Radial View of Continuous Cellular Automata

Continuous cellular automata (or coupled map lattices) are cellular automata where the state of the cells are real values in [0, 1] and the local transition rule is a real function. The classical observation medium for cellular automata, whether Boolean or continuous, is the space-time diagram, where successive rows correspond to successive configurations in time. In this paper we introduce a different way to visualize the evolution of continuous cellular automata called Radial Representation and we employ it to observe a particular class of continuous cellular automata called fuzzy cellular automata (FCA), where the local rule is the “fuzzification” of the disjunctive normal form that describes the local rule of the corresponding Boolean cellular automata. Our new visualization method reveals interesting dynamics that are not easily observable with the space-time diagram. In particular, it allows us to detect the quick emergence of spatial correlations among cells and to observe that all circular FCA from random initial configurations appear to converge towards an asymptotic periodic behavior. We propose an empirical classification of FCA based on the length of the observed periodic behavior: interestingly, all the minimum periods that we observe are of lengths one, two, four, or n (where n is the size of a configuration).