Recovering Trajectories of Chaotic Piecewise Linear Dynamical Systems

M. A. Aziz Alaoui
Depart. of Math., L.M., LeHavre University

Fedorenko, A.
Institute of Math., National Academy of Sciences, kiev

Lozi R.
Labo. of Math., Univrsity of Nice

Sharkovsky, A.N.
Institute of Math., National Academy of Sciences, kiev

Control of Oscillations and Chaos, Proc. IEEE, vol. 2, 230-233, eds. F.L. Chernousko, A.L. Fradkov, (1997)

ABSTRACT

The purpose of this paper is to present some results on recovering trajectories of chaotic dynamical systems from their epsilon-perturbed trajectories. Such a mathematical problem appears to be equivalent to the extraction of an information bearing signal from an input signal, to the reduction of a noise in an input signal, etc.

Ref: M.A. Aziz Alaoui et al , Control of Oscillations and Chaos, Proc. IEEE, vol. 2, 380-383, eds. F.L. Chernousko, A.L. Fradkov, (2000).

Introduction

When almost each trajectory of a dynamical system is Lyapunov stable, the problem of extraction of efficient signal from input signal in receiving system is well known and detailed methods of solutions are available. For a chaotic dynamical system, where there is a sensitive dependence on initial data and every trajectory on the strange attractor is unstable, the situation is absolutely different. The elaboration of methods of extraction of efficient signal from input signal (in order to reduce the noise in an input signal, to encode/decode electronic messages for secure communication, etc.) for chaotic dynamical system is still a very attractive problem. There are different approaches to investigate such a type of task (for instance, see Kocarev et al., Int. J. Bifurcation and Chaos, 2(3), 1992, 709-713, and Lozi-Aziz). Let us point out that this problem is equivalent to the problem of recovering a trajectory {x_i}_{i=0,n} of the system from its epsilon-perturbed trajectory. {\overline{x}}_{i=0}^n. This is the purpose of our paper in which we present some results on recovering the trajectories of chaotic dynamical systems (in particular, systems given by piecewise linear maps) from their epsilon-perturbed trajectories.

paper in ps format To cite this paper : Aziz-Alaoui M.A., Fedorenko A., Lozi R. and Sharkovsky A.N. . Control of Oscillations and Chaos,Proc. IEEE, vol. 2, 230-233, eds. F.L. Chernousko, A.L. Fradkov, (1997)


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