Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 759363, 16 pages
doi:10.1155/2009/759363
Research Article

Double Hopf Bifurcation with Strong Resonances in Delayed Systems with Nonlinerities

J. Xu1 and K. W. Chung2

1School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
2Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received 25 February 2009; Revised 6 September 2009; Accepted 11 October 2009

Academic Editor: Oleg Gendelman

Copyright © 2009 J. Xu and K. W. Chung. 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

An efficient method is proposed to study delay-induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension-three and codimension-two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.