Discrete Dynamical Systems Embedded in Cantor Sets

Details Discrete Dynamical Systems Embedded in Cantor Sets Discrete Dynamical Systems Embedded in Cantor Sets

2005-11-26

arxiv.org/abs/nlin/0511057v2

Nonlinear Sciences

Chaotic Dynamics

36 pages, 13 figures: minor text amendments in places, time running top to bottom in figures, to appear in J. Math. Phys

Scientific paper

10.1063/1.2171518

While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with $ N $ variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit $N\to\infty$. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error-profile. We made explicit calculations both numerical and analytic for well known discrete dynamical models.

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