On a codimension-three unfolding of the interaction of degenerate Hopf andpitchfork bifurcations

We consider a codimension-three unfolding for the Hopf-zero linear degenera cy with Z(2)-symmetry, in the presence of an additional nonlinear degenerac y. In fact, we assume that the term rho(3) in the radial component of the n ormal form vanishes. We find up to 12 distinct topological equivalence classes for the unfolding , obtaining a rich variety of dynamical and bifurcation behaviors. In parti cular, we point out the existence of two codimension-two bifurcations of pe riodic 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 electron ic circuit. Numerical continuation of the bifurcation curves in the vicinit y of the degenerate Hopf-pitchfork shows agreement with the analysis.