A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation

A codimension-three unfolding for the Z(2)-symmetric Hopf-pitchfork bifurca tion, in the presence of an additional nonlinear degeneracy, is analysed. Up to ten distinct topological equivalence classes for the unfolding are fo und. A rich variety of dynamical and bifurcation behaviours is pointed out. Beyond the bifurcations present in the nondegenerate case, we show that th e following bifurcations appear locally: Takens-Bogdanov of periodic orbits , degenerate pitchfork of periodic orbits, and global connections involving equilibria and/or periodic orbits. The local results achieved, extended by means of numerical continuation met hods, are used to understand the dynamics of a modified van der Pol-Duffing electronic oscillator, for a certain range of the parameters.