Zero-Hopf bifurcation for van der Pol’s oscillator with delayed feedback

In this paper, we study the dynamical behaviors of the following van der Pol oscillator with delay ẍ+ε(x2−1)ẋ+x=εg(x(t−τ)). In the case that its associated characteristic equation has a simple zero root and a pair of purely imaginary roots (zero-Hopf singularity), the normal form is obtained by performing a center manifold reduction and by using the normal form theory developed by Faria and Magalhães. A critical value ε0ε0 of εε in (0,2) is obtained to predict the bifurcation diagrams from which saddle–node bifurcation, pitchfork bifurcation, Hopf bifurcation (the existence and stability of the periodic solutions), and heteroclinic bifurcation are determined. Some examples are given to confirm the theoretical results.