A Unified Approach to Piecewise Linear Hopf and Hopf-Pitchfork Bifurcations

We propose for symmetric three-dimensional piecewise linear systems with three zones a unified approach to analyze both Hopf and Hopf-pitchfork bifurcations. For the equilibrium at the origin, the crossing of a complex eigenvalue pair through the imaginary axis of complex plane, with the possible simultaneous crossing of a real eigenvalue, is considered. Some results related to the bifurcation of limit cycles are provided, and an illustrative example is included.