Analytical properties and optimization of time-delayed feedback control.

Time-delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. If the equations governing the system dynamics are known, the success of the method can be predicted by a linear stability analysis of the desired orbit. Unfortunately, the usual procedures for evaluating the Floquet exponents of such systems are rather intricate. We show that the main stability properties of the system controlled by time-delayed feedback can be simply derived from a leading Floquet exponent defining the system behavior under proportional feedback control. Optimal parameters of the delayed feedback controller can be evaluated without an explicit integration of delay-differential equations. The method is valid for low-dimensional systems whose unstable periodic orbits are originated from a period doubling bifurcation and is demonstrated for the Rössler system and the Duffing oscillator.

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

[2]  Kestutis Pyragas.  Control of chaos via extended delay feedback , 1995 .

[3]  Reibold,et al.  Influence of stable Floquet exponents on time-delayed feedback control , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  S. Boccaletti,et al.  The control of chaos: theory and applications , 2000 .

[5]  E. Kostelich,et al.  Characterization of an experimental strange attractor by periodic orbits. , 1989, Physical review. A, General physics.

[6]  Kestutis Pyragas.  Control of chaos via an unstable delayed feedback controller. , 2001, Physical review letters.

[7]  G. Pfister,et al.  Control of chaotic Taylor-Couette flow with time-delayed feedback. , 2001, Physical review letters.

[8]  Gauthier,et al.  Stabilizing unstable periodic orbits in a fast diode resonator using continuous time-delay autosynchronization. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Gauthier,et al.  Stabilizing unstable periodic orbits in fast dynamical systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[11]  Robert G. Harrison,et al.  Instabilities and Tracking of Travelling Wave Patterns in a Three-Level Laser , 1998 .

[12]  Glorieux,et al.  Controlling unstable periodic orbits by a delayed continuous feedback. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Takashi Hikihara,et al.  An experimental study on stabilization of unstable periodic motion in magneto-elastic chaos , 1996 .

[14]  O. Rössler.  An equation for continuous chaos , 1976 .

[15]  L. Glass,et al.  DYNAMIC CONTROL OF CARDIAC ALTERNANS , 1997 .

[16]  Grebogi,et al.  Detecting unstable periodic orbits in chaotic experimental data. , 1996, Physical review letters.

[17]  I Z Kiss,et al.  Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  A. Piel,et al.  Continuous control of ionization wave chaos by spatially derived feedback signals , 1997 .

[19]  Visarath In,et al.  REAL-TIME EXPERIMENTAL CONTROL OF A SYSTEM IN ITS CHAOTIC AND NONCHAOTIC REGIMES , 1997 .

[20]  Antonia J. Jones,et al.  The control of higher dimensional chaos: comparative results for the chaotic satellite attitude control problem , 2000 .

[21]  Kestutis Pyragas,et al.  Experimental control of chaos by delayed self-controlling feedback , 1993 .

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

[23]  Celso Grebogi,et al.  Using small perturbations to control chaos , 1993, Nature.

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

[25]  Piotr Fronczak,et al.  Limits of time-delayed feedback control , 1999 .

[26]  G. Benettin,et al.  Kolmogorov entropy of a dynamical system with an increasing number of degrees of freedom , 1979 .

[27]  Celso Grebogi,et al.  Extracting unstable periodic orbits from chaotic time series data , 1997 .

[28]  H. Nakajima,et al.  Limitation of generalized delayed feedback control , 1998 .

[29]  I. Shimada,et al.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems , 1979 .

[30]  W. Müller,et al.  Biotinylated Carbohydrate Markers -A Novel Tool for Lectin Research , 1994 .