Global stabilization of periodic orbits in chaotic systems by using symbolic dynamics

2008 ◽  
Vol 41 (2) ◽  
pp. 15131-15136
Author(s):  
Masayasu Suzuki ◽  
Noboru Sakamoto
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Global Stabilization

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.

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.

On the Potential for Entangled States Between Chaotic Systems

2014 ◽  
Vol 24 (06) ◽  
pp. 1450077 ◽  
Author(s):  
Matthew A. Morena ◽  
Kevin M. Short
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Entangled States ◽  
Future Research ◽  
Unstable Periodic Orbits ◽  
Qualitative Information ◽  
Chaotic Dynamical System ◽  
Naturally Occurring ◽  
External Intervention

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

Chaotic scattering on a double well: Periodic orbits, symbolic dynamics, and scaling

1993 ◽  
Vol 3 (4) ◽  
pp. 475-485 ◽  
Author(s):  
Vincent Daniels ◽  
Michel Vallières ◽  
Jian‐Min Yuan
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Scattering

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

scholarly journals Tracking Control of a Class of Chaotic Systems

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 568 ◽  
Author(s):  
Anqing Yang ◽  
Linshan Li ◽  
Zuoxun Wang ◽  
Rongwei Guo
Keyword(s):  
Control Problem ◽  
Periodic Orbits ◽  
Chaotic System ◽  
Tracking Control ◽  
Reference System ◽  
Chaotic Systems ◽  
Asymptotic Tracking ◽  
The Given

This paper investigates the asymptotic tracking control problem of the chaotic system. Firstly, a reference system is presented, the output of which can asymptotically track a given command. Then, a both physically implementable and simple controller is designed, by which the given chaotic system synchronizes the reference system, and thus the output of such chaotic systems can asymptotically track the given command. It should be pointed out that the output of the given chaotic system can asymptotically track arbitrary desired periodic orbits. Finally, several illustrative examples are taken as example to show the validity and effectiveness of the obtained results.

Map Dynamics Study of the Lorenz Equations

1997 ◽  
Vol 07 (02) ◽  
pp. 373-382 ◽  
Author(s):  
Olivier Michielin ◽  
Paul E. Phillipson
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Strange Attractor ◽  
Good Accuracy ◽  
Simple Solution ◽  
Lorenz Equations ◽  
Computer Solution ◽  
One Dimensional

The Lorenz equations [Lorenz, 1963], in addition to a strange attractor, display sequences of periodic and aperiodic orbits. Approximate one-dimensional map solutions are heuristically constructed, supplementing previous symbolic dynamics studies, which closely reproduce these sequences. A relatively simple solution reproduces the sequence topology to good accuracy. A second more refined solution reproduces to higher accuracy both the topology and scale of the attractor. The second solution is sufficiently accurate to predict periodic orbits not previously observed and difficult to extract directly from computer solution of the Lorenz equations.

TIME-DELAY EMBEDDINGS OF IFS ATTRACTORS

Fractals ◽  
1999 ◽  
Vol 07 (02) ◽  
pp. 133-138
Author(s):  
SONYA BAHAR
Keyword(s):  
Time Series ◽  
Time Delay ◽  
Periodic Orbits ◽  
Topological Structure ◽  
Chaotic Systems ◽  
Iterated Function System ◽  
Function System ◽  
Chaotic Attractors ◽  
Topological Characterization ◽  
Iterated Function

A modified type of iterated function system (IFS) has recently been shown to generate images qualitatively similar to "classical" chaotic attractors. Here, we use time-delay embedding reconstructions of time-series from this system to generate three-dimentional projections of IFS attractors. These reconstructions may be used to access the topological structure of the periodic orbits embedded within the attractor. This topological characterization suggests an approach by which a rigorous comparison of IFS attractors and classical chaotic systems may be attained.

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