Complex Symbolic Dynamics of Bernoulli Shift Cellular Automata Rule

2008 ◽  
Author(s):  
Lin Chen ◽  
Fangyue Chen ◽  
Fangfang Chen ◽  
Weifeng Jin
Keyword(s):  
Cellular Automata ◽  
Symbolic Dynamics ◽  
Bernoulli Shift

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.

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.

scholarly journals Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yangjun Pei ◽  
Qi Han ◽  
Chao Liu ◽  
Dedong Tang ◽  
Junjian Huang
Keyword(s):  
Cellular Automata ◽  
Symbolic Dynamics ◽  
Bernoulli Shift ◽  
Global Attractors ◽  
Chaotic Attractors ◽  
Transition Matrices ◽  
Topological Mixing ◽  
Dynamical Behaviors ◽  
Strongly Connected ◽  
Bernoulli Measure

The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulliστ-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 classε43of 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.

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.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE. PART XIII: BERNOULLI στ-SHIFT RULES

2010 ◽  
Vol 20 (07) ◽  
pp. 1859-2003 ◽  
Author(s):  
LEON O. CHUA ◽  
GIOVANNI E. PAZIENZA
Keyword(s):  
Nonlinear Dynamics ◽  
Cellular Automata ◽  
Majority Rule ◽  
Bernoulli Shift ◽  
Testing Signal ◽  
The Difference ◽  
Shift Dynamics ◽  
Science Part

More than one third of the 88 globally-independent Cellular Automata rules exhibit robust simple Bernoulli-shift dynamics. Among them we find rule [Formula: see text], which we proved to be chaotic in the previous episodes of our chronicle, and rule [Formula: see text], the famous global majority rule. Therefore, we cannot overstate the importance of the Bernoulli στ-shift rules which we will present in two parts of our continuing odyssey on the Nonlinear Dynamics Perspective of Cellular Automata. This paper covers the first 15 of the 30 Bernoulli στ-shift rules. In this paper, after recalling the main concepts of Bernoulli rules — such as the role of the three Bernoulli parameters σ, τ and β — we will display the basin tree diagrams of these rules together with a convenient summary of the results extracted from them. Then, we will show that the superstring [Formula: see text] is an excellent testing signal to find the robust behavior of a given rule. Finally, we will conclude this paper with a discussion about the difference between robust and nonrobust ω-limit orbits of the Bernoulli στ-shift rules.

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 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.

SOME NONROBUST BERNOULLI-SHIFT RULES

2009 ◽  
Vol 19 (10) ◽  
pp. 3407-3415 ◽  
Author(s):  
LIN CHEN ◽  
FANGYUE CHEN ◽  
WEIFENG JIN ◽  
FANGFANG CHEN ◽  
GUANRONG CHEN
Keyword(s):  
Cellular Automata ◽  
Topological Entropy ◽  
Sequence Space ◽  
Bernoulli Shift ◽  
Binary Sequence ◽  
Class I ◽  
Positive Topological Entropy ◽  
Elementary Cellular Automata ◽  
Topologically Mixing ◽  
Shift Dynamics

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chua's robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.

Sign in / Sign up

Export Citation Format

Share Document