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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)

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ABSTRACT :

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.

DEFINITION:

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).

DEFINITION:

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).

DEFINITION:

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)

DEFINITION:

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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