Abstract
Hopf bifurcation exists commonly in time-delay systems. The local dynamics of delayed systems near a Hopf bifurcation is usually investigated by using the center manifold reduction that involves a great deal of tedious symbolic and numerical computation. In this paper, the delayed oscillator of concern is considered as a system slightly perturbed from an undamped oscillator, then as a combination of the averaging technique and the method of Lyapunov's function, the energy analysis concludes that the local dynamics near the Hopf bifurcation can be justified by the averaged power function of the oscillator. The computation is very simple but gives considerable accurate prediction of the local dynamics. As an illustrative example, the local dynamics of a delayed Lienard oscillator is investigated via the present method.
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Hu, H. Y. and Wang, Z. H., Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer-Verlag, Heidelberg, 2002.
Niculescue, S.-I., Delay Effects on Stability: A Robust Approach, Springer-Verlag, London, 2001.
Qin, Y. X., Liu, Y. Q., Wang, L., and Zhen, Z.X., Stability of Dynamic Systems with Delays, Science Press, Beijing, 1989.
Campbell, S. A., Belair, J., Ohira, T., and Milton, J., ‘Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback,’ Chaos 5, 1995, 640–645.
Stepan, G., ‘Modelling nonlinear regenerative effects in metal cutting,’ Philosophical Transactions of the Royal Society of London 359, 2001, 739–757.
Xu, J. and Lu, Q. S., ‘Hopf bifurcation of time-delay Lienard equations,’ International Journal of Bifurcation and Chaos 9, 1999, 939–951.
Das, S. L. and Chatterjee, A., ‘Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations,’ Nonlinear Dynamics 30, 2002, 323–335.
Hu, H. Y., Dowell, E. H., and Virgin, L. N., ‘Resonances of a harmonically forced Duffing oscillator with time delay state feedback,’ Nonlinear Dynamics 15, 1998, 311–327.
Wang, H. L., Hu, H. Y., and Wang, Z. H., ‘Global dynamics of Duffing oscillator with delayed displacement feedback,’ International Journal of Bifurcation and Chaos 14, 2004, 2753–2775.
Hale, J. K. and Kocak, H., Dynamics and Bifurcations, Springer-Verlag, New York, 1991.
Hale, J. K. and Lunel, S. M. V., Introduction to Functional Diffrential Equations, Springer-Verlag, New York, 1993.
Hale, J. K., ‘Averaging methods for differential equations with retarded arguments and a small parameter,’ Journal of Differential Equations 2, 1966, 57–73.
Sato, K. Yamamoto, S., Okimura, T., and Watanabe, S., ‘Dynamic motion of a nonlinear mechanical system with time delay,’ Transactions of JSME, Series C, 61, 1995, 101–106.
Yoshitake, Y., Inoue, J., and Sueoka, A., Vibration of a forced self-excited system with time delay, Transactions of JSME, Series C, 49, 1984, 298–305.
Minorsky, N., Nonlinear Oscillations, D. Van Nostrand Company Inc., Princetion, 1962.
Wang, Z. H., Hu, H. Y., and Wang, H. L., ‘Robust stabilization to nonlinear delayed systems via delayed state feedback: The averaging method,’ Journal of Sound and Vibration 279, 2005, 937–953.
Wang, Z. H. and Hu, H. Y., An energy analysis of nonlinear oscillators with time delays, International Journal of Bifurcation and Chaos 2006, 16(7): accepted for publication
Wang, Z. H. and Hu, H. Y., Pseudo-energy analysis of scalar nonlinear dynamics with time delays near Hopf bifurcation, International Journal of Bifurcation and Chaos, 2005.
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Wang, Z.H., Hu, H.Y. An Energy Analysis of the Local Dynamics of a Delayed Oscillator Near a Hopf Bifurcation. Nonlinear Dyn 46, 149–159 (2006). https://doi.org/10.1007/s11071-006-9020-5
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DOI: https://doi.org/10.1007/s11071-006-9020-5