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

Abstract

A codimension-three unfolding for the 2-symmetric Hopf-pitchfork bifurcation, in the presence of an additional nonlinear degeneracy, is analysed. Up to ten distinct topological equivalence classes for the unfolding are found. A rich variety of dynamical and bifurcation behaviours is pointed out. Beyond the bifurcations present in the nondegenerate case, we show that the 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 methods, are used to understand the dynamics of a modified van der Pol - Duffing electronic oscillator, for a certain range of the parameters.