Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 569141, 16 pages
http://dx.doi.org/10.1155/2011/569141
Research Article

Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays

Xiaobing Zhou,1 Murong Jiang,1 and Xiaomei Cai2

1School of Information Science and Engineering, Yunnan University, Kunming 650091, China
2Bureau of Asset Management, Yunnan University, Kunming 650091, China

Received 5 December 2010; Revised 7 February 2011; Accepted 2 March 2011

Academic Editor: Leonid Shaikhet

Copyright © 2011 Xiaobing Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis.