Topology-induced phase transitions in totalistic cellular automata

Notwithstanding numerous papers are dedicated to the intriguing dynamics of cellular automata (CAs), which are mathematical models in which the time, space and state domain are discrete, in most of them the discussion is confined to the sensitivity of a CA’s dynamical properties to the imposed initial conditions. Since the dynamics of a CA relies on both the states of its spatial entities and the interconnections between them, to which we refer as a CA’s topology, it is striking that the influence of the latter on the evolved dynamics has received only minor attention. To fill this gap, we investigate in this paper how a CA’s topology affects its dynamical properties by relying on Lyapunov exponents. We demonstrate the existence of so-called topology-induced phase transitions and topological bifurcation points, which lead us to discriminate between intrinsically complex CAs, on the one hand, and topology-mediated complex CAs on the other hand.