A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol-Duffing oscillator

摘要

We consider a modified van der Pol-Duffing electronic circuit, focusing on the case where a Hopf-pitchfork bifurcation takes places. The analysis of this bifurcation is a simple way to detect and characterize purely three-dimensional behaviour (an oscillatory regime in three variables, quasiperiodic motion, etc.). The normal form analysis provides the classification of different kinds of Hopf-pitchfork bifurcation, organized according to some degenerate cases. One of these degenerate cases is analyzed, by considering a codimension-three unfolding of a reflectionally symmetric planar vector field. Later, the implications for three-dimensional flows are presented. Unlike another degenerate Hopf-pitchfork bifurcations exhibited by the system, the one studied here does not involve quasiperiodic behaviour, so that the complexity related to quasiperiodic motion is not present. [References: 20]