Response of a Parametrically Excited Duffing-Van der Pol Oscillator With Delayed Feedback

LI Xin-ye; CHEN Yu-shu; WU Zhi-qiang; SONG Tao

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2006 > 27(12): 1387-1396

LI Xin-ye, CHEN Yu-shu, WU Zhi-qiang, SONG Tao. Response of a Parametrically Excited Duffing-Van der Pol Oscillator With Delayed Feedback[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1387-1396.

Citation:

PDF( 1198 KB)

Response of a Parametrically Excited Duffing-Van der Pol Oscillator With Delayed Feedback

1.
School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, P. R. China;

Received Date: 2005-07-02
Rev Recd Date: 2006-07-24
Publish Date: 2006-12-15

Abstract

Abstract

The dynamical behaviour of a parametrically excited Duffing-Van der Poloscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciation choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original systems. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.
- Duffing-Van der Pol oscillator,
- principal parametric resonance,
- time delay,
- feedback control,
- bifurcation

FullText(HTML)

References(17)

References

[1]	Ferdinand Verhulst.Nonlinear Differential Equations and Dynamical Systems[M].Berlin：Springer-Verlag, 1990.
[2]	Jackson E Atlee.Perspectives of Nonlinear Dynamics[M].New York: Cambridge University Press, 1991.
[3]	Guckenheimer J, Holmes P.Nonlinear Oscillation and Bifurcation of Vector Fields[M].New York:Springer-Verlag, 1993.
[4]	Holmes P,Rand D.Phase portraits and bifurcation of the nonlinear oscillator:+(α+γx2)+βx+δx3=0[J].International Journal of Nonlinear Mechanics,1980,15(6):449—458. doi: 10.1016/0020-7462(80)90031-1
[5]	Tsuda Y,Tamura H,Sueoka A,et al.Chaotic behaviour of a nonlinear vibrating system with a retarded argument[J].JSME International Journal ,Series Ⅲ,1992,35(2):259—267.
[6]	Szemplinska-Stupnicka, Rudowski J. The coexistence of periodic, almost-periodic and chaotic attractions in the Van der Pol-Duffing oscillator[J].Journal of Sound and Vibration,1997,199(2):165—175. doi: 10.1006/jsvi.1996.0648
[7]	Maccari Attilio.Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation[J].Nonlinear Dynamics,1998,15(4):329—343. doi: 10.1023/A:1008235820302
[8]	Algaba A,Fernandez-Sanchez E,Freire E,et al.Oscillation-sliding in a modified Van der Pol-Duffing electronic oscillator[J].Journal of Sound and Vibration,2002,249(5):899—907. doi: 10.1006/jsvi.2001.3931
[9]	XU Jian,Chung K W.Effects of time delayed position feedback on a Van der Pol-Duffing oscillator[J].Physica D,2003,180(1):17—39. doi: 10.1016/S0167-2789(03)00049-6
[10]	Moukam Kakmeni F M,Bowong S,Tchawoua C,et al.Strange attractors and chaos control in a Duffing-Van der Pol oscillator with two external periodic forces[J].Journal of Sound and Vibration,2004,277(4/5):783—799. doi: 10.1016/j.jsv.2003.09.051
[11]	陈予恕.非线性振动系统的分叉和混沌理论[M]. 北京: 高等教育出版社,1993.
[12]	Fofana M S,Ryba P B.Pramertic stability of nonlinear time delay equations[J].International Journal of Nonlinear Mechanics,2004,39(1):79—91. doi: 10.1016/S0020-7462(02)00139-7
[13]	Ji J C,Leung A Y T. Bifurcation control of parametrically excited Duffing system[J].Nonlinear Dynamics,2002,27(4):411—417. doi: 10.1023/A:1015221422293
[14]	Hairer E,Norsett S P, Wanner G.Solving Ordinary Differential Equations Ⅰ: Nonstiff Problems[M].Berlin:Springer-Verlag,1987.
[15]	Hiroshi Yabuno. Bifurcation control of parametrically excited Duffing system by a combined linear-plus-nonlinear feedback control[J].Nonlinear Dynamics,1997,12(3):263—274. doi: 10.1023/A:1008270815516
[16]	Ji J C, Hansen C H. Nonlinear oscillations of a rotor in active magnetic bearings[J].Journal of Sound and Vibration,2001,240(4):599—612. doi: 10.1006/jsvi.2000.3257
[17]	丁千，陈予恕，叶敏，等. 一类非自治滞后-自激系统的主共振与锁模现象[J].力学学报，2002,34(1):123—130.