Topological chaos for elementary cellular automata

Abstract

We apply the definition of chaos given by Devaney for discrete time dynamical systems to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A discrete time dynamical system is chaotic according to the Devaney's definition of chaos if it is topologically transitive, is sensitive to initial conditions, and has dense periodic orbits. We enucleate an easy-to-check property of the local rule on which a cellular automaton is based which is a necessary condition for chaotic behavior. We prove that this property is also sufficient for a large class of elementary cellular automata. The main contribution of this paper is the formal proof of chaoticity for many non additive elementary cellular automata. Finally, we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata.