Bifurcation analysis in van der Polʹs oscillator with delayed feedback

In this paper, a classical van der Polʹs equation with generally delayed feedback is considered. It is shown that there are Bogdanov–Takens bifurcation, triple zero and Hopf-zero singularities by analyzing the distribution of the roots of the associated characteristic equation. In the situation that the zero is as a simple eigenvalue, the normal forms of the reduced equations are obtained by the center manifold theory and normal form method for functional differential equation, and hence the stability of the fixed point is determined, and transcritical and pitchfork bifurcations are found.