Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback

摘要

In this paper, we consider the traditional Van der Pol Oscillator with aforcing dependent on a delay in feedback. The delay is taken to be a nonlinearfunction of both position and velocity which gives rise to many different typesof bifurcations. In particular, we study the Zero-Hopf bifurcation that takesplace at certain parameter values using methods of centre manifold reduction ofDDEs and normal form theory. We present numerical simulations that have beenaccurately predicted by the phase portraits in the Zero-Hopf bifurcation toconfirm our numerical results and provide a physical understanding of theoscillator with the delay in feedback.