Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:368

Article

Wang, Hongbin¹  Jiang, Weihua¹ 

[1] Harbin Inst Technol, Dept Math, Harbin 150001, Peoples R China

关键词: Van der Pol's equation;    Hopf-pitchfork bifurcation;    Normal form;    Quasi-periodic motion;    Delayed feedback;   

DOI  :  10.1016/j.jmaa.2010.03.012

来源: Elsevier

【 摘 要 】

First, we identify the critical values for Hopf-pitchfork bifurcation. Second, we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point, by the normal form method and center manifold theory. Then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free