Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback

In this paper, we study a classical van der Pol’s equation with delayed feedback. Triple-zero bifurcation is investigated by using center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the triple-zero bifurcation and show that the model can exhibit transcritical bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. Some numerical simulations are given to support the analytic results.

► We discuss triple-zero singularity of codimension three in van der Pol’s equation with delayed feedback. ► We give the versal unfolding of the norm forms at the triple-zero bifurcation point. ► We numerically find the complicated dynamics near the triple-zero point.