Dynamics and topological entropy of 1D Greenberg-Hastings cellular automata

Abstract

In this paper we analyse the non-wandering set of 1D-Greenberg-Hastings cellular automata models for excitable media with $e⩾1$ excited and $r⩾1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney-chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift-dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$.

33 pages, 12 figures