Analysis of Hopf and Takens-Bogdanov bifurcations in a modified van der Pol-Duffing oscillator

We analyze a modified van der Pol-Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z(2)-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens-Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf-Shil'nikov singularity is also shown.