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

2007 ◽  
Vol 17 (12) ◽  
pp. 4245-4259 ◽  
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.

scholarly journals Chaotic Behavior of One-Dimensional Cellular Automata Rule 24

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zujie Bie ◽  
Qi Han ◽  
Chao Liu ◽  
Junjian Huang ◽  
Lepeng Song ◽  
...  
Keyword(s):  
Cellular Automata ◽  
Characteristic Function ◽  
Symbolic Dynamics ◽  
Bernoulli Shift ◽  
Chaotic Behavior ◽  
Class Ii ◽  
Chaotic Attractors ◽  
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 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.

Topological Conjugacy Classification of Elementary Cellular Automata with Majority Memory

2017 ◽  
Vol 27 (14) ◽  
pp. 1750217 ◽  
Author(s):  
Haiyun Xu ◽  
Fangyue Chen ◽  
Weifeng Jin
Keyword(s):  
Cellular Automata ◽  
Vector Space ◽  
Symbolic Dynamics ◽  
Continuous Functions ◽  
Equivalence Classes ◽  
Topological Conjugacy ◽  
Symbolic Sequence ◽  
And Topology ◽  
Elementary Cellular Automata

The topological conjugacy classification of elementary cellular automata with majority memory (ECAMs) is studied under the framework of symbolic dynamics. In the light of the conventional symbolic sequence space, the compact symbolic vector space is identified with a feasible metric and topology. A slight change is introduced to present that all global maps of ECAMs are continuous functions, thereafter generating the compact dynamical systems. By exploiting two fundamental homeomorphisms in symbolic vector space, all ECAMs are furthermore grouped into 88 equivalence classes in the sense that different mappings in the same global equivalence are mutually topologically conjugate.

scholarly journals Complex Dynamic Behaviors in Cellular Automata Rule 14

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
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.

A Study of Chaos in Cellular Automata

2018 ◽  
Vol 28 (03) ◽  
pp. 1830008 ◽  
Author(s):  
Supreeti Kamilya ◽  
Sukanta Das
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Equivalence Class ◽  
Binary Relation ◽  
Equivalence Relation ◽  
De Bruijn Graph ◽  
One Dimensional ◽  
De Bruijn ◽  
Better Than

This paper presents a study of chaos in one-dimensional cellular automata (CAs). The communication of information from one part of the system to another has been taken into consideration in this study. This communication is formalized as a binary relation over the set of cells. It is shown that this relation is an equivalence relation and all the cells form a single equivalence class when the cellular automaton (CA) is chaotic. However, the communication between two cells is sometimes blocked in some CAs by a subconfiguration which appears in between the cells during evolution. This blocking of communication by a subconfiguration has been analyzed in this paper with the help of de Bruijn graph. We identify two types of blocking — full and partial. Finally a parameter has been developed for the CAs. We show that the proposed parameter performs better than the existing parameters.

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

2002 ◽  
Vol 13 (06) ◽  
pp. 837-855 ◽  
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.

CNN Genes for One-Dimensional Cellular Automata: A Multi-Nested Piecewise-Linear Approach

1998 ◽  
Vol 08 (10) ◽  
pp. 1987-2001 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Cellular Automata ◽  
Boolean Functions ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Truth Table ◽  
Local Rule ◽  
Mathematical Formula ◽  
One Dimensional ◽  
Linear Approach ◽  
Piecewise Linear Approach

This paper introduces a novel CNN cell which guarantees the implementation of any local rule on three variables defined by a Boolean truth table. Moreover, since the output of the cell is completely specified by a simple mathematical formula, it is possible to develop a systematic theory for locating those regions in the CNN genes parameter space where complex behaviors may occur. The output cell formula is a simple piecewise-linear function, and for the case of a one-dimensional CNN the entire set of 256 CNN genes associated with the corresponding local Boolean functions are listed in this paper.

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