Weifeng Jin

Chaos of elementary cellular automata rule 42 of Wolfram's class II.

In this paper, the dynamics of elementary cellular automata rule 42 is investigated in the bi-infinite symbolic sequence space. Rule 42, a member of Wolframs class II which was said to be simply as periodic before, actually defines a chaotic global attractor; that is, rule 42 is topologically mixing on its global attractor and possesses the positive topological entropy. Therefore, rule 42 is chaotic in the sense of both Li-Yorke and Devaney. Meanwhile, the characteristic function and the basin tree diagram of rule 42 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 other rules, especially the 112 Bernoulli-shift rules of the elementary cellular automata.

SOME NONROBUST BERNOULLI-SHIFT RULES

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chuas robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.

Topological Entropy and Complexity of One Class of Cellular Automata Rules

In this paper, some complex dynamics of one equivalence class of elementary cellular automata are characterized via symbolic dynamics on the space of bi-infinite symbolic sequences. By establishing a topologically conjugate relationship with a 2-order subshift of finite type of symbolic dynamical systems, it is rigorously proved that the four rules N=119, 63, 17, and 3 are topologically mixing on their global attractors. Meanwhile, it is shown that they are chaotic both in the sense of Li-Yorke and Devaney on their global attractors. Furthermore, the topological entropies of these rules on Sigma2 are computed.

Chaos emerged on the ‘edge of chaos’

Rule 110 is a complex cellular automaton (CA) in Wolframs system of identification, capable of supporting universal computation. It has been suggested that a universal CA should be on the ‘edge of chaos’, which means that the dynamical behaviour of such a system is neither simple nor chaotic. There is no doubt that the dynamical property of Rule 110 is extremely complex and still not well understood. This paper proves the existence of subsystems on which this rule is chaotic in the sense of Devaney.

Gliders, Collisions and Chaos of Cellular Automata Rule 62

This paper provides a systematic analysis of glider dynamics and interactions in rule 62, including a catalog of glider collisions. Based on these empirical observations, it is proved that rule 62 defines a subsystem with complicated dynamical properties in the bi-infinite symbolic sequence space, such as topologically mixing and positive topological entropy. Meanwhile, the phenomena of glider collisions provide an intriguing and valuable bridge for researching the symbolic dynamics of rule 62 in the bi-infinite case, especially for proving that the union of period-3 attractor and Bernoulli attractors is not the global attractor.

Topologically Mixing and Chaos of One Class of Bernoulli-Shift Cellular Automata Rules

This paper is devoted to an in-depth study of Chuas Bernoulli-shift rules 11, 14, 43 and 142 from the viewpoint of symbolic dynamics. It is shown that each of these four rules identifies two chaotic dynamical subsystems and presents very rich and complicated dynamical properties. In particular, they are topologically mixing and possess the positive topological entropies on their two subsystems. Therefore, they are chaotic in the sense of both Li-Yorke and Devaney on the subsystems. The method proposed in this work is also gives some support for investigating the dynamics of subsystems of other rules, especially the hyper-Bernoulli-shift rules therein.