On Synchronization and Anti-Synchronization in Piecewise Linear Systems

M. A. Aziz Alaoui
Depart. of Math., Labo. of Mechanics, LeHavre University

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

proc. Intern. Conf. ICNBC'96, 83-86. eds. Awrejcewicz J., Lamarque, C. H., (1996)


We present some results on synchronization and anti-synchronization of chaotic systems. A new algorithm is developed to recover a trajectory of a chaotic piecewise linear system from its epsilon-trajectory . Such a problem may be applied to the extraction of information bearing signal from an input signal. (To realize secure communications via a strange attractor.) The problem of extracting the information bearing signal s_i, (i=0,1 ...) from the input signal \bar{x}_i = x_i + s_i , where x_i is some unknown trajectory of one of the receiving systems

is aquivalent to

the problem of recovering the trajectory x_i$ of the system from its epsilon-trajectory \bar x_i.


The sequence \bar{x}_0, \bar{x}_1, ..., \bar{x}_N is an {\bf epsilon-trajectory} of a dynamical system x_{n+1}=f(x_n), if |\bar{x}_{i+1} - f(\bar{x}_i)| < \varepsilon (for i=0,1, ..., N-1).


A dynamical system x_{n+1}=f(x_n) has the {shadowing property} if there exists epsilon_0 > 0 such that for any epsilon-trajectory \bar{x}_0, \bar{x}_1, ... , \bar{x}_N with epsilon < epsilon_0, the dynamical system has trajectory x_0, x_1, ... , x_N such that |\bar{x}_i - x_i| < epsilon (for i=0,1, ... , N).


Two trajectories x_0, x_1, ... , x_N and x'_0, x'_1, ... , x'_N are said epsilon-closed if |x_i - x'_i| < epsilon (for i=0,1, ... , N)


A dynamical system x_{n+1}=f(x_n) has the {ANTI-SYNCHRONIZATION} property if there exist epsilon_0 > 0 and 0<\lambda <1, such that for any two epsilon-closed (epsilon \leq \varepsilon_0) trajectories x_0, x_1, ... , x_N and x'_0, x'_1, ... , x'_N, we have |x_0 - x'_0| < constant.\lambda^N.

This definition may be generalized in an obvious way to systems of any dimension and to continuous-time systems.

HOW TO RECOVER The TRAJECTORY FROM ITS epsilon-TRAJECTORY ? Algorithms in D1 and D2 are given in this paper.


the useful signal ...


the bearing chaotic signal ...


the transmitted signal ...


the retrieved signal ...

See a photo from the ICNBC'96.

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