Chaos control via TDFC in time-delayed systems: The harmonic balance approach

Abstract This Letter deals with the problem of designing time-delayed feedback controllers (TDFCs) to stabilize unstable equilibrium points and periodic orbits for a class of continuous time-delayed chaotic systems. Harmonic balance approach is used to select the appropriate controller parameters: delay time and feedback gain. The established theoretical results are illustrated via a case study of the well-known Logistic model.

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

[2]  Jigui Jian,et al.  Stability, bifurcation and a new chaos in the logistic differential equation with delay , 2006 .

[3]  Giacomo Innocenti,et al.  A global qualitative view of bifurcations and dynamics in the Rössler system , 2008 .

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

[5]  A. Sharkovsky,et al.  Chaos from a time-delayed Chua's circuit , 1993 .

[6]  Jorge L. Moiola,et al.  On period doubling bifurcations of cycles and the harmonic balance method , 2006 .

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

[8]  Eckehard Schöll,et al.  Handbook of Chaos Control , 2007 .

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

[10]  Earl H. Dowell,et al.  A novel harmonic balance analysis for the Van Der Pol oscillator , 2007 .

[11]  K. Ikeda,et al.  Study of a high-dimensional chaotic attractor , 1986 .

[12]  R. Lang,et al.  External optical feedback effects on semiconductor injection laser properties , 1980 .

[13]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[14]  Alberto Tesi,et al.  Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics , 1996, Autom..

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

[16]  A. Sedigh,et al.  Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation , 2008 .

[17]  Peter A. Cook,et al.  Nonlinear dynamical systems , 1986 .

[18]  C. Piccardi.  Controlling chaotic oscillations in delay-differential systems via peak-to-peak maps , 2001 .

[19]  A. Tesi,et al.  A frequency approach for analyzing and controlling chaos in nonlinear circuits , 1993 .

[20]  M. N. Hamdan,et al.  BIFURCATIONS OF APPROXIMATE HARMONIC BALANCE SOLUTIONS AND TRANSITION TO CHAOS IN AN OSCILLATOR WITH INERTIAL AND ELASTIC SYMMETRIC NONLINEARITIES , 2001 .

[21]  Fotios Giannakopoulos,et al.  Local and Global Hopf Bifurcation in a Scalar Delay Differential Equation , 1999 .

[22]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .