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 equilibri a give rise to several dynamical behaviours, including periodic, homoc linic and heteroclinic orbits. The local analysis provides, in first a pproximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a gl obal understanding of the bifurcation sets. The study of the normal fo rm of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four H opf bifurcation is also pointed out. In the case of the Takens-Bogdano v 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.