes 240, 257 cross-over temperature 280, 281 current state 59, 60
Debye-relaxation 276 degenerate tokamap 388, 389,
392–394, 397, 400, 403 deleterious mutation 198, 206

408

Index

deterministic scale-free network 77,
79–81, 83, 88, 89, 92–95 dimension 112, 132, 133, 137, 275, 276,
283, 290, 291, 318, 349, 373, 376 dimensional 137, 138, 142, 143, 275,
342, 374 dimensionless 137, 179 discrete non-Markov random process
177, 178, 194 discrete non-Markov stochastic process
175
disordered systems 99, 101 distribution function 225, 261, 262,
264–266, 317, 318, 321, 325, 332,
333
dynamic crack growth 304, 314 dynamical relaxation 341 dynamical state amplitude 77 ecology 21, 29, 31, 214, 218 economic growth 259, 260 elastodynamic crack growth condition
301, 302, 306 environment 3, 7, 29, 34, 161, 167, 168,
176, 199, 208, 213, 224, 225, 235,
288
environment capacity terms 154 environmental capacity terms 150 environmental changes 147 environmental open ﬂow 118 equilibrium state 62, 64, 121, 148, 350 ergogic states 348 exponentail growth 101 exponential growth 18, 32, 35, 40, 49,
99, 100, 102, 105, 108, 111, 197, 259 exponential growth model 6 extrinsic mortality 162
Fibonacci numbers 99, 290
Fibonacci sequence 99–101, 109, 290
Finsler spaces 370 ﬁrst-order phase transition to coherence
93
ﬁrst-order transition to coherence 93,
95
ﬁtness 101, 147, 148, 212–215, 224, 225 fractal dimension 318, 322–324,
326–328, 348, 349 full synchronized state 80

fully synchronized (coherent) states
77
fully synchronized state 80 future state 139 future states 60 genotype 168 geodesic instability 372 geodesics 369–372, 374, 375, 378, 381,
382
geometric indicator of chaos 369, 376,
378–380, 382 goal-seeking growth 35
Gompertz 162, 200
Gompertz law 197, 200, 201
Gompertz model 7 growth 3, 5, 6, 8, 9, 21, 29, 35, 36, 41,
43–47, 53, 59, 99, 101–113, 117,
118, 124, 147, 150, 151, 256, 287,
288, 318, 326, 327, 339, 350 growth archetype 30 growth archetypes 29 growth coeﬃcient 18, 23 growth curve 7, 32 growth exponent 283, 288, 320, 321,
325, 328, 329, 333 growth goal 36 growth mechanism 46 growth model 3, 5–7 growth parameter 23, 34, 53, 54, 66, 74 growth pattern 31, 32, 35, 40, 45 growth process 5, 7, 8, 31, 35 growth rate 6, 8, 31, 41, 42, 47, 54,
55, 119, 123, 125, 147, 149, 150,
152–155, 157, 300, 375 growth speed 6, 8, 290 growth time scale 119 harmonic oscillator 360, 363, 365, 366 high-temperature conditions 281 high-temperature limit 281
Hill equation 22, 375 incursive map 60 infant mortality 162, 165, 166,
168–170
inﬁnite-dimensional 291 instability 44, 117, 124–126, 176, 275,
284, 287, 295–298, 304, 305, 309,

Index
311–314, 369, 370, 372, 374–376,
392
intermediate growth 111, 112 internal transport barrier 385 intrinsic mortality 162, 167, 170
Irwin–Mott crack growth dynamic condition 302
Jacobi equation 369, 370, 372, 375,
378, 381, 382
Jacobi metric 369, 373 life expectancies 207, 209 life expectancy 162–164, 167–170, 172 limiting growth factor 54 limiting state 353 linear growth 105, 111 log-periodic 239, 241, 243, 251, 254,
255, 257 logistic fractal 24, 27 logistic function 8, 13, 17, 19, 22, 24,
26, 239–243, 245, 247, 249, 251,
253, 255–257 logistic growth 5, 29–31, 35, 36, 40, 49,
117, 118, 122, 239 logistic growth model 7, 8 logistic growth process 9 logistic map 34, 40, 77, 83, 84, 95,
223, 295, 307, 308, 310, 312–314,
339–342, 348, 352, 353 low temperature limit 275 low total mortality 168
Lyapunov exponent 82, 84, 87, 226,
227, 233, 234, 348, 355, 369, 370,
375, 376, 378–382
Malthus 6, 18, 225, 259
Malthus equation 53 marine environment 131 marine environments 143
Maupertuis least action principle 371 mean ﬁeld approximation 225, 264,
265
menopause 206–210 merging 281 merging parameter 225 metastable state 281
Methuselah life expectancy 168 mortality 6, 38, 161–165, 167–172

409

mortality data 171 mortality dependence on age 170 mortality dynamics 161 mortality plasticity 169 mortality plateau 209 mortality rate 6, 162, 172 mortality tables 4 mortality variable 169 mutation accumulation 198, 199, 201 mutation mechanism 157 natural selection 161, 167, 168 network 77–84, 88–96, 147, 150, 153,
157, 209, 260 network of interactions 147–150, 153,
154, 156, 157 neural networks 242 non universal extrinsic mortality 171 non-dimensional 119 non-twist map 390–394, 396, 397, 403,
404
non-universal mortality 170, 172 nonequilibrium 178, 275, 276, 278,
280, 287, 291, 292 nonergodic state 348, 353 one-dimensional 22, 101, 148, 326 oscillatory 23, 195, 239, 313 parametric instability 374 parametric resonance 374, 381, 382 parametric stability 374
Parkinson’s disease 175, 177, 182–185,
194, 195 percolation transition 153, 157 phase portrait 385, 390–392, 396, 397,
403
phase space 32, 77, 81, 84, 85, 99, 110,
112, 118, 127, 340, 348–350, 352,
355–358, 370, 390, 394, 397 phase transition 280 phenotype 168, 211, 213–215 plateau 40, 350, 351, 353 population growth 3, 5, 6, 8, 17–21,
117, 147, 339 power law 79–81, 85, 86, 89, 90, 95,
241, 276, 279, 283 power-law 109, 111, 112, 152, 259–261,
267, 341, 343, 350–353

410

Index

power-law growth 101, 109 predicted state 59
Probability distribution 99, 101, 102,
111
probability distribution 149, 151, 154,
156, 256 program 198, 203 pseudo-fractal network 81, 89, 90,
92–96
quantitative factor of quality of the medical treatment 181 quantitative factor of quality of the treatment 182 quantum chaos 355, 356, 362
Quetelet 4, 5, 13–20, 22 random process 156, 292 random scale-free network 83, 88, 95,
96
random sequence 106 random sequences 99, 101, 103,
105–107, 109, 111–113 reconnection 386, 391, 400–404 reconnection threshold 392, 400, 403 relaxation 175, 178–180, 193, 195, 241,
275, 276, 282, 284, 285, 287, 288,
291, 292, 295–298, 304, 306, 310,
311, 350, 351 relaxation mode 284 relaxation parameter 185, 191–195 relaxation time 179, 295, 296, 304,
307, 314, 339, 341, 349–351
Renyi dimensions 343 replicative senescence 161, 170 rescaling 100, 243, 244 restricted three-body problem 369,
370, 374–376, 382 rev-tokamap 387 reversed tokamap 387, 391–395,
397–404
Riemann curvature tensor 379, 381
Riemannian curvature 380
Riemannian geometry 370, 382
Riemannian manifold 369–371, 382
Riemannian metric 370 rugged crack growth 304 scalar curvature

371, 374, 382

scale-free network 77–81, 83, 88, 93,
95, 96, 209 scaling 55, 163, 165, 170, 201, 203, 239,
255, 318, 321, 323–327, 329, 333,
334
Selection 162 selection 78, 155, 156, 158, 161, 162,
167, 170 selection pressure 198, 200, 201, 224,
225
selection trait 158, 214 selection traits 211 separation vector 135, 371, 372, 374 shift map 54, 56, 57, 74 shock 241, 243, 251, 257 short mortality memory 169 sigmoid growth curve 20 single state 350 slow growth conditions 285 stability 24, 33, 148, 227, 229, 238, 254,
256, 278, 369, 371, 374–376, 382,
389, 390, 400 stability analysis 80, 83, 84, 90, 125 stability criterion 55, 58, 61, 62, 370 stability intervals 226 stability time 226, 227, 229, 231 stabilization diagram 53, 54, 67, 69–74 stable state 154, 157 state 38, 54, 59, 60, 78, 117, 121, 124,
127, 154, 262, 351 state dependent strategy 225 state variable 39 state variables 36 state with glassy properties 353 stationary state 58 steady state 53, 55, 58, 139, 149, 150,
152, 153, 155, 157, 301, 312 stochastic process 101, 177, 260, 283 stress relaxation 285, 288 sympatric speciation 210 symplectic 370, 378 synchronization 78, 80, 82, 83, 88, 89,
93–95, 276, 288 synchronous state 82 telomere 199, 210, 211 temperature 169, 275–277, 280, 281,
298, 299, 350 three-dimensional 290, 318, 319, 333

Index time delay – initial concentration resonance 238 tokamap 387–393, 396, 402 total mortality 168 transition 80, 83, 84, 86, 87, 99, 101,
112, 121, 124, 125, 152, 153, 181,
182, 348, 349, 352 transition curve 85, 86 transition to a periodic organisation
124
transition to chaos 339 transition to coherence 77, 83–89, 93,
94, 96 transition to full synchronization 83 transition to percolation 78 trend 40, 42, 44, 45, 47, 137, 144, 239,
267
two-dimensional 118, 119, 122, 321,
326, 370, 375 two-step relaxation 339, 341, 353

Verhulst 3–5, 7, 8, 10, 13–27, 29–31,
33, 35–37, 39–41, 43, 45, 47, 49, 51,
53, 54, 56, 57, 74, 117, 122, 128,
147, 197, 199, 201, 203, 206, 212,
216, 223, 225, 259, 275, 276, 289,
339
Verhulst chaos map 53–57, 59–61, 64,
74
Verhulst diﬀerential equation 58, 59,
74
Verhulst diﬀerential growth equation
54
Verhulst equation 29–31, 122
Verhulst incursive map 57, 58, 60, 74
Verhulst logistic equation 124
Verhulst map 22, 53, 54, 56, 57, 60, 61,
74
von Neuman neighbourhood 224 yeast mortality

uniform description 356, 358 universal intrinsic mortality 171 universal mortality 169–171 universal mortality law 163 unstable state 60

411

168

zero mortality 168 zero universal mortality 168 zooplankton mortality 119

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