scholarly journals ON THE STABILIZATION OF PERIODIC ORBITS FOR DISCRETE TIME CHAOTIC SYSTEMS BY USING SCALAR FEEDBACK

2007 ◽  
Vol 17 (12) ◽  
pp. 4431-4442 ◽  
Author(s):  
ÖMER MORGÜL
Keyword(s):  
Periodic Orbits ◽  
Discrete Time ◽  
Chaotic Systems ◽  
Delayed Feedback ◽  
Stabilization Problem ◽  
Unstable Periodic Orbits ◽  
Scalar Input ◽  
Simulation Results ◽  
The Stability ◽  
Stability Results

In this paper we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems by using a scalar input. We use a simple periodic delayed feedback law and present some stability results. These results show that all hyperbolic periodic orbits as well as some nonhyperbolic periodic orbits can be stabilized with the proposed method by using a scalar input, provided that some controllability or observability conditions are satisfied. The stability proofs also lead to the possible feedback gains which achieve stabilization. We will present some simulation results as well.

scholarly journals A NEW GENERALIZATION OF DELAYED FEEDBACK CONTROL

2009 ◽  
Vol 19 (01) ◽  
pp. 365-377 ◽  
Author(s):  
ÖMER MORGÜL
Keyword(s):  
Periodic Orbits ◽  
Chaotic Systems ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Stabilization Problem ◽  
Unstable Periodic Orbits ◽  
One Dimensional ◽  
Feedback Scheme ◽  
The Stability ◽  
Stability Results

In this paper, we consider the stabilization problem of unstable periodic orbits of one-dimensional discrete time chaotic systems. We propose a novel generalization of the classical delayed feedback law and present some stability results. These results show that for period 1 all hyperbolic periodic orbits can be stabilized by the proposed method; for higher order periods the proposed scheme possesses some inherent limitations. However, some more improvement over the classical delayed feedback scheme can be achieved with the proposed scheme. The stability proofs also give the possible feedback gains which achieve stabilization. We will also present some simulation results.

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.

DELAYED FEEDBACK CONTROL OF CHAOTIC SYSTEMS WITH DRY FRICTION

2002 ◽  
Vol 12 (08) ◽  
pp. 1877-1883 ◽  
Author(s):  
UGO GALVANETTO
Keyword(s):  
Dry Friction ◽  
Chaotic Systems ◽  
Control Method ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Numerical Techniques ◽  
The Stability ◽  
Time Delayed Feedback

This paper describes some numerical techniques to control unstable periodic orbits embedded in chaotic attractors of a particular discontinuous mechanical system. The control method is a continuous time delayed feedback that modifies the stability of the orbit but does not affect the orbit itself.

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 Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

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.

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