FROM SYMBOLIC DYNAMICS TO A DIGITAL APPROACH

2001 ◽  
Vol 11 (06) ◽  
pp. 1683-1694 ◽  
Author(s):  
K. KARAMANOS
Keyword(s):  
Symbolic Dynamics ◽  
Chaotic Attractors ◽  
Unimodal Maps ◽  
Quantitative Manner ◽  
The Asymmetry

We show that the numbers generated by the symbolic dynamics of Feigenbaum attractors are transcendental. Due to the asymmetry of the chaotic attractors of unimodal maps around the maximum in the general case, a standard conjecture, that the occurrence of chaos is related to the transcendence of the number defined by the corresponding symbolic dynamics is reassessed and formulated in a quantitative manner. It is concluded that transcendence may provide an appropriate measure of complexity.

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 WINDING NUMBERS FOR PERIOD-DOUBLING CASCADES

1996 ◽  
Vol 06 (01) ◽  
pp. 185-187
Author(s):  
CARSTEN KNUDSEN
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
Accumulation Point ◽  
Winding Number ◽  
Unimodal Maps ◽  
Forced Oscillators ◽  
Essential Properties ◽  
Limiting Value

We define the topological winding number for unimodal maps that share the essential properties of that of winding numbers for forced oscillators exhibiting period-doubling cascades. It is demonstrated how this number can be computed for any of the periodic orbits in the first period-doubling cascade. The limiting winding number at the accumulation point of the first period-doubling cascade is also derived. It is shown that the limiting value for the winding number ω∞ can be computed as the Farey sum of any two neighbouring topological winding numbers in the period-doubling cascade. The derivations are all based on symbolic dynamics and simple combinatorics.

scholarly journals Robust chaos and the continuity of attractors

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Paul A Glendinning ◽  
David J W Simpson
Keyword(s):  
Normal Form ◽  
Parameter Space ◽  
Hausdorff Metric ◽  
Piecewise Smooth ◽  
Chaotic Attractors ◽  
Unimodal Maps ◽  
Smooth Maps ◽  
Tent Maps ◽  
Piecewise Smooth Maps

Abstract As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for many families of piecewise-smooth maps it provides a way to think about changing structures within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map and the border-collision normal form.

A GEOMETRIC MODEL FOR THE DUFFING OSCILLATOR

1993 ◽  
Vol 03 (03) ◽  
pp. 685-691 ◽  
Author(s):  
J.W.L. McCALLUM ◽  
R. GILMORE
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Attractor ◽  
Geometric Model ◽  
Duffing Oscillator ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Full Period ◽  
Parameter Values

A geometric model for the Duffing oscillator is constructed by analyzing the unstable periodic orbits underlying the chaotic attractors present at particular parameter values. A template is constructed from observations of the motion of the chaotic attractor in a Poincaré section as the section is swept for one full period. The periodic orbits underlying the chaotic attractor are found and their linking numbers are computed. These are compared with the linking numbers from the template and the symbolic dynamics of the orbits are identified. This comparison is used to validate the template identification and label the orbits by their symbolic dynamics.

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.

BETA ENCODERS: SYMBOLIC DYNAMICS AND ELECTRONIC IMPLEMENTATION

2012 ◽  
Vol 22 (09) ◽  
pp. 1230031 ◽  
Author(s):  
TOHRU KOHDA ◽  
YOSHIHIKO HORIO ◽  
YOICHIRO TAKAHASHI ◽  
KAZUYUKI AIHARA
Keyword(s):  
Characteristic Equation ◽  
Symbolic Dynamics ◽  
Electronic Circuit ◽  
Electric Circuit ◽  
Quantization Error ◽  
Chaotic Attractors ◽  
Analog To Digital ◽  
New Class ◽  
Circuit Technique ◽  
Electronic Implementation

A new class of analog-to-digital (A/D) and digital-to-analog (D/A) converters that uses a flaky quantizer, known as a β-encoder, has been shown to have exponential bit rate accuracy and a self-correcting property for fluctuations of the amplifier factor β and quantizer threshold ν. The probabilistic behavior of this flaky quantizer is explained by the deterministic dynamics of a multivalued Rényi–Parry map on the middle interval, as defined here. This map is eventually locally onto map of [ν - 1, ν], which is topologically conjugate to Parry's (β, α)-map with α = (β - 1)(ν - 1). This viewpoint allows us to obtain a decoded sample, which is equal to the midpoint of the subinterval, and its associated characteristic equation for recovering β, which improves the quantization error by more than 3 dB when β > 1.5. The invariant subinterval under the Rényi–Parry map shows that ν should be set to around the midpoint of its associated greedy and lazy values. Furthermore, a new A/D converter referred to as the negative β-encoder is introduced, and shown to further improve the quantization error of the β-encoder. Then, a switched-capacitor (SC) electronic circuit technique is proposed for implementing A/D converter circuits based on several types of β-encoders. Electric circuit experiments were used to verify the validity of these circuits against deviations and mismatches of the circuit parameters. Finally, we demonstrate that chaotic attractors can be observed experimentally from these β-encoder circuits.

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.

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