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

TL;DR

This paper investigates the complex dynamics of 1D Greenberg-Hastings cellular automata, revealing the structure of their non-wandering set and calculating their topological entropy, which relates to chaotic behavior in excitable media.

Contribution

It characterizes the non-wandering set of 1D Greenberg-Hastings automata and explicitly computes its topological entropy, linking it to chaotic invariant subsets and Markov systems.

Findings

Non-wandering set contains a chaotic subset of colliding waves.

Topological entropy is positive and varies with parameters.

Remaining non-wandering set is a Markov system with lower entropy.

Abstract

In this paper we analyse the non-wandering set of 1D-Greenberg-Hastings cellular automata models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ 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 $e,r$.