Hopf bifurcation in delayed van der Pol oscillators

Abstract

In this paper, we consider a classical van der Pol equation with a general delayed feedback. Firstly, by analyzing the associated characteristic equation, we derive a set of parameter values where the Hopf bifurcation occurs. Secondly, in the case of the standard Hopf bifurcation, the stability of bifurcating periodic solutions and bifurcation direction are determined by applying the normal form theorem and the center manifold theorem. Finally, a generalized Hopf bifurcation corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance) is analyzed by using a normal form approach.