On the dynamical behavior of chaotic cellular automata

The shift (bi-infinite) cellular automaton is a chaotic dynamical system ac cording to all the definitions of deterministic chaos given for discrete ti me dynamical systems (e.g., those given by Devaney [6] and by Knudsen [10]) . The main motivation to this fact is that the temporal evolution of the sh ift cellular automaton under finite description of the initial state is unp redictable. Even tough rigorously proved according to widely accepted. form al definitions of chaos, the chaoticity of the shift cellular automaton rem ains quite counterintuitive and in some sense unsatisfactory. The space-tim e patterns generated by a shift cellular automaton do not correspond to tho se one expects from a chaotic process. In this paper we propose a new definition of strong topological chaos for d iscrete time dynamical systems which fulfills the informal intuition of cha otic behavior that everyone has in mind. We prove that under this new defin ition, the bi-infinite shift is no more chaotic. Moreover, we put into rela tion the new definition of chaos and those given by Devaney and Knudsen. In the second part of this paper we focus our attention on the class of add itive cellular automata (those based on additive local rules) and are prove that essential transformations [2] preserve the new definition of chaos gi ven in the first part of this paper and many other aspects of their global qualitative dynamics. (C) 1999-Elsevier Science B.V. All rights reserved.