Henon Attractor Application in Real Life
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
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
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|