A Lyapunov View on the Stability of Two-State Cellular Automata

Although cellular automata (CAs) have been invented only about three quarter of a century ago, much has been written about the fascinating space-time patterns these utter discrete dynamical systems can bring forth, which can be largely attributed to the fact that it is striking to notice that the dynamics of these overly simple dynamical systems can be so intriguing. Driven by the advances in the theory of dynamical systems of continuous dynamical systems, several quantitative measures have been proposed to grasp the stability of CAs, among which the Lyapunov exponent (LE) has received particular attention. Originally, the latter was understood as the Hamming distance between configurations evolved by the same CA from different initially close configurations, but it suffers from the important drawback that it can grow only linearly over time. In this paper, it will be shown how one should determine the LE of a two-state CA, in such a way that its meaning is in line with the framework of continuous dynamical systems. Besides, the proposed methodology will be exemplified for two CA families.