Henon attractor application in real life

The Henon map is one of the many 2-Dimensional maps. There are at least two maps known as Henon map. One of which is the 2-D dissipative quadratic map, given by the following X and Y equations that produce a fractal made up of strands [3] :

xn+1 = 1 - axn2 + byn yn+1 = xn

The Henon map can also be written in terms of a single variable with two time delays,
Since the second equation above can be written as yn = xn-1:

xn+1 = 1 - axn2 + bxn-1

It’s a simple invertible iterated map that showed a chaotic attractor and it’s a simplified version of the Poincare map for the Lorenz model. It was named after its discoverer, the French mathematician and astronomer Michele Henon.[2] [5]

[pic]

Fig. 1 Henon map with parameters a = 1.4 and b = 0.3.

The chaotic behavior of the attractor has many physical applications. Such as:

▪ Application to the transverse betatron motion in cyclic accelerators ▪ Application of the Henon Chaotic Model on to design of low density parity ▪ Application to Financial Markets ▪ Application on area-conserving ▪ Deterministic chaos in financial time series by recurrence plots ▪ Application to the motion of stars

Application in air bubble formation

Introduction

Below the explanation of how the Henon attractor effects a real life application is presented, which is based on the bubble formation.
This experiment took place in order to detect the chaotic dynamics that give the bubble shape and motion.
By using the methodology that is described below, observations using topological characterization, a chaotic region where some reconstructed attractors resemble Henon-like attractors, which visualize a possible route to chaos in bubbling dynamics.
The formation of air bubbles was studied submerging nozzle in a water/glycerol solution inside a cylindrical tube, and then was submitted to a sound wave perturbation. Observations showed a route to chaos due to the effect of period doubling, as a function of the sound wave amplitude. A flip saddle can then be localized; allowing us to establish relations of Henon-like dynamics to the construction of symbolic planes. [1]

Methodology

Keeping the air flow rate (~36.6 bubbles/s) and the sound wave frequency(150Hz) constant, but increasing the driven voltage (in the loudspeaker), a period doubling occurs around 2V and the bubbles are issued in pairs until about 3V when there is a two band behavior that presents chaotic behavior which is getting replaced by a large emerged chaotic attractor at 3.5V. The attractor dimensions for driven voltages of 3.5V and 4V are close to the dimensions of the henon map. After comparing the bubble formation dynamics by keeping fix six driven voltages, the best similarity with henon attractor occurs for the driven voltage of 3.5V. [1]

It’s stated that there is a conjecture which is valid for Henon attractor and it is checked on reconstructed attractors. This conjecture relates the Lyapunov spectrum (λn) and the information dimension by the Kaplan-Yorke dimension DKY[4]. Taking as driven voltages 3.5V, 4V, 4.5V the Kaplan-Yorke dimensions agree with the information dimensions. The first two chaotic attractors (3.5V and 4V) have a Lyapunov spectrum with one positive and one negative exponent, while the third one has one positive and two negative exponents. So it is observed the first two chaotic attractors have kind of the same dimensions with the Henon map. Because it is a 2 dimensional attractor the cue when the driven voltage is 4.5V can not be untangled. [1][5]

After drawing the bisectors to find the crossing points with the reconstructed attractors, it was reported that for a driven voltage of 4V the flipping behaviors were similar for all three cases. Therefore even when the voltage is at 4.5V (three components for Lyapunov spectra), there is a flip saddle. [1]

By using a = 1.55 and b = 0.1 as henon map parameter values instead of the classical ones a = 1.4 and b = 0.3, it is achieved similarity between the experimental attractors and the Henon one. Symbolic sequences were generated and it was observed that the pattern of the experimental symbolic planes resembles the Henon symbolic plane. [1]

Conclusion

Overall I can conclude that the use of 2D attractors to simulate bubble formation was a success. The Henon attractor allowed the simulation of the random notion of the formation and production of the bubbles along with. The main disadvantage of using this was the increased complexity compared to the 1D attractor. However this is necessary when dealing with more advanced applications. The 2D attractor can be used to predict and determine random bubble formations however a more complex attractor (3D) will have to be used for an application such as gas formation. This creates a limiting factor in the use of the 2D attractor but for the application I studied it more than sufficed.

Bibliography

[1] A. Tufaile, J.C. Sartorelli. “Henon-like attractor in air bubble formation”. Physics Letters A 275 )2000) 211–217. |http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6TVM-41DHXG0-7-23&_cdi=5538&_user=122878&_orig=search&_coverDate=10%2F09%2F2000&_sk=997249996&view=c&wchp=dGLzVlz-zSkWz&md5=81f2df70fd1c7aa846bc1554bfd68817&ie=/sdarticle.pdf|

[2] G. Elert. “The chaos hyper text book”. |http://hypertextbook.com/chaos/21.shtml|

[3] M. Henon, (1976). Communications Of Mathematical Physics. MAIK
Nauka/Interperiodica, 1976.

[4] Konstantinos E. Chlouverakis, J.C. Sprott. “A comparison of correlation and Lyapunov dimensions”. Physica D 200 (2005) 156–164. |http://sprott.physics.wisc.edu/pubs/paper289.pdf|

[5] A. Cenys, “Lyapunov Spectrum of the Maps Generating Identical Attractors”(1993) Europhys. Lett. 21 407-411

[6] J. C. Sprott. “Henon Map Maximum Lyapunov Exponent and Kaplan-Yorke Dimension”(2007). | http://sprott.physics.wisc.edu/chaos/henondky.htm|