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Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 304297, 8 pages
http://dx.doi.org/10.1155/2014/304297
Research Article

Chaotic Behavior of One-Dimensional Cellular Automata Rule 24

1Internet Data Center, Chongqing University of Science and Technology, Chongqing 401331, China
2School of Electrical and Information Engineering, Chongqing University of Science and Technology, Chongqing 401331, China
3Department of Mathematics and Information Engineering, Chongqing University of Education College, Chongqing 400065, China

Received 21 January 2014; Accepted 11 April 2014; Published 15 May 2014

Academic Editor: Zhen Jin

Copyright © 2014 Zujie Bie 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

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 24, which is Bernoulli -shift rule and is member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of four rules, whether they possess chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 24 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 24 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Furthermore, we prove that four rules of global equivalence of cellular automata are topologically conjugate. Then, we use diagrams to explain the attractor of rule 24, where characteristic function is used to describe the fact that all points fall into Bernoulli-shift map after two iterations under rule 24.