Synchronization and maximum Lyapunov exponents of cellular automata

Abstract

We study the synchronization of totalistic one-dimensional cellular automata (CA). The CA with a nonzero synchronization threshold exhibit complex nonperiodic space time patterns and vice versa. This synchronization transition is related to directed percolation. We also study the maximum Lyapunov exponent for CA, defined analogous to continuous dynamical systems as the exponential rate of expansion of the linear map induced by the evolution rule of CA, constructed with the aid of the Boolean derivatives. The synchronization threshold is strongly correlated to the maximum Lyapunov exponent and we propose approximate relations between these quantities. The value of this threshold can be used to parametrize the space time complexity of CA.