`a`
Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay

Pages: 503 - 516, Volume 13, Issue 2, March 2010      doi:10.3934/dcdsb.2010.13.503

 
       Abstract        Full Text (240.5K)       Related Articles

Xiaoqin P. Wu - Department of Mathematics, Computer & Information Sciences, Mississippi Valley State University, Itta Bena, MS 38941, United States (email)
Liancheng Wang - Department of Mathematics & Statistics, Kennesaw State University, Kennesaw, GA 30144, United States (email)

Abstract: In this paper, we study the following system of two coupled relaxation oscillators of the van der Pol type with delay

ε$\ddot{x}_1-(1-x_1^2)\dot{x}_1+x_1=h_1(x_2(t-\tau)-x_1(t-\tau)),$
ε$\ddot{x}_2-(1-x_2^2)\dot{x}_2+x_2=h_2(x_1(t-\tau)-x_2(t-\tau)),$

where $h_1$ and $h_2$ are nonlinear functions. It is shown that this system can exhibit Hopf bifurcation as the time delay $\tau$ passes certain critical values. The distribution of the eigenvalues of the linearized system is studied thoroughly in terms of the parameter $\ep$ and the linear parts of functions $h_1$ and $h_2$. The normal form theory for general retarded functional equations developed by Faria and Magalhães is applied to perform center manifold reduction and hence to obtain the explicit normal form Hopf bifurcation which can be used to determine the stability of the bifurcating periodic solutions and and the direction of Hopf bifurcation. Examples are given to confirm the theoretical results.

Keywords:  van der Pol oscillators, Hopf bifurcation, normal form, stability.
Mathematics Subject Classification:  Primary: 34K18.

Received: February 2009;      Revised: October 2009;      Available Online: December 2009.