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Chaotic Behavior of Cellular Automata

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

A discrete time dynamical system (DTDS) is a pair \( { \left\langle X,F \right\rangle } \) where X is a set equipped witha distance d and \( { F\colon X\mapsto X} \) is a mapping which is continuous on X with respect to themetric d. The set X and the function F are called the state space and the next state map. At the very beginning of the seventies, the notion of chaotic behavior forDTDS has been introduced in experimental physics [46]. Successively, mathematicians startedinvestigating this new notion finding more and more complex examples. Although a general universally accepted theory of chaos has not emerged, atleast some properties are recognized as basic components of possible chaotic behavior. Among them one can list: sensitivity to initial conditions,transitivity, mixing, expansively etc. [5,6,21,22,28,29,38,41,42,58,59,60].

In the eighties, S. Wolfram started studying some of these properties in the context of cellular automata (CA) [64]. These...

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Abbreviations

Equicontinuity:

All points are equicontinuity points (in compact settings).

Equicontinuity point:

A point for which the orbits of nearby points remain close.

Expansivity:

From two distinct points, orbits eventually separate.

Injectivity:

The next state function is injective.

Linear CA:

A CA with additive local rule.

Regularity:

The set of periodic points is dense.

Sensitivity to initial conditions:

For any point x there exist arbitrary close points whose orbits eventually separate from the orbit of x.

Strong transitivity:

There always exist points which eventually move from any arbitrary neighborhood to any point.

Surjectivity:

The next state function is surjective.

Topological mixing:

There always exist points which definitely move from any arbitrary neighborhood to any other.

Transitivity:

There always exist points which eventually move from any arbitrary neighborhood to any other.

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Acknowledgments

This work has been supported by the Interlink/MIUR project “Cellular Automata: Topological Properties, Chaos and Associated FormalLanguages”, by the ANR Blanc Project “Sycomore” and by the PRIN/MIUR project “Formal Languages and Automata: Mathematical andApplicative Aspects”.

Author information

Authors and Affiliations

  1. Laboratoire d'Informatique de l'Institut Gaspard‐Monge, Université Paris‐Est, Marne la Vallée, France

    Julien Cervelle

  2. Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano‐Bicocca, Milan, Italy

    Alberto Dennunzio

  3. Laboratoire I3S, Université de Nice‐Sophia Antipolis, Sophia Antipolis, France

    Enrico Formenti

Authors
  1. Julien Cervelle
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Cervelle, J., Dennunzio, A., Formenti, E. (2009). Chaotic Behavior of Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_65

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