Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Chaotic Behavior of Cellular Automata

  • Julien Cervelle
  • Alberto Dennunzio
  • Enrico Formenti
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_65-4



All points are equicontinuity points (in compact settings).

Equicontinuity point

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


From two distinct points, orbits eventually separate.


The next state function is injective.

Linear CA

A CA with additive local rule.


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.


The next state function is surjective.

Topological mixing

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


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

Definition of the Subject

A discrete time dynamical system (DTDS) is a pair 〈X, F〉 where Xis a set...

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This work has been supported by the Interlink/MIUR project “Cellular Automata:

Topological Properties, Chaos and Associated Formal Languages”, by the ANR Blanc Project “Sycomore” and by the PRIN/MIUR project “Formal Languages and Automata: Mathematical and Applicative Aspects”.


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Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Julien Cervelle
    • 1
  • Alberto Dennunzio
    • 2
  • Enrico Formenti
    • 3
  1. 1.Laboratoire d’Informatique de l’Institut Gaspard–MongeUniversité Paris-EstMarne- la-ValléeFrance
  2. 2.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-BicoccaMilanItaly
  3. 3.Laboratoire I3SUniversité de Nice-Sophia AntipolisSophia AntipolisFrance