Abstract
The paper presents a study on the validity of perturbation methods, suchas the method of multiple scales, the Lindstedt–Poincaré method and soon, in seeking for the periodic motions of the delayed dynamic systemsthrough an example of a Duffing oscillator with delayed velocityfeedback. An important observation in the paper is that the method ofmultiple scales, which has been widely used in nonlinear dynamics, worksonly for the approximate solutions of the first two orders, and givesrise to a paradox for the third-order approximate solutions of delaydifferential equations. The same problem appears when theLindstedt–Poincaré method is implemented to find the third-orderapproximation of periodic solutions for delay differential equations,though it is effective in seeking for any order approximation ofperiodic solutions for nonlinear ordinary differential equations. Apossible explanation to the paradox is given by the results obtained byusing the method of harmonic balance. The paper also indicates thatthese perturbation methods, despite of some shortcomings, are stilleffective in analyzing the dynamics of a delayed dynamic system sincethe approximate solutions of the first two orders already enable one togain an insight into the primary dynamics of the system.
Similar content being viewed by others
References
Hu, H. Y. and Wang, Z. H., Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer, Berlin, 2002.
Hu, H. Y., ‘Abundant dynamic features of a nonlinear system under delayed feedback control’, in Proceedings of the 6th Asia-Pacific Vibration Conference, Vol. 1, Jilin Science and Technology Press, Changchun, 2001, pp. 11–15.
Pyragas, K., ‘Continuous control of chaos by self-controlling feedback’, Physical Review Letters 78(2), 1992, 421–428.
Yamamoto, S., Hino, T., and Ushio. T., ‘Delayed feedback control with a minimal-order observer for stabilization of chaotic discrete-time systems’, International Journal of Bifurcation and Chaos 12(5), 2002, 1047–1055.
MacDonald, N., Biological Delay Systems: Linear Stability Theory, Cambridge University Press, New York, 1989.
Kuang, Y., Delay Differential Equations with Applications to Population Dynamics, Academic Press, New York, 1993.
Beuter, A., Belair, J., and Labrie, C., ‘Feedback and delays in neurological diseases: A modelling study using dynamical systems’, Bulletin of Mathematical Biology 55, 1993, 525–541.
Stépán, G., ‘Modeling nonlinear regenerative effects in metal cutting’, Philosophical Transactions of the Royal Society of London A359, 2000, 739–757.
Perioux, D., Erneux, T., Gavrielides, A., and Kovanis, V., ‘Hopf bifurcation subject to a large delay in a laser system’, SIAM Journal of Applied Mathematics 61(3), 2000, 966–982.
Shayer, L. and Campbell, S. A., ‘Stability, bifurcation, and multi-stability in a system of two coupled neurones with multiple time delays’, SIAM Journal of Applied Mathematics 61, 2000, 673–700.
Plaut, R. H. and Hsieh, J. C., ‘Nonlinear structural vibrations involving a time delay in damping’, Journal of Sound and Vibration 117, 1987, 497–510.
Moiola, J. L. and Chen, G. R., ‘Hopf bifurcations in time-delayed nonlinear feedback control systems’, in Proceedings of the 34th Conference on Decision and Control, New Orleans, LA, December 13–15, 1995, pp. 942–948.
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(4), 1998, 311–327.
Maccari, A., ‘The response of a parametrically excited van der Pol oscillator to a time delay state feedback’, Nonlinear Dynamics 26(3), 2001, 105–119.
Hale, J. K., Theory of Functional Differential Equations, Springer, New York, 1977.
Diekmann, O., van Gils, S. A., Lunel, S. M. V., and Walther, H. O., Delay Equations, Functional, Complex, and Nonlinear Analysis, Springer, New York, 1995.
Hale, J. K., Magalhães, L. T., and Oliva, W. M., Dynamics in Infinite Dimensions, 2nd edition, Springer, Berlin, 2002.
Wang, H. L., Wang, Z. H., and Hu, H. Y., ‘Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback’, Acta Mechanica Sinica, 2002, submitted.
Morris, H. C., ‘A perturbative approach to periodic solutions of delay differential equation’, Journal of the Institute of Mathematics and Its Application 18, 1976, 15–24.
Casal, A. and Freedman, M., ‘A Poincaré–Lindstedt approach to bifurcation problems for differential-delay equations’, IEEE Transactions on Automatic Control 25(5), 1980, 967–973.
Iooss, G. and Joseph, D. D., Elementary Stability and Bifurcation Theory, 2nd edition, Springer, New York, 1989.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Zhang, J. Y., The Geometric Theory and Bifurcation Problem in Ordinary Differential Equations, 2nd edition, High Education Press, Beijing, 1987.
Wang, H. L. and Hu, H. Y., ‘Bifurcation analysis of a delayed dynamic system via method of multiple scales and shooting technique’, International Journal of Bifurcation and Chaos, 2003, submitted.
MacDonald, N., ‘Harmonic balance in delay-differential equations’, Journal of Sound and Vibration 186(4), 1995, 649–656.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wang, H., Hu, H. Remarks on the Perturbation Methods in Solving the Second-Order Delay Differential Equations. Nonlinear Dynamics 33, 379–398 (2003). https://doi.org/10.1023/B:NODY.0000009957.42817.4f
Issue Date:
DOI: https://doi.org/10.1023/B:NODY.0000009957.42817.4f