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Mathematical Problems in Engineering
Volume 2014, Article ID 834268, 9 pages
http://dx.doi.org/10.1155/2014/834268
Research Article

Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

1School of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing 401331, China
2Department of Mathematics and Information Engineering, Chongqing University of Education College, Chongqing 400065, China

Received 14 April 2014; Accepted 25 June 2014; Published 22 July 2014

Academic Editor: Chuandong Li

Copyright © 2014 Yangjun Pei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulli -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 class of 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.