Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

Author:

Pei Yangjun1,Han Qi1ORCID,Liu Chao1,Tang Dedong1,Huang Junjian2ORCID

Affiliation:

1. School of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing 401331, China

2. Department of Mathematics and Information Engineering, Chongqing University of Education College, Chongqing 400065, China

Abstract

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulliστ-shift rule, which is members of Wolfram’s class II, and it was said to be simple as periodic before. It is worthwhile to study dynamical behaviors of rule 58 and whether it possesses chaotic attractors or not. It is shown that there exist two Bernoulli-measure attractors of rule 58. The dynamical properties of topological entropy and topological mixing of rule 58 are exploited on these two subsystems. According to corresponding strongly connected graph of transition matrices of determinative block systems, we divide determinative block systems into two subsets. In addition, it is shown that rule 58 possesses rich and complicated dynamical behaviors in the space of bi-infinite sequences. Furthermore, we prove that four rules of global equivalence classε43of CA are topologically conjugate. We use diagrams to explain the attractors of rule 58, where characteristic function is used to describe that some points fall into Bernoulli-shift map after several times iterations, and we find that these attractors are not global attractors. The Lameray diagram is used to show clearly the iterative process of an attractor.

Funder

Chongqing University

Publisher

Hindawi Limited

Subject

General Engineering,General Mathematics

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Beyond Solvable Elementary Rules;Understanding Complex Systems;2023

2. Graphs Related to Reversibility and Complexity in Cellular Automata;Cellular Automata;2018

3. Graphs Related to Reversibility and Complexity in Cellular Automata;Encyclopedia of Complexity and Systems Science;2017

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