the useful signal ...
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.
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 ...
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