Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback
This paper deals with a van der Pol oscillator with delayed state feedback \[ x''(t)-\mu \omega_{0}(1-\delta x(t)^2)x'(t)+\omega_{0}^2 x(t)=u\omega_{0}^2x(t-\tau)+v\omega_{0}x'(t-\tau), \tag{1} \] where \(\mu, \omega_{0}\) and \(\delta\) are positive real constants. The study is confined to the case of small damping. The authors first consider the stability switches of the trivial equilibrium. Then center manifold reduction and the normal form method are used to study the Hopf bifurcation. To study the global diagrams of the bifurcated periodic motions as the delay varies to a large scale, they use the method of multiple scales to obtain the global view of Hopf bifurcations with respect to time delay. Both the analysis results of Hopf bifurcations and global view of those bifurcations are validated via a collocation scheme implemented on the DDE-Biftool.
scientific article; zbMATH DE number 5770135