Topological perturbations and their effect on the dynamics of totalistic cellular automata

Abstract

Although several studies addressed the dynamical properties of cellular automata (CAs) in general and the sensitivity to the initial condition from which they are evolved in particular, only minor attention has been paid to the interference between a CA's dynamics and its underlying topology, by which we refer to the whole of a CA's spatial entities and their interconnection. Nevertheless, some preliminary studies highlighted the importance of this issue. Henceforth, in contrast to the sensitivity to the initial conditions, which is frequently quantified by means of Lyapunov exponents, to this day no methodology is available for grasping this so-called topological sensitivity. Inspired by the concept of classical Lyapunov exponents, we elaborate on the machinery that is required to grasp the topological sensitivity of CAs, which consists of topological Lyapunov exponents and Jacobians. By relying on these concepts, the topological sensitivity of a family of 2-state irregular totalistic CAs is characterized. © 2012 Springer-Verlag Berlin Heidelberg.