ON A CODIMENSION-THREE UNFOLDING OF THE INTERACTION OF DEGENERATE HOPF AND PITCHFORK BIFURCATIONS

Abstract

We consider a codimension-three unfolding for the Hopf-zero linear degeneracy with ℤ2-symmetry, in the presence of an additional nonlinear degeneracy. In fact, we assume that the term ρ3 in the radial component of the normal form vanishes. We find up to 12 distinct topological equivalence classes for the unfolding, obtaining a rich variety of dynamical and bifurcation behaviors. In particular, we point out the existence of two codimension-two bifurcations of periodic orbits, and also of Arnold's resonance zones with bananas and banana split shapes. The results obtained are applied to a modified van der Pol-Duffing electronic circuit. Numerical continuation of the bifurcation curves in the vicinity of the degenerate Hopf-pitchfork shows agreement with the analysis.