Symbolic dynamics based method for rigorous study of the existence of short cycles for chaotic systems

2009 ◽  
Author(s):  
Zbigniew Galias ◽  
Warwick Tucker
Keyword(s):  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Rigorous Study

Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850155 ◽  
Author(s):  
Chengwei Dong
Keyword(s):  
Nonlinear System ◽  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Building Blocks ◽  
Topological Classification ◽  
One Dimensional ◽  
Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

Symbolic Dynamics Generated by a Hybrid Chaotic Systems

2016 ◽  
Vol 18 (2) ◽  
pp. 1-12 ◽  
Author(s):  
Carlos Ramos ◽  
Ana Santos ◽  
Sandra Vinagre
Keyword(s):  
Symbolic Dynamics ◽  
Chaotic Systems

scholarly journals Chaotic Entanglement: Entropy and Geometry

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1254
Author(s):  
Matthew A. Morena ◽  
Kevin M. Short
Keyword(s):  
Coherent Structures ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Entanglement Entropy ◽  
Future Research ◽  
Unstable Periodic Orbits ◽  
Research Directions ◽  
Period Orbit ◽  
Future Research Directions

In chaotic entanglement, pairs of interacting classically-chaotic systems are induced into a state of mutual stabilization that can be maintained without external controls and that exhibits several properties consistent with quantum entanglement. In such a state, the chaotic behavior of each system is stabilized onto one of the system’s many unstable periodic orbits (generally located densely on the associated attractor), and the ensuing periodicity of each system is sustained by the symbolic dynamics of its partner system, and vice versa. Notably, chaotic entanglement is an entropy-reversing event: the entropy of each member of an entangled pair decreases to zero when each system collapses onto a given period orbit. In this paper, we discuss the role that entropy plays in chaotic entanglement. We also describe the geometry that arises when pairs of entangled chaotic systems organize into coherent structures that range in complexity from simple tripartite lattices to more involved patterns. We conclude with a discussion of future research directions.

Unstable cycles for the Burke–Shaw system via variational approach

2019 ◽  
Vol 33 (21) ◽  
pp. 1950240
Author(s):  
Chengwei Dong ◽  
Huihui Liu
Keyword(s):  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Topological Structure ◽  
Variational Approach ◽  
Chaotic Systems ◽  
Dissipative System ◽  
Section Method ◽  
Varied Structure ◽  
Low Dimensional

In this paper, the systematical calculations of the unstable cycles for the Burke–Shaw system (BSS) are presented. In contrast to the Poincaré section method used in previous studies, we used the variational method for the cycle search and established appropriate symbolic dynamics on the basis of the topological structure of the cycles. The variational approach made it easy to continuously track the periodic orbits when the parameters were varied. Structure of the whole cycle in the dissipative system demonstrated that the methodology could be effective in most low-dimensional chaotic systems.

ROBUSTNESS OF SYMBOLIC DYNAMICS AND SYNCHRONIZATION PROPERTIES

2000 ◽  
Vol 10 (04) ◽  
pp. 811-818
Author(s):  
ZBIGNIEW GALIAS
Keyword(s):  
Interval Arithmetic ◽  
Coupling Strength ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Parameter Mismatch ◽  
Topological Methods ◽  
Henon Maps ◽  
Hénon Maps

In this paper we introduce the method for investigation of coupled chaotic systems using topological methods. We show that if the coupling is small then there exists independent symbolic dynamics for every coupled subsystem and in consequence the systems are not synchronized. As an example we consider coupled Hénon maps. Using computer interval arithmetic we find parameter mismatch and perturbation range for which the symbolic dynamics in the Hénon system is sustained. For coupled Hénon maps we compute the value of coupling strength for which the symbolic dynamics in every subsystem survives.

D-test: A New Test for Analyzing Scale Invariance Using Symbolic Dynamics and Symbolic Entropy

Methodology ◽  
2011 ◽  
Vol 7 (3) ◽  
pp. 88-95 ◽  
Author(s):  
Jose A. Martínez ◽  
Manuel Ruiz Marín
Keyword(s):  
Symbolic Dynamics ◽  
Scale Invariance ◽  
Rating Scales ◽  
Rating Scale ◽  
Marketing Research ◽  
Response Patterns ◽  
Measurement Scales ◽  
Improve Measurement ◽  
Entropy Measures ◽  
The Difference

The aim of this study is to improve measurement in marketing research by constructing a new, simple, nonparametric, consistent, and powerful test to study scale invariance. The test is called D-test. D-test is constructed using symbolic dynamics and symbolic entropy as a measure of the difference between the response patterns which comes from two measurement scales. We also give a standard asymptotic distribution of our statistic. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. We applied D-test to a real marketing research to study if scale invariance holds when measuring service quality in a sports service. We considered a free-scale as a reference scale and then we compared it with three widely used rating scales: Likert-type scale from 1 to 5 and from 1 to 7, and semantic-differential scale from −3 to +3. Scale invariance holds for the two latter scales. This test overcomes the shortcomings of other procedures for analyzing scale invariance; and it provides researchers a tool to decide the appropriate rating scale to study specific marketing problems, and how the results of prior studies can be questioned.

Sign in / Sign up

Export Citation Format

Share Document