Synchronization and Maximum Lyapunov Exponents of Cellular Automata

TL;DR

This paper investigates the synchronization phenomena and maximum Lyapunov exponents in one-dimensional cellular automata, revealing their relationship and implications for the complexity of CA dynamics.

Contribution

It introduces a novel analysis linking synchronization thresholds with Lyapunov exponents in cellular automata, providing approximate relations and insights into their complexity.

Findings

Synchronization transition relates to directed percolation.

Maximum Lyapunov exponent correlates with synchronization threshold.

Threshold values help parametrize CA space-time complexity.

Abstract

We study the synchronization of totalistic one dimensional cellular automata (CA). The CA with a non zero synchronization 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. 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.