Stability switches of a controlled van der Pol-Duffing oscillator

Abstract

The trivial equilibrium of a van der Pol-Duffing oscillator under a nonlinear feedback control may change its stability via either a single or a double Hopf bifurcation if the time delay involved in the feedback reaches certain values. The conditions for switches between stability and instability of the zero solution and the occurrence of Hopf bifurcations are established based on the linear stability analysis. For a double Hopf bifurcation, it is found that the trivial equilibrium may lose its stability via a subcritical or supercritical Hopf bifurcation and regain its stability via a reverse subcritical or supercritical Hopf bifurcation when the time delay is increased. A stable limit cycle appears after a supercritical Hopf bifurcation occurs and disappears through a reverse supercritical Hopf bifurcation.