Harmonic balance-based approach for optimal time delay to control unstable periodic orbits of chaotic systems

2020 ◽  
Vol 36 (4) ◽  
pp. 918-925
Author(s):  
Y. M. Chen ◽  
Q. X. Liu ◽  
J. K. Liu
Keyword(s):  
Time Delay ◽  
Periodic Orbits ◽  
Harmonic Balance ◽  
Chaotic Systems ◽  
Optimal Time ◽  
Unstable Periodic Orbits

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.

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.

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.

A multi-dimensional scheme for controlling unstable periodic orbits in chaotic systems

2006 ◽  
Vol 349 (1-4) ◽  
pp. 116-127 ◽  
Author(s):  
Niranjan Chakravarthy ◽  
Kostas Tsakalis ◽  
Leon D. Iasemidis ◽  
Andreas Spanias
Keyword(s):  
Periodic Orbits ◽  
Chaotic Systems ◽  
Unstable Periodic Orbits

scholarly journals STABILIZATION OF UNSTABLE PERIODIC ORBITS FOR DISCRETE TIME CHAOTIC SYSTEMS BY USING PERIODIC FEEDBACK

2006 ◽  
Vol 16 (02) ◽  
pp. 311-323 ◽  
Author(s):  
ÖMER MORGÜL
Keyword(s):  
Periodic Orbits ◽  
Discrete Time ◽  
Chaotic Systems ◽  
Unstable Periodic Orbits ◽  
Discrete Time Systems ◽  
Feedback Scheme ◽  
Higher Dimensional ◽  
Periodic Feedback ◽  
Time Systems ◽  
Stability Results

We propose a periodic feedback scheme for the stabilization of periodic orbits for discrete time chaotic systems. We first consider one-dimensional discrete time systems and obtain some stability results. Then we extend these results to higher dimensional discrete time systems. The proposed scheme is quite simple and we show that any hyperbolic periodic orbit can be stabilized with this scheme. We also present some simulation results.

STABILIZING UNSTABLE PERIODIC ORBITS OF CHAOTIC SYSTEMS WITH UNKNOWN PARAMETERS

2000 ◽  
Vol 10 (03) ◽  
pp. 611-620 ◽  
Author(s):  
YU-PING TIAN ◽  
XINGHUO YU
Keyword(s):  
Periodic Orbits ◽  
Chaotic Systems ◽  
Control Method ◽  
Uncertain System ◽  
Parameter Estimates ◽  
System State ◽  
Unstable Periodic Orbits ◽  
Unknown Parameters ◽  
Model Following ◽  
Delayed System

A novel adaptive time-delayed control method is proposed for stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems with unknown parameters. We first explore the inherent properties of chaotic systems and use the system state and time-delayed system state to form an asymptotically stable invariant manifold so that when the system state enters the manifold and stays in it thereafter, the resulting motion enables the stabilization of the desired UPOs. We then use the model following concept to construct an identifier for the estimation of the uncertain system parameters. We shall prove that under the developed scheme, the system parameter estimates will converge to their true values. The effectiveness of the method is confirmed by computer simulations.

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