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.

STABILIZATION OF UNSTABLE PERIODIC ORBITS OF CHAOTIC DISCRETE-TIME SYSTEMS USING PREDICTION-BASED FEEDBACK CONTROL

2002 ◽  
Vol 12 (02) ◽  
pp. 439-446 ◽  
Author(s):  
TORU HINO ◽  
SHIGERU YAMAMOTO ◽  
TOSHIMITSU USHIO
Keyword(s):  
Fixed Point ◽  
Feedback Control ◽  
Periodic Orbits ◽  
Discrete Time ◽  
Delayed Feedback Control ◽  
Necessary Condition ◽  
Unstable Periodic Orbits ◽  
Discrete Time Systems ◽  
Unstable Fixed Point ◽  
Time Systems

In this paper, we consider feedback control that stabilizes unstable periodic orbits (UPOs) of chaotic discrete-time systems. First, we show that there exists a strong necessary condition for stabilization of the UPOs when we use delayed feedback control (DFC) that is known as one of the useful methods for controlling chaotic systems. The condition is similar to that in the fixed point stabilization problem, in which it is impossible to stabilize the target unstable fixed point if the Jacobian matrix of the linearized system around it has an odd number of real eigenvalues greater than unity. In order to stabilize UPOs which cannot be stabilized by the standard DFC, we adopt prediction-based control. We show a necessary and sufficient condition for the stabilization of the UPOs with arbitrary period.

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.

EXPERIMENTAL RESULTS ON CHUA'S CIRCUIT ROBUST SYNCHRONIZATION VIA LMIs

2007 ◽  
Vol 17 (09) ◽  
pp. 3199-3209 ◽  
Author(s):  
C. D. CAMPOS ◽  
R. M. PALHARES ◽  
E. M. A. M. MENDES ◽  
L. A. B. TORRES ◽  
L. A. MOZELLI
Keyword(s):  
Discrete Time ◽  
Chaotic Systems ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Chua’S Oscillator ◽  
Laboratory Setup ◽  
Robust Synchronization ◽  
Discrete Time Systems ◽  
Main Ideas ◽  
Time Systems

This paper investigates the synchronization of coupled chaotic systems using techniques from the theory of robust [Formula: see text] control based on Linear Matrix Inequalities. To deal with the synchronization of a class of Lur'e discrete time systems, a project methodology is proposed. A laboratory setup based on Chua's oscillator circuit is used to demonstrate the main ideas of the paper in the context of the problem of information transmission.

Efficient switching between controlled unstable periodic orbits in higher dimensional chaotic systems

1995 ◽  
Vol 51 (5) ◽  
pp. 4169-4172 ◽  
Author(s):  
Ernest Barreto ◽  
Eric J. Kostelich ◽  
Celso Grebogi ◽  
Edward Ott ◽  
James A. Yorke
Keyword(s):  
Periodic Orbits ◽  
Chaotic Systems ◽  
Unstable Periodic Orbits ◽  
Higher Dimensional

THE GENERALIZED HÉNON MAPS: EXAMPLES FOR HIGHER-DIMENSIONAL CHAOS

2002 ◽  
Vol 12 (06) ◽  
pp. 1371-1384 ◽  
Author(s):  
HENDRIK RICHTER
Keyword(s):  
Discrete Time ◽  
Finite Dimension ◽  
Initial Conditions ◽  
Henon Maps ◽  
Hénon Maps ◽  
Discrete Time Systems ◽  
Higher Dimensional ◽  
Parameter Values ◽  
Time Systems

The generalized Hénon maps (GHM) are discrete-time systems with given finite dimension, which show chaotic and hyperchaotic behavior for certain parameter values and initial conditions. A study of these maps is given where particularly higher-dimensional cases are considered.

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