Optimization of Hénon chaotic coefficients based on Lyapunov exponents

Prof. Antonios Andreatos (1) and Prof. Apostolos Leros (2)

(1) Div. of Computer Engineering & Information ScienceHellenic Air Force Academy

Dekeleia, Attica, TGA-1010, GREECE aandreatos.hafa@haf.gr, aandreatos@gmail.com

(2) Department of Automation EngineeringSchool of Technological Applications

Technological Educational Institute of Sterea Hellas34400 Psachna, Evia, GREECE

lerosapostolos@gmail.com

v p7

Βελτιστοποίηση των χαοτικών συντελεστών Hénon βάσει του εκθέτη Lyapunov

Α. Ανδρεάτος1 & Α. Λερός2

1Τομέας Πληροφορικής και Υπολογιστών, Σχολή Ικάρων

2 Τμήμα Αυτοματισμού, ΤΕΙ Στερεάς Ελλάδος

3rd International Conference CryCybIW

27 May 2016

Abstract

In this study we demonstrate a simple method for finding optimal Hénon's parameters in order to get good quality chaotic behavior.

Lyapunov exponent is used as the criterion for assessing chaotic behavior.

The method has been simulated in Matlab.

Keywords: Hénon map, chaotic map, Lyapunov exponent, optimization, simulation, Matlab.

Introduction - Hénon map

The Hénon map is produced by the solution of two coupled first-order differential equations:

x' = 1-a*x^2+y

y' = b*x

where a, b are chosen parameters.

Typical values are: a = 1.4 & b = 0.3.

Difference equations

For numerical solution, the following set of difference equations is used:

x(i+1)= y(i) + 1 - ax(i)^2

y(i+1) = bx(i)

with initial conditions [x(0), y(0)] = [0, 0] and constant coefficients a and b.

Attractors and strange attractors (ελκυστές)

In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist.

https://en.wikipedia.org/wiki/Attractor#Strange_attractor

Hénon attractor

Depending on the initial point (x0,y0) and the parameters a & b, the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange strange chaotic attractorchaotic attractor.

Next figure shows the characteristic Hénon attractor obtained by 20000 iterations for a=1.4 and b=0.3, starting from the initial point x0 = 0, y0 = 0.

Hénon Strange Attractor

Lyapunov exponent

In mathematics the Lyapunov exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories.

Quantitatively, two trajectories in phase space with initial separation δZο diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by |δZ(t)| ≈ eλt|δZο|, where λ is the Lyapunov exponent.

https://en.wikipedia.org/wiki/Lyapunov_exponent

AIMIn this study, given an initial point (x0, y0) and initial values for the parameters a0 and b0, we demonstrate a simple method for finding optimal Hénon's parameters abest and bbest, in order to get good quality chaotic behavior. Lyapunov exponent is used as the criterion for assessing chaotic behavior.

The proposed method during the first round optimizes the a parameter (finds abest) while keeping b constant; the abest value is that which gives the relatively maximum Lyapunov exponent.

Using abest constant, the process is repeated for finding bbest which maximizes the Lyapunov exponent.

Matlab simulation

The whole research is simulated in Matlab. From the simulations it was clear that there are countless parameter combinations which produce good chaotic behavior, as well as, other sets which produce no chaos at all (indicated by zero or negative Lyapunov exponent).

Using this two-phase method we achieve an increase of the Lyapunov exponent from about 1.5 to up to 2 or even higher. The higher the Lyapunov exponent the more chaotic the signal; Lyapunov exponent values around 3 are considered good.

Simulation results

a) Optimizing a

Starting values:

aο = 1.3175

bο = 0.3

da = 0.0125

i = 1

i = 5

i = 6

i = 8

a values for 10 steps

Results after 1st round

max Lyapunov exponent = 1.7089

for a best = 1.355

a) Optimizing b

Starting values:

aο = 1.355

bο = 0.293

db = 0.001

j = 16

j = 22

j = 30

j = 41

b values for 42 steps

Final result

max Lyapunov exponent = 1.9376

for

a best = 1.355

b best = 0.3290

Achievement

Double optimisation loop (for a & b) produces considerably better results than single loop:

Lyapunov exponent initially λ ≈ 1.52.

Lyapunov exponent after 1st round λ = 1.7089.

Lyapunov exponent after 2nd round λ = 1.9376.

In general we have elevated the Lyapunov exponent from values as low as 1.5 to 2 or even 2.5;

Simulation code is a good tool for studying the behaviour of Hénon chaotic systems.

Hénon attractor for a=1.355 & b=0.329 ==>

Hénon attractor for a=1.355 & b=0.329

ComparisonLeft: a = 1.4 & b = 0.3; Right: a=1.355 & b=0.329

Example good sets of (a,b)

x0 = 0; y0 = 0; (a = 1.4, b = 0.3)

a0 = 1.3175; INITIAL VALUE b0 = 0.29; da = 0.0125

a0 = best_a; b0 = 0.29; db = 0.001; n = 400;

max_lyap_b_value = 1.8797 at best_b value of 0.2940

x0 = 0; y0 = 0; (a = 1.4, b = 0.3)

a0 = 1.3175; INITIAL VALUE b0 = 0.3; da = 0.0125

a0 = best_a; b0 = 0.293; db = 0.001; n = 400;

max_lyap_b_value = 1.9376 at best_b value of 0.3290

Studying the behaviour of Hénon's chaotic map

Example1:Transition from chaotic to periodic behaviour

Example2: Choose diff. starting value instead of best a (1.8): (same result)

Example3: Changing initial value of a (a0 = 1.3168): (diff result)

Conclusions

Using this simulation tool we can investigate the behaviour of Hénon's chaotic map.

System is parameterizable (Xo,Yo,a,b,da,db,n).

We can optimise the produced chaos.

We can find various sets of good parameters, around the default (Hénon's) values (a=1.4, b=0.3), for instance: (a=1.355, b=0.329) and and (a=1.0, b=0.54).

There are countless good pairs (a,b), since a and b can be any real numbers other than zero.

Applications

Chaotic numbers find various applications including:

* Cryptography

* Image encryption

* Steganography

* Etc.

BibliographyHénon M., A Two-dimensional Mapping with a Strange Attractor. Commun.

Mathematical Physics 50, 69—77 (1976), Springer-Verlag.

Ching-Kun Chen, Chun-Liang Lin and Yen-Ming Chiu, Data Encryption Using ECG Signals with Chaotic Hénon Map, International Conference on Information Science and Applications (ICISA), 21-23 April 2010.

Wadia Faid Hassan Al-Shameri, Dynamical Properties of the Hénon Mapping, Int. Journal of Math. Analysis, Vol. 6, 2012, no. 49, 2419 - 2430.

A. Leontitsis, Chaotic Systems Toolbox. 11 Apr. 2002 (Updated 26 Aug 2004). Available from: http:// www. mathworks.com /matlabcentral/ fileexchange/1597-chaotic-systems-toolbox.

Leros A. and Andreatos A. (2013) On the optimisation of Chua chaotic attractors. In Proceedings of ICACM '13, 2nd WSEAS International Conference on Applied and Computational Mathematics. Vouliagmeni, Athens, Greece, May 14-16, 2013. http: // www.wseas.org/ wseas/cms.action?id=2574.

Andreatos A. S. & Leros A. P. (2014) Audio Steganography Telecom System Based οn Hénon Chaotic Map. Presented at the 2nd International Conference on Cryptography, Network Security and Applications in the Armed Forces. Hellenic Military Academy. April 2, 2014.

Bibliography (2)

Ahmed BenSaida, A practical test for noisy chaotic dynamics, SoftwareX 3-4 (2015), 1-5.

Robert L. V. Taylor, Attractors: Nonstrange to Chaotic. SIAM, 72-80.

Wadia Faid Hassan Al-Shameri, Dynamical Properties of the Hénon Mapping. Int. Journal of Math. Analysis, Vol. 6, 2012, no. 49, 2419-2430.

Zonghua Liu, Chaotic Time Series Analysis, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Volume 2010, Article ID 720190, 1-31.

Baojie Zhang and Hongxing Li, A New Four-Dimensional Autonomous Hyper-chaotic System and the Synchronization of Different Chaotic Systems by Using Fast Terminal Sliding Mode Control. Hindawi Publishing Corporation, Mathematical Problems in Engineering, Volume 2013, Article ID 179428, 1-8.

Santo Banerjee, Chaos Synchronization and Cryptography for Secure Communications: Applications for Encryption. InformatIon scIence reference, IGI Global, 2011.

38

END of the PRESENTATION

Thank you for your attention.

– Any questions?

Downloaded from:http://t-h.wikispaces.com/