Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control

International Journal of Non-Linear Mechanics

Volumn 45, Issue 6, 2010, Pages 621-627

  Yang, X L ^a   Sun, Z K ^b  

^a SHAANXI NORMAL UNIVERSITY   (China)

^b NORTHWESTERN POLYTECHNICAL UNIVERSITY   (China)

Author keywords

Multiple time delayed feedback; Parametric resonance; Random noise

Indexed keywords

CIRCUIT OSCILLATIONS; DYNAMICAL SYSTEMS; EXCITED STATES; FEEDBACK; FEEDBACK CONTROL; LYAPUNOV METHODS; METHOD OF MOMENTS; OSCILLATORS (MECHANICAL); PARAMETRIC DEVICES; PARAMETRIC OSCILLATORS; RESONANCE; STOCHASTIC SYSTEMS;

LARGEST LYAPUNOV EXPONENT; MULTIPLE TIME DELAYED FEEDBACKS; NUMERICAL CALCULATION; PARAMETRIC RESONANCE; RANDOM NOISE; STEADY STATE SOLUTION; TIME DELAYED FEEDBACK CONTROL; VAN DER POL OSCILLATOR;

TIME DELAY;

EID: 84855291609     PISSN: 00207462     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijnonlinmec.2010.03.005     Document Type: Conference Paper

References (22)

1. W. Wischert, A. Wunderlin, A. Pelster, M. Olivier, J. Groslambert, Delay-induced instabilities in nonlinear feedback systems, Phys. Rev. E 49 (1994) 203-219.
2. K. Miyakawa, K. Yamada, Entrainment in coupled salt-water oscillators, Physica D 127 (1999) 177-186. (Pubitemid 129586903)
3. P. Tass, J. Kurths, M.G. Rosenblum, G. Guasti, H. Hefter, Delay-induced transitions in visually guided movements, Phys. Rev. E 54 (1996) R2224-R2227. (Pubitemid 126589581)
4. A. Destexhe, Stability of periodic oscillations in a network of neurons with time delay, Phys. Lett. A 187 (1994) 309-316.
5. Z.K. Sun, W. Xu, X.L. Yang, T. Fang, Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback, Chaos, Solitons Fractals 27 (3) (2006) 705-714. (Pubitemid 41084204)
6. A. Maccari, The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dyn. 26 (2001) 105-119.
7. A. Maccari, Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback, Int. J. Nonlinear Mech. 38 (2003) 123-131.
8. J. Xu, K.W. Chung, Effects of time delayed position feedback on a van der Pol-Duffing oscillator, Physica D 180 (2003) 17-39.
9. J.C. Ji, C.H. Hansen, Stability and dynamics of a controlled van der Pol-Duffing oscillator, Chaos, Solitons Fractals 28 (2006) 555-570. (Pubitemid 41491963)
10. T.M. Morrison, R.H. Rand, 2:1 Resonance in the delayed nonlinear Mathieu equation, Nonlinear Dyn. 50 (2007) 341-352. (Pubitemid 47327550)
11. T. Mori, H. Kokame, Stability of x(t) = Ax(t)+Bx(t-T), IEEE Trans. Autom. Control 34 (1989) 460-462.
12. H.Y. Hu, Z.H. Wang, Stability of analysis of damped SDof systems with two time delays in state feedback, J. Sound Vib. 214 (1998) 213-225. (Pubitemid 128379392)
13. Z.K. Sun, W. Xu, X.L Yang, Response of nonlinear system to random narrow-band excitation with time delay state feedback, J. Vib. Eng. 19 (2006) 57-64. (Pubitemid 43819301)
14. Y.F. Jin, H.Y. Hu, Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation, Nonlinear Dyn. 50 (2007) 213-227.
15. W.V. Weding, Invariant measures and Lyapunov exponents for generalized parameter fluctuations, Struct. Saf. 8 (1990) 13-25.
16. A.H. Hayfeh, B. Blachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, John Wiley & Sons, Inc., New York, 1995.
17. H.Y. Hu, Z.H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer-Verlag, Berlin Heidelberg, 2002.
18. R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen and den Rijn, 1980.
19. V.I. Oseledec, A multiplicative ergodic theorem Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (2) (1968) 197-231.
20. G. Schmidt, A. Tondl, Nonlinear Vibrations, Cambridge University Press, Cambridge, 1986.
21. M. Shinozuka, Digital simulation of random processes and its applications, J. Sound Vib. 25 (1972) 111-128.
22. W.Q. Zhu, Random Vibration, Beijing Science Press, China, 1992.