Complex Symbolic Dynamics of Chua's Period-2 Rule 37
In this paper, the dynamical behaviors of cellular automata rule 37 are investigated from the viewpoint of symbolic dynamics in the bi-infinite sequence space. It is shown that rule 37, a member of Chua's period-2 rules and Wolfram's class 11, defines two chaotic subsystems and exhibits very rich and complicated dynamics; that is, rule 37 is topologically mixing on each subsystem and possesses the positive topological entropy. Meanwhile, the basin tree diagrams of rule 37 are explored for some finite length of binary strings, which reveal its Bernoulli-shift characteristics. It is interesting to find the collisions between propagating pixels of opposite colors in the space-time evolution of rule 37, which can also mimic Dirac's annihilation of matter and anti-matter. The method presented in this work is also applicable to studying the dynamics of other rules, especially the 112 Bernoulli-shift rules of the elementary cellular automata.
symbolic dynamics topologically mixing topological entropy basin tree diagram cellular automata (ca) chaos