Time-Delayed Feedback Control of Time-Delay Chaotic Systems

This paper addresses time-delayed feedback control (DFC) of time-delay chaotic systems. To extend the DFC approach to time-delay chaotic system, alter having been successfully used in chaotic systems without time-delays, the standard feedback control (SFC) method is firstly employed to show the main control technique in this paper based on one error control system. Then sufficient conditions for stabilization and tracking problems via DFC are derived from the results based on SFC. Also, the systematic and analytic controller design method can be obtained to stabilize the system to an unstable fixed point and to tracking an unstable periodic orbit, respectively. Some numerical examples are provided to demonstrate the effectiveness of the presented method.

[1]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[2]  Hsueh-Chia Chang,et al.  Transition to Chaos from a Two-Torus in a Delayed Feedback System , 1991 .

[3]  Yudong Liu,et al.  Control of Lorenz chaos , 1994 .

[4]  Glorieux,et al.  Stabilization and characterization of unstable steady states in a laser. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[5]  Guanrong Chen,et al.  On time-delayed feedback control of chaotic systems , 1999 .

[6]  José Manuel Gutiérrez,et al.  SUPPRESSION OF CHAOS THROUGH CHANGES IN THE SYSTEM VARIABLES THROUGH POINCARÉ AND LORENZ RETURN MAPS , 1996 .

[7]  E. A. Jackson.  On the control of complex dynamic systems , 1991 .

[8]  L. Chua,et al.  Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication , 1997 .

[9]  H. Nakajima.  On analytical properties of delayed feedback control of chaos , 1997 .

[10]  G. A. Johnson,et al.  Maintaining stability in Chua's Circuit Driven into Regions of oscillation and Chaos , 1993, J. Circuits Syst. Comput..

[11]  Antonia J. Jones,et al.  PERIODIC RESPONSE TO EXTERNAL STIMULATION OF A CHAOTIC NEURAL NETWORK WITH DELAYED FEEDBACK , 1999 .

[12]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[13]  Joshua E. S. Socolar,et al.  Stability of periodic orbits controlled by time-delay feedback , 1995, chao-dyn/9510019.

[14]  Güémez,et al.  Stabilization of chaos by proportional pulses in the system variables. , 1994, Physical review letters.

[15]  Marco Gilli,et al.  Predicting chaos through an harmonic balance technique: an application to the time-delayed Chua's circuit , 1996 .

[16]  Zhenya He,et al.  A chaos-generator: analyses of complex dynamics of a cell equation in delayed cellular neural networks , 1998 .

[17]  Guanrong Chen,et al.  On feedback control of chaotic continuous-time systems , 1993 .

[18]  Roy,et al.  Tracking unstable steady states: Extending the stability regime of a multimode laser system. , 1992, Physical review letters.

[19]  Guido Herrmann,et al.  A robust delay adaptation scheme for Pyragas' chaos control method , 2001 .

[20]  Jaros,et al.  Absorption in p-type Si-SiGe strained quantum-well structures. , 1994, Physical review. B, Condensed matter.

[21]  Guanrong Chen,et al.  FEEDBACK CONTROL OF A QUADRATIC MAP MODEL OF CARDIAC CHAOS , 1996 .

[22]  J. D. Farmer,et al.  Chaotic attractors of an infinite-dimensional dynamical system , 1982 .

[23]  B. Huberman,et al.  Dynamics of adaptive systems , 1990 .

[24]  Andrew V. Shobukhov.  Complex Dynamic Behavior in a Model of Electro-Optic Device with Time-Delayed Feedback , 1998 .

[25]  E. Atlee Jackson,et al.  The entrainment and migration controls of multiple-attractor systems , 1990 .

[26]  Guanrong Chen,et al.  FUZZY PREDICTIVE CONTROL OF UNCERTAIN CHAOTIC SYSTEMS USING TIME SERIES , 1999 .

[27]  Furong Gao,et al.  Adaptive control of chaotic continuous-time systems with delay , 1998 .

[28]  Kestutis Pyragas,et al.  An electronic analog of the Mackey-Glass system , 1995 .

[29]  Tong Kun Lim,et al.  An experimental study of storing information in a controlled chaotic system with time-delayed feedback , 1998 .

[30]  G. Duan,et al.  Robust guaranteed cost control for discrete-time uncertain systems with delay , 1999 .

[31]  Wolfram Just,et al.  MECHANISM OF TIME-DELAYED FEEDBACK CONTROL , 1996, chao-dyn/9611012.

[32]  Zonghua Liu,et al.  Control of Coupled Standard Map , 1998 .

[33]  Keiji Konishi,et al.  LEARNING CONTROL OF TIME-DELAY CHAOTIC SYSTEMS AND ITS APPLICATIONS , 1998 .

[34]  Celso Grebogi,et al.  Using chaos to target stationary states of flows , 1992 .

[35]  Singer,et al.  Controlling a chaotic system. , 1991, Physical review letters.

[36]  Chiappinelli.  Georgi-Weinberg theorem at q2=MZ2. , 1994, Physical review. D, Particles and fields.

[37]  E. R. Hunt,et al.  Derivative control of the steady state in Chua's circuit driven in the chaotic region , 1993 .

[38]  Guanrong Chen,et al.  Controlling the dynamical behavior of a circle map model of the human heart , 2004, Biological Cybernetics.

[39]  Guanrong Chen,et al.  Bifurcation Dynamics in Control Systems , 1999 .

[40]  Goldhirsch,et al.  Taming chaotic dynamics with weak periodic perturbations. , 1991, Physical review letters.

[41]  Kestutis Pyragas,et al.  Stabilization of an unstable steady state in a Mackey-Glass system , 1995 .

[42]  Zhenya He,et al.  Chaotic behavior in first-order autonomous continuous-time systems with delay , 1996 .

[43]  Th. Meyer,et al.  Recovery of the time-evolution equation of time-delay systems from time series , 1997, chao-dyn/9907009.