Classifying 1D elementary cellular automata with the 0-1 test for chaos

We utilise the 0-1 test to automatically classify elementary cellular automata. The quantitative results of the 0-1 test reveal a number of advantages over Wolfram’s qualitative classification. For instance, while almost all rules classified as chaotic by Wolfram were confirmed as such by the 0-1 test, there were two rules which were revealed to be non-chaotic. However, their periodic nature is hidden by the high complexity of their spacetime patterns and not easy to see without looking very carefully. Comparing each rule’s chaoticity (as quantified by the 0-1 test) against its intrinsic complexity (as quantified by its Chua complexity index) also reveals a number of counter-intuitive discoveries; i.e. non-chaotic dynamics are not only found in simpler rules, but also in rules as complex as chaos

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Lyapunov Exponents Vs Expansivity and Sensitivity in Cellular Automata

AbstractWe establish a connection between the theory of Lyapunov exponents and the properties of expansivity and sensitivity to initial conditions for a particular class of discrete time dynamical systems; cellular automata (CA). The main contribution of this paper is the proof that all expansive cellular automata have positive Lyapunov exponents for almost all the phase space configurations. In addition, we provide an elementary proof of the non-existence of expansive CA in any dimension greater than 1. In the second part of this paper we prove that expansivity in dimension greater than 1 can be recovered by restricting the phase space to asuitablesubset of the whole space. To this extent we describe a 2-dimensional CA which is expansive over adense uncountablesubset of the whole phase space. Finally, we highlight the different behavior of expansive and sensitive CA for what concerns the speed at which perturbations propagate