Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback

This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback. Based on a detailed analysis of the stability switches of the trivial equilibrium of the system, the stability charts are given in a parameter space consisting of the time delay and the feedback gains. The center manifold reduction and the normal form method are used to study Hopf bifurcations with respect to the time delay. To gain an insight into the persistence of a Hopf bifurcation as the time delay varies farther away from its critical value, the method of multiple scales is used to obtain the global view of Hopf bifurcations with respect to the time delay. Both the analytical results of Hopf bifurcations and global view of those bifurcations are validated via a collocation scheme implemented on DDE-Biftool. The most important discovery in this paper is the well-structured global view of Hopf bifurcations for the system of concern, showing the generality of the persistence of Hopf bifurcations.