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 Bernoulliστ-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 equivalenceε52of 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.
Highlights
Cellular automata (CA) was first introduced by von Neumann in 1951 [1]
We prove that four rules of global equivalence ε52 of cellular automata are topologically conjugate
In terms of the transition matrix of determinative block system of subsystem of rule 24, we obtain the value of topological entropy of subsystem
Summary
Cellular automata (CA) was first introduced by von Neumann in 1951 [1]. CA is a mathematical model consisting of large numbers of simple identical components with local interactions [2]. A cellular automaton consists of a number of cells which evolve by a simple local rule (identical rule). If every row of an image is evolved according to the rule 24, every column of the image is evolved according to the rule 24 and their results do bitwise XOR (exclusive OR); the image edges can be obtained. It is worth researching the dynamical behaviors of rule 24. Complex dynamical behaviors of rule 24 are studied in detail.
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