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.

ON THE STRUCTURE OF REAL-VALUED ONE-DIMENSIONAL CELLULAR AUTOMATA

2011 ◽  
Vol 21 (05) ◽  
pp. 1265-1279 ◽  
Author(s):  
XU XU ◽  
STEPHEN P. BANKS ◽  
MAHDI MAHFOUF
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Cellular Systems ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
New Type ◽  
Complex Patterns

It is well-known that binary-valued cellular automata, which are defined by simple local rules, have the amazing feature of generating very complex patterns and having complicated dynamical behaviors. In this paper, we present a new type of cellular automaton based on real-valued states which produce an even greater amount of interesting structures such as fractal, chaotic and hypercyclic. We also give proofs to real-valued cellular systems which have fixed points and periodic solutions.

CHAOTIC ATTRACTORS IN ONE-DIMENSION GENERATED BY A SINGULAR SHILNIKOV ORBIT

2001 ◽  
Vol 11 (12) ◽  
pp. 3059-3083 ◽  
Author(s):  
KRISTA J. TAYLOR ◽  
BO DENG
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
One Dimension ◽  
Chaotic Attractors ◽  
Singularly Perturbed Systems ◽  
One Dimensional ◽  
Physical Systems ◽  
Return Maps ◽  
Singular Limits ◽  
Natural Measures

Chaotic attractors containing Shilnikov's saddle-focus homoclinic orbits have been observed in many physical systems. Past and current researches of this type of Shilnikov homoclinic phenomena have focused on the orbit and nearby structures only. In this paper we will look at the role such orbits play in a type of attractor, which arises from one-dimensional return maps at the singular limits of some singularly perturbed systems. Results on symbolic dynamics, natural measures, and Lyapunov exponents are obtained for a sequence of a one-parameter caricature family of such attractors.

scholarly journals TOPOLOGICAL COMPLEXITY AND PREDICTABILITY IN THE DYNAMICS OF A TUMOR GROWTH MODEL WITH SHILNIKOV'S CHAOS

2013 ◽  
Vol 23 (07) ◽  
pp. 1350124 ◽  
Author(s):  
JORGE DUARTE ◽  
CRISTINA JANUÁRIO ◽  
CARLA RODRIGUES ◽  
JOSEP SARDANYÉS
Keyword(s):  
Tumor Growth ◽  
Tumor Cells ◽  
Symbolic Dynamics ◽  
Chaotic Attractors ◽  
Effector Cells ◽  
One Dimensional ◽  
Tumor Growth Model ◽  
Iterated Maps ◽  
Theoretical Investigations ◽  
Dynamical Systems Modeling

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

scholarly journals A No-Equilibrium Hyperchaotic System and Its Fractional-Order Form

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Duy Vo Hoang ◽  
Sifeu Takougang Kingni ◽  
Viet-Thanh Pham
Keyword(s):  
Fractional Order ◽  
Chaotic Behavior ◽  
Chaotic Attractors ◽  
Dimensional System ◽  
Hyperchaotic System ◽  
Equilibrium System ◽  
Coexisting Attractors ◽  
Amplitude Control ◽  
Dynamical Behaviors ◽  
Point Attractor

No-equilibrium system with chaotic behavior has attracted considerable attention recently because of its hidden attractor. We study a new four-dimensional system without equilibrium in this work. The new no-equilibrium system exhibits hyperchaos and coexisting attractors. Amplitude control feature of the system is also discovered. The commensurate fractional-order version of the proposed system is studied using numerical simulations. By tuning the commensurate fractional-order, the proposed system displays a wide variety of dynamical behaviors ranging from coexistence of quasiperiodic and chaotic attractors and bistable chaotic attractors to point attractor via transient chaos.

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

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.

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