Chaos control and synchronization of a @F^6-Van der Pol oscillator [rapid communication]

Abstract The dynamic behavior of a Φ 6 -Van der Pol oscillator subjected to an external disturbance is studied. Numerical analysis of phase diagrams are presented to observe periodic and chaotic motions and sliding mode control design is used to control chaos effectively. Assuming that the exact model of the oscillator is not known and that position is the only state available for measurements, the controller comprises a sliding-like feedback and an uncertainty estimator. As a potential application of the proposed control strategy, we used it to study the synchronization of two Φ 6 -Van der Pol oscillators with uncertainties. Numerical simulations are provided to illustrate the performance of the proposed control strategy.

[1]  M. Lakshmanan,et al.  Chaos in Nonlinear Oscillators: Controlling and Synchronization , 1996 .

[2]  M. Lakshmanan,et al.  Bifurcation and chaos in the double-well Duffing–van der Pol oscillator: Numerical and analytical studies , 1997, chao-dyn/9709013.

[3]  J. Reade,et al.  Experiments of light and shadow made by means of the prism , 1830 .

[4]  Norio Akamatsu,et al.  Chaotically transitional phenomena in the forced negative-resistance oscillator , 1980 .

[5]  H. Sira-Ramírez.  On the dynamical sliding mode control of nonlinear systems , 1993 .

[6]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[7]  Hong-Jun Zhang,et al.  Phase effect of two coupled periodically driven Duffing oscillators , 1998 .

[8]  C. Tchawoua,et al.  Exponential stabilization of two nonlinearly coupled oscillators by an estimated state feedback , 2002 .

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

[10]  Tao Yang,et al.  Synchronizing chaotic dynamics with uncertainties based on a sliding mode control design. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Chyun-Chau Fuh,et al.  Experimental and analytical study of dither signals in a class of chaotic systems , 1997 .

[12]  Fatihcan M. Atay,et al.  VAN DER POL'S OSCILLATOR UNDER DELAYED FEEDBACK , 1998 .

[13]  Wanda Szemplińska-Stupnicka,et al.  Neimark bifurcation, almost-periodicity and chaos in the forced van der Pol-Duffing system in the neighbourhood of the principal resonance , 1994 .

[14]  A. Teel,et al.  Tools for Semiglobal Stabilization by Partial State and Output Feedback , 1995 .

[15]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[16]  Parlitz,et al.  Period-doubling cascades and devil's staircases of the driven van der Pol oscillator. , 1987, Physical review. A, General physics.

[17]  P. Kokotovic,et al.  The peaking phenomenon and the global stabilization of nonlinear systems , 1991 .

[18]  Leon O. Chua,et al.  ADAPTIVE SYNCHRONIZATION OF CHUA'S OSCILLATORS , 1996 .

[19]  Balth. van der Pol,et al.  VII. Forced oscillations in a circuit with non-linear resistance. (Reception with reactive triode) , 1927 .

[20]  Nan-Sheng Huang,et al.  CONTROL AND SYNCHRONIZATION OF DISCRETE-TIME CHAOTIC SYSTEMS VIA VARIABLE STRUCTURE CONTROL TECHNIQUE , 1997 .