INTRODUCTION TO CHAOS THEORY
BY T.R.RAMAMOHAN C-MMACS BANGALORE -560037

SOME INTERESTING QUOTATIONS
* “PERHAPS THE NEXT GREAT ERA OF UNDERSTANDING WILL BE DETERMINING THE QUALITATIVE CONTENT OF EQUATIONS; TODAY WE DO NOT KNOW WHETHER THE EQUATIONS OF FLUID FLOW CONTAIN THE BARBER POLE STRUCTURE OF TURBULENCE; TODAY WE DO NOT KNOW WHETHER THE SCHRODINGER EQUATION CONTAINS FROGS, MUSICAL COMPOSERS OR MORALITY OR WHETHER SOMETHING BEYOND IT LIKE GOD IS NEEDED OR NOT “ PARAPHRASED FROM R.P.FEYNMAN * “ONLY A CORNER OF THE VEIL HAS BEEN LIFTED, BUT PERHAPS WE HAVE STARTED ON THE ABOVE JOURNEY ” PARAPHRASED FROM FLORIS TAKENS

WHAT IS CHAOS THEORY?
* CHAOS THEORY IS BASED ON THE OBSERVATION THAT SIMPLE RULES WHEN ITERATED CAN GIVE RISE TO APPARENTLY COMPLEX BEHAVIOR. * EG. LET US CONSIDER
X0
RULE

X1 X2
RULE
| | | |

X1

XN

RULE

X N+1

THE BASIC OBSERVATION IS ;

IF IN THE IMPLEMENTATION OF THE ABOVE RULE X 0 IS CHANGED BY A SMALL AMOUNT, THE RESULTING SEQUENCE WILL BE VERY DIFFERENT. AFTER SOME ITERATIONS MATHEMATICIANS HAVE STILL NOT AGREED ON A DEFINITION OF CHAOS THAT IS ACCEPTABLE TO ALL

HOW EVER AN OPERATIONAL (MEANING ONE THAT CAPTURES MOST ESSENTIAL FEATURES) DEFINITION OF A CHAOTIC SYSTEM IS ; 1) THE SOLUTION MUST BE APERIODIC 2) THE SOLUTION MUST BE BOUNDED 3) THE SOLUTION MUST BE EXPONENTIALLY SENSITIVE TO INITIAL CONDITIONS

THE RULE THAT GOVERNS A CHAOTIC SYSTEM MUST SATISFY CERTAIN PROPERTIES 1) THE RULE MUST BE NONLINEAR 2) IN THE CASE OF MAPS, IT MUST BE NON-INVERTIBLE IF IT IS ONE DIMENSIONAL 3) IN THE CASE OF ODES IT MUST BE ATLEAST THREE DIMENSIONAL

THE THREE CHARACTERISTICS WE HAVE MENTIONED ARE ALL ESSENTIAL FOR A SOLUTION TO BE CALLED CHAOTIC NONLINEARITY IS ESSENTIAL WHAT IS MEANT BY A SYSTEM BEING NONLINEAR IS : THE WHOLE IS GREATER THAN OR LESS THAN THE SUM OF ITS PARTS. IN A LINEAR SYSTEM THE WHOLE IS EQUAL TO THE SUM OF ITS PARTS

A TYPICAL EXAMPLE OF A CHAOTIC SYSTEM IS ;
X N +1 = 10 X N (mod1)

O ≤ XN ≤ 1 TO SHOW CHAOS IN ACTION IN THIS SYSTEM; CONSIDER : THEN WE SEE AND

X 0 = 0.1123456…….
X 0 = 0.1123598…….
|

X 1 = 0.123456……
X 0 = 0.123598…….
X 2= 0.23456…….
| X 2 = 0.23598……. |

WE SEE THAT THE ITERATES DIVERGE VERY RAPIDLY

CERTAIN PROPERTIES OF CHAOTIC SYSTEMS MAKE THEM INTERESTING : (1) CHAOTIC SOLUTIONS UNDERGO A STRETCHING AND FOLDING IN PHASE SPACE BECAUSE OF EXPONENTIAL DIVERGENCE AND BOUNDEDNESS. (2) CHAOTIC SYSTEMS CAN EITHER BE “DISSIPATIVE” OR “ CONSERVATIVE” IN DISSIPATIVE SYSTEMS PHASE SPACE VOLUMES CONTRACT IN CONSERVATIVE SYSTEMS PHASE SPACE VOLUMES ARE CONSERVED

(3) CHAOTIC SYSTEMS CAN BE CONTROLLED. A CHAOTIC SOLUTION CONTAINS AN INFINITE NUMBER OF UNSTABLE PERIODIC SOLUTIONS OF ARBITRARY PERIOD. AT LEAST, ‘IN PRINCIPLE’, THE CHAOTIC SYSTEM CAN BE CONTROLLED ON TO ONE OF THESE PERIODIC SOLUTIONS. (4) CHAOTIC SYSTEMS CAN BE “SYNCHRONISED “ TO EACH OTHER, ‘ AT LEAST ‘, IN PRINCIPLE TWO OR MORE CHAOTIC SYSTEMS CAN BE COUPLED TO EACH OTHER SO THAT • THEIR OUTPUTS DIFFER BY A CONSTANT • THEIR OUTPUTS LAG EACH OTHER BY A TIME LAG • THEIR PHASES ARE SYNCHRONIZED • THEIR OUTPUTS ARE RELATED TO EACH OTHER BY AN INVERTIBLE MAP

(5) IN SPITE OF THEIR APPARENT COMPLEXITY THERE ARE CERTAIN INVARIANTS ASSOCIATED WITH CHAOTIC SYSTEMS THESE CAN BE CLASSIFIED ROUGHLY INTO (a) DYNAMICAL INVARIANTS TYPICAL DYNAMICAL INVARIANTS INCLUDE THE ‘LYAPUNOV EXPONENTS’ WHICH MEASURE THE AVERAGE RATE OF DIVERGENCE OF THE SOLUTION. (b) METRICAL INVARIANTS TYPICAL METRICAL INVARIANTS INCLUDE “ DIMENSIONS” WHICH MEASURE IN SOME SENSE THE DISTRIBUTION OF POINTS ON THE ATTRACTOR (c) TOPOLOGICAL INVARIANTS TYPICAL TOPOLOGICAL INVARIANTS INCLUDE THE “TEMPLATE” OF THE SYSTEM

ALGORITHMS HAVE BEEN DEVELOPED BY A NUMBER OF AUTHORS TO MEASURE APPROXIMATIONS TO THESE INVARIANTS ALL THE ALGORITHMS NEED AN INFINITE AMOUNT OF NOISE FREE DATA TO BE EXACT HOWEVER, CERTAIN HEURISTIC GUIDELINES HAVE BEEN DEVELOPED TO COMPUTE APPROXIMATIONS TO THESE INVARIANTS FROM NOISY FINITE TIME SERIES 6) THE OUTPUT OF CHAOTIC SYSTEMS MAY APPEAR IRREGULAR BUT THEY ARE GENERATED BY DETERMINISTIC RULES

SINCE THE OUTPUT OF A CHAOTIC SYSTEM IS GENERATED BY A DETERMINISTIC RULE. THE RULE CAN IN MANY CASES BE EXTRACTED FROM THE OUTPUT A NUMBER OF SUCH TOOLS HAVE BEEN DEVELOPED BY A NUMBER OF AUTHORS. THE RULE THAT IS THUS EXTRACTED CAN BE USED TO DETERMINE IMPORTANT FEATURES OF THE SYSTEM.

THE RULE WHICH HAS BEEN EXTRACTED FROM THE SYSTEM CAN BE USED: (a) TO MAKE ACCURATE SHORT TERM PREDICTIONS OF THE SYSTEM OUTPUT (b) TO GENERATE ENOUGH DATA TO MORE ACCURATELY ESTIMATE THE DYNAMICAL, METRICAL AND TOPOLOGICAL INVARIAN TS OF THE SYSTEM (c) TO FINE TUNE ALGORITHMS FOR “CONTROL OF CHAOS” (d) TO FINE TUNE ALGORITHMS FOR ‘SYNCHRONIZING’ TWO OR MORE CHAOTIC SYSTEMS (d) TO STUDY THE EFFECTS OF NOISE IN SUCH SYSTEMS

THE OUTPUT OF A CHAOTIC SYSTEM CAN BE CONTAMINATED WITH NOISE USUALLY TWO TYPES OF NOISE CAN CORRUPT THE OUTPUT (a) MEASUREMENT NOISE THIS IS NOISE THAT ARISES FROM INSTRUMENTAL ERRORS IN MEASURING THE OUTPUT OF THE SYSTEM IF 'F(XN )' IS THE RULE THIS TYPE OF NOISE OCCURS AS
X N +1 = 10 X N + ε

HERE ‘ε ‘ IS THE NOISE

b) DYNAMICAL NOISE THIS OCCURS AS A RESULT OF UNCERTAINTIES IN THE SYSTEM WE MEASURE IF F(X N)IS THE RULE, THEN THIS IS REPRESENTED AS

X N+1 = F(X N + ε )

HERE ‘ ‘ IS THE NOISE ε THE THEORY OF MEASUREMENT RELATIVELY WELL DEVELOPED NOISE IS

THE THEORY OF DYNAMICAL NOISE IS STILL IN ITS INFANCY

IN SPITE OF THE VARIETY OF CHAOTIC SYSTEMS THERE ARE CERTAIN “UNIVERSAL” FEATURES OF SUCH SYSTEMS WHICH MAKE THEM INTERESTING SOME EXAMPLES OF SUCH “UNIVERSAL FEATURES” INCLUDE a) ROUTES TO CHAOS THE APPROACH OF A SYSTEM TO CHAOTIC BEHAVIOUR IS IN MANY CASES SIMILAR (SOMETIMES EVEN QUANTITATIVELY) THREE MAJOR ROUTES TO CHAOS HAVE BEEN IDENTIFIED . MANY OTHER ROUTES HAVE BEEN IDENTIFIED, THOUGH THEY APPEAR IN SPECIAL CIRCUMSTANCES

THE MAJOR ROUTES ARE (i) THE PERIOD DOUBLING ROUTE TO CHAOS IN THIS ROUTE, THE SYSTEM UNDERGOES A NUMBER OF SUB-HARMONIC BIFURCATIONS WHICH EVENTUALLY ACCUMALATE AT A CRITICAL VALUE. FOR LOCALLY QUADRATIC, UNIMODAL MAPS THERE ARE QUANTITATIVE NUMBERS THAT ARE OBSERVED IN THIS ROUTE (ii) THE QUASI-PERIODIC ROUTE TO CHAOS IN THIS ROUTE TO CHAOS, THERE ARE A NUMBER OF PERIODIC SOLUTIONS WITH IRRATIONALLY RELATED FREQUENCIES THAT LEAD TO CHAOS.

AN INTRESTING FEATURE OF THESE FREQUENCIES IS THAT IN MANY CASES AFTER TWO OR THREE SUCH FREQUENCIES ARE EXCITED, THE SYSTEM BECOMES CHAOTIC. (iii) THE INTERMITTENCY ROUTE TO CHAOS. IN THIS ROUTE TO CHAOS, THERE IS BEHAVIOUR THAT IS ALMOST PERIODIC WHICH IS INTERSPERSED WITH REGIMES OF IRREGULAR BEHAVIOUR. THE OCCURENCE OF THESE IRREGULAR BURSTS INCREASE UNTIL THE SYSTEM BECOMES APPARENTLY IRREGULAR CERTAIN SCALING LAWS HOLD FOR THE DURATION OF THESE BURSTS. WHICH ARE OFTEN OBSERVED IN MANY SYSTEMS.

IN ADDITION TO THESE MAJOR ROUTES TO CHAOS, THERE ARE SOME OTHER ROUTES WHICH OCCUR IN A LIMITED NUMBER OF SITUATIONS. IN A NONLINEAR SYSTEM WHERE THE STATE VARIABLES ARE COUPLED TO EACH OTHER, IT IS QUITE CLEAR THAT A MEASUREMENT OF ANY VARIABLE CONTAINS INFORMATION ABOUT ALL THE OTHER VARIABLES. IN DISSIPATIVE SYSTEMS, THE SYSTEM’S LONG TERM BEHAVIOUR IS GOVERNED BY FEWER VARIABLES THAN THE NUMBER OF INITIAL VARIABLES.

THIS IMPLIES THAT THE SYSTEM’S LONG TERM BEHAVIOUR RESTS ON AN “ATTRACTOR” IN MANY SITUATIONS. THE STRETCHING AND FOLDING OF THE PHASE SPACE ASSOCIATED WITH THE COMBINATION OF EXPONENTIAL DIVERGENCE AND BOUNDEDNESS IMPLIES THAT THE ”ATTRACTOR” MAY HAVE STRUCTURE AT ALL SCALES. THE DIMENSION OF THE ATTRACTOR MAY THUS BE A NON-INTEGER. IN THIS CASE, THE ATTRACTOR IS CALLED “ STRANGE”.

SOME TYPICAL ATTRACTORS

MANY MORE SIMILAR IMAGES CAN BE FOUND ON THE INTERNET

THE FACT THAT IN DISSIPATIVE SYSTEMS, THE SYSTEMS LONG TERM BEHAVIOUR RESIDES IN A SPACE OF MUCH LOWER DIMENSION THAN THE INTIAL DIMENSION OF THE SYSTEM COUPLED WITH THE FACT THAT THE EVOLUTION OF ANY ONE VARIABLE CONTAINS INFORMATION ABOUT THE EVOLUTION OF ALL THE VARIABLES GOVERNING THE SYSTEM OPENS UP MANY POSSIBILITIES

THIS LEADS TO THE FAMOUS EMBEDDING THEOREM WHICH STATES THAT IF THE EVOLUTION OF A SYSTEM IS GOVERNED BY ‘D’ VARIABLES THAN AN EMBEDDING DIMENSION OF LESS THAN ‘2D+1’ DIMENSIONS IS SUFFICIENT TO COMPLETELY DETERMINE THE BHAVIOUR OF THE SYSTEM THIS WHEN COUPLED WITH THE FACT THE EVOLUTION OF ANY ONE VARIABLE CONTAINS INFORMATION ABOUT ALL THE VARIABLES, SUGGESTS THAT THE MEASUREMENT OF ANY ONE VARIABLE CAN BE USED TO DETERMINE THE BASIC FEATURES OF THE SYSTEM UNDER CONSIDERATION.

THE MATHEMATICAL THEOREM WHICH CAPTURES THE ESSENCE OF THE ABOVE STATEMENT REQUIRES AN INFINTE AMOUNT OF NOISE FREE DATA IN PRACTICE, HOWEVER HEURISTIC RULES HAVE BEEN DEVELOPED THAT CAN ESTIMATE THE BASIC FEATURES OF THE SYSTEM, IN MANY CASES, FROM THE EVOLUTION OF A SINGLE VARIABLE. THE ABOVE FACTS LEAD TO ONE OF THE MAJOR APPLICATIONS OF CHAOS THEORY

THIS IS “ TIME SERIES ANALYSIS” MANY OF THE COMPLEX EVOLUTION BEHAVIOUR WE SEE IN NATURE AND ENGINEERING IN THE FORM OF A MEASUREMENT OF A SINGLE VARIABLE WITH TIME HAVE BEEN LOOKED AT THROUGH THIS CONCEPTUAL LENS WITH SOME SUCCESS. TWO MAJOR FACTORS THAT HAVE TO BE CONSIDERED IN ANALYZING A TIME SERIES ARE THE “TIME DELAY” AND THE “DIMENSION”. THE TIME DELAY IS ARBITRARY IN THEORY, BUT IN PRACTICE IT IS NECESSARY TO USE HEURISTIC GUIDELINES TO ESTIMATE IT.

SOME SUCH GUIDELINES ARE : (i) THE FIRST FUNCTION ZERO OF THE AUTOCORRELATION

(ii) THE FIRST MINIMUM OF THE AVERAGE MUTUAL INFORMATION FUNCTION THE ABOVE FUNCTIONS MEASURE THE EXTENT TO WHICH SUCCESSIVE MEASUREMENTS ARE RELATED TO EACH OTHER THE “DIMENSION” OF A SYSTEM MEASURES THE NUMBER OF VARIABLES REQUIRED TO DESCRIBE, TO A LARGE EXTENT, THE SYSTEM BEHAVIOUR IN THEORY THIS IS “ 2D + 1’ WHERE ‘D’ IS THE NUMBER OF VARIABLES GOVERNING THE SYSTEM

IN PRACTICE, THERE ARE A NUMBER OF HEURISTIC GUIDELINES TO ESTIMATE THIS. SOME OF THESE ARE: PRINCIPAL COMPONENT ANALYSIS THIS IS A MEASURE OF THE NUMBER OF VARIABLES REQUIRED TO LARGELY DESCRIBE THE SYSTEM THE FALSE NEAREST NEIGHBOUR TECHNIQUE THIS IS A MEASURE OF THE NUMBER OF DIMENSIONS REQUIRED TO COMPLETELY UNFOLD THE ATTRACTOR, I.E.TO REMOVE SELF-INTERSECTIONS IN PHASE SPACE

ANOTHER MAJOR APPLICATION OF THE IDEAS OUTLINED SO FAR IS “SECURE COMMUNICATION”. HERE A MESSAGE TO BE TRANSMITTED IN A SECURE FASHION IS CODED USING CERTAIN FEATURES OF THE CHAOTIC SYSTEM SUCH AS THE FACT THAT WE MENTIONED EARLIER THAT CHAOTIC SYSTEMS CAN BE SYNCHRONISED.

MANY FEATURES AND APPLICATIONS STILL REMAIN TO BE DISCOVERED ONLY YOUR IMAGINATION IS THE LIMITING FACTOR

GOOD LUCK AND THANK YOU