Synchronization and Maximum Lyapunov Exponents of Cellular Automata
Abstract
We study the synchronization of totalistic one dimensional cellular automata (CA). We find a synchronization threshold for all the CA considered. The CA with a non zero threshold exhibit complex non periodic space time patterns and conversely. This synchronization transition is related to directed percolation. We study also the maximum Lyapunov exponent for CA, defined in analogy with 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. We find that the synchronization threshold is strongly correlated to the maximum Lyapunov exponent and we propose approximate relations between these quantities. The synchronization threshold is a numerical parameter that can be used to characterize the space time complexity of CA.Keywords
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