Investigating topological chaos by elementary cellular automata dynamics

Abstract

We apply the two different definitions of chaos given by Devaney and by Knudsen for general discrete time dynamical systems (DTDS) to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A DTDS is chaotic according to the Devaney's definition of chaos iff it is topologically transitive, has dense periodic orbits, and it is sensitive to initial conditions. A DTDS is chaotic according to the Knudsen's definition of chaos iff it has a dense orbit and it is sensitive to initial conditions. We enucleate an easy-to-check property (left or rightmost permutivity) of the local rule associated with a cellular automaton which is a sufficient condition for D-chaotic behavior. It turns out that this property is also necessary for the class of 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.