Stability and bifurcation analysis in Van der Pol's oscillator with delayed feedback

The classical Van der Pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered. Linear stability of the equations is investigated by analyzing the associated characteristic equations. It is found that there exist the stability switches when delay varies, and the Hopf bifurcation occurs when the delay passes through a sequence of critical values. The bifurcation diagram is drawn in (ε,k)-plane, and the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem.