Symbolic dynamics of glider guns for some one-dimensional cellular automata

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.

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Complex Dynamic Behaviors in Cellular Automata Rule 14

Discrete Dynamics in Nature and Society ◽

10.1155/2012/258309 ◽

2012 ◽

Vol 2012 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Qi Han ◽

Xiaofeng Liao ◽

Chuandong Li

Keyword(s):

Cellular Automata ◽

Characteristic Function ◽

Fixed Points ◽

Symbolic Dynamics ◽

Bernoulli Shift ◽

Chaotic Attractors ◽

Complex Dynamic ◽

One Dimensional ◽

Dynamical Behaviors ◽

Shift Map

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 14, which is Bernoulliστ-shift rule and is a member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 14, whether it possesses chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 14 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 14 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Then, we prove that there exist fixed points in rule 14. Finally, we use diagrams to explain the attractor of rule 14, where characteristic function is used to describe that all points fall into Bernoulli-shift map after two iterations under rule 14.

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EXTENDING CHUA'S GLOBAL EQUIVALENCE THEOREM ON WOLFRAM'S NEW KIND OF SCIENCE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407019925 ◽

2007 ◽

Vol 17 (12) ◽

pp. 4245-4259 ◽

Cited By ~ 25

Author(s):

JUNBIAO GUAN ◽

SHAOWEI SHEN ◽

CHANGBING TANG ◽

FANGYUE CHEN

Keyword(s):

Cellular Automata ◽

Equivalence Class ◽

Symbolic Dynamics ◽

Equivalence Theorem ◽

Equivalence Classes ◽

Local Rule ◽

One Dimensional ◽

Symbolic Space ◽

Symbolic Sequences

We establish the relation between the extended (i.e. I = ∞) one-dimensional binary Cellular Automata (1D CA) and the bi-infinite symbolic sequences in symbolic dynamics. That is, the 256 local rules of 1D CA correspond to 256 local rule mappings in the symbolic space. By employing the two homeomorphisms T† and [Formula: see text] from [Chua et al., 2004] for finite I, we classify these 256 local rule mappings into the same 88 equivalence classes identified in [Chua et al., 2004] and [Chua, 2006]. Different mappings in the same equivalence class are mutually topologically conjugate.

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TRANSITIVE BEHAVIOR IN REVERSIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA WITH A WELCH INDEX 1

International Journal of Modern Physics C ◽

10.1142/s0129183102003607 ◽

2002 ◽

Vol 13 (06) ◽

pp. 837-855 ◽

Cited By ~ 1

Author(s):

JUAN CARLOS SECK TUOH MORA

Keyword(s):

Cellular Automata ◽

Cellular Automaton ◽

Symbolic Dynamics ◽

Transitive Closure ◽

Matrix Representation ◽

Global Dynamics ◽

One Dimensional ◽

Interesting Topic

The problem of knowing and characterizing the transitive behavior of a given cellular automaton is a very interesting topic. This paper provides a matrix representation of the global dynamics in reversible one-dimensional cellular automata with a Welch index 1, i.e., those where the ancestors differ just at one end. We prove that the transitive closure of this matrix shows diverse types of transitive behaviors in these systems. Part of the theorems in this paper are reductions of well-known results in symbolic dynamics. This matrix and its transitive closure were computationally implemented, and some examples are presented.

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