Abstract
The van der Pol equation with a distributed time delay is analyzed. Itslinear stability is investigated by employing the Routh–Hurwitzcriteria. Moreover, the local asymptotic stability conditions are alsoderived. By using the mean time delay as a bifurcation parameter, themodel is found to undergo a sequence of Hopf bifurcations. The directionand the stability criteria of the bifurcating periodic solutions areobtained by the normal form theory and the center manifold theorem. Somenumerical simulation examples for justifying the theoretical analysisare also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Guckenheimer J. and Holmes P. J., Nonlinear Oscillations, Dynamical System, and Bifurcations of Vector Fields, Springer-Verlag, Berlin, 1983.
Schuster, H. G., Deterministic Chaos, Physik-Verlag, Weinheim, 1984.
Flaherty, J. E. and Hoppensteadt, F. C., Studies in Applied Mathematics, Vol. 5, Blackwell, Cambridge, MA, 1978, p. 58.
Venkatasubramanian, K. and Vaithianathan, P., ‘Singularity induced bifurcation and the van der Pol oscillator’, IEEE Transactions on Circuits and Systems 41, 1994, 765–769.
Frey, M. and Douglas, R., ‘A class of relaxation algorithms for finding the periodic steady-state solution’, IEEE Transactions on Circuits and Systems 45, 1998, 659–663.
Buonomo, A., ‘The periodic solution of van der Pol's equation’, SIAM Journal on Applied Mathematics 59, 1998, 156–171.
Murakaimi, K., ‘Bifurcated periodic solutions for delayed van der Pol equation’, Neural, Parallel & Scientific Computations 7, 1999, 1–16.
Hopfield, J. J., ‘Neurons with graded response have collective computational properties like those of twostate neurons’, Proceedings of National Academic Sciences, USA 81, 1984, 3088–3092.
Marcus, C.M. and Westervelt, R. M., ‘Stability of analog neural network with delay’, Physical Review A 39, 1989, 347–359.
Belair, J. and Dufour, S., ‘Stability in a three-dimensional system of delay-differential equation’, Canadian Applied Mathematics Quarterly 4, 1996, 135–156.
Belair, J., Campbell, S., and van den Driessche, P., ‘Frustration, stability and delay-induced oscillations in a neural network model’, SIAM Journal Applied Mathematics 56, 1996, 245–255.
De Vries, B. and Principle, J. C., ‘The gamma model - A new neural model for temporal processing’, Neural Networks 5, 1992, 565–576.
Gopalsamy, K. and Leung, I., ‘Convergence under dynamical thresholds with delays’, IEEE Transactions on Neural Networks 8, 1997, 341–348.
Liao, X., Wu, Z., and Yu, J., ‘Stability switches and bifurcation analysis of a neural network with continuously delay’, IEEE Transactions on Sytems, Man and Cybernetics - Part A 29, 1999, 692–696.
Stepan, G., ‘Great delay in a predator-prey model’, Nonlinear Analysis 10, 1986, 913–929.
Stepan, G., Retarded Dynamical System: Stability and Characteristic Functions, Pitman Research Notes in Mathematics Series, Vol. 210, Academic Press, New York, 1989.
Marsden, J. E. and McCracken, M., The Hopf Bifurcation and Its Applications, Applied Mathematics Sciences, Vol. 19, Springer-Verlag, Heidelberg, 1976.
Kung, Y., Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1992.
MacDonald, N., Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge, 1989.
Hassard, B. D., Kazarinoff, N. D., and Wan, Y. H., Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
Hale, J. and Verduyn Lunel, S., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Boston, MA, 1992.
Iooss, G. and Joiseph, D. D., Elementary Stability and Bifurcation Theory, 2nd edition, Springer-Verlag, New York, 1989.
Wu, J., Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liao, X., Wong, Kw. & Wu, Z. Hopf Bifurcation and Stability of Periodic Solutions for van der Pol Equation with Distributed Delay. Nonlinear Dynamics 26, 23–44 (2001). https://doi.org/10.1023/A:1012927603832
Issue Date:
DOI: https://doi.org/10.1023/A:1012927603832