Extending the symbolic dynamics of Chua's Bernoulli-shift rule 56

In this paper, the dynamical behaviors of Chua's Bernoulli-shift rule 56 are investigated from the viewpoint of symbolic dynamics on the bi-infinite sequence space. It is shown that rule 56, a member of Wolfram's class II, defines two chaotic dynamical subsystems and possesses very rich and complicated dynamical properties. In addition, the topological entropy of rule 56 is calculated on its subsystems and the global attractor of rule 56 is characterized. Meanwhile, the isles of Eden of rule 56 are explored for some finite length of binary strings, which reveal its Bernoulli characteristics. The method presented in this work is also applicable to studying the dynamics of subsystems of other rules, especially the 112 Bernoulli-shift rules of the elementary cellular automata. © 2009 Old City Publishing, Inc.