Phenomenological study of irregular cellular automata based on Lyapunov exponents and Jacobians
Jan Marcel Baetens, Bernard De Baets · Chaos An Interdisciplinary Journal of Nonlinear Science · 2010
Originally, cellular automata (CA) have been defined upon regular tessellations of the n-dimensional Euclidean space, while CA on irregular tessellations have received only little attention from the scientific community, notwithstanding serious shortcomings are associated with the former manner of subdividing Rn. In this paper we present a profound phenomenological study of two-state, two-dimensional irregular CA from a dynamical systems viewpoint. We opted to exploit properly defined quantitative measures instead of resorting to qualitative methods for discriminating between behavioral classes. As such, we employ Lyapunov exponents, measuring the divergence rate of close trajectories in phase space, and Jacobians, formulated using Boolean derivatives and expressing the sensitivity of a cellular automaton to its inputs. Both are stated for two-state CA on irregular tessellations, enabling us to characterize these discrete dynamical systems, and advancing us to propose a classification scheme for this CA family. In addition, a relationship between these quantitative measures is established in extension of the insights already developed for the classical CA paradigm. Finally, we discuss the repercussions on the CA dynamics that arise when the geometric variability of the spatial entities is taken into account during the CA simulation.