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.

Weak and strong interaction of excitation kinks in scalar parabolic equations

“…Motivated by studies of the Greenberg-Hastings cellular automaton (GHCA) as a caricature of excitable systems, in this paper we have considered the θ-equations describing oscillatory phase dynamics, as the perhaps simplest PDE model of excitable media. Since the non-wandering set of GHCA in essence consists of certain excitation pulse sequences [KRU20], we have focussed on the analogue of such data in θ-equations, which consists of kinks and antikinks. Moreover, in GHCA the topological entropy can be related to kink-antikink collisions.…”

“…The motivation to consider kink-antikink dynamics in the θ-equations emerges from our analysis of GHCA and their (topological) complexity [KRU20]. The understanding of this complexity mainly relies on the observation that for GHCA the original and somehow abstract definitions of topological entropy [AKM65, Bow71] can be substantiated by considering simpler but equivalent definitions.…”

“…The understanding of this complexity mainly relies on the observation that for GHCA the original and somehow abstract definitions of topological entropy [AKM65, Bow71] can be substantiated by considering simpler but equivalent definitions. In this regard, one combinatorial approach is to count possible realizations of bounded space-time windows and to determine the exponential growth rate as the size of these windows increases [DS91,KRU20]. On the other hand, the decomposition of the non-wandering set reveals that the topological complexity is completely determined by the invariant subsystem consisting of bi-infinite configurations which are composed of counter propagating pulses,…”

“…Pulse trains can be generated from local pulses by placing these at arbitrary positions; they will maintain their relative distance until a possible annihilation. In fact, the topological entropy of the GHCA results from a Devaney-chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating local pulses [DS91,KRU20]. From a dynamical systems viewpoint the non-wandering set is of prime relevance, and it turns out that for the GHCA it can be decomposed into invariant sets with different wave dynamics [KRU20].…”