Unfolding a Codimension-Two, Discontinuous, Andronov-Hopf Bifurcation

TL;DR

This paper analyzes a complex bifurcation scenario involving simultaneous discontinuous and Hopf bifurcations in a planar system, deriving scaling laws and exploring the bifurcation structure.

Contribution

It provides a detailed unfolding of a codimension-two bifurcation combining discontinuous and Hopf bifurcations in piecewise-smooth systems.

Findings

Hopf cycle undergoes a grazing bifurcation

Grazing bifurcation may be followed by a saddle-node bifurcation

Scaling laws for bifurcation curves are derived

Abstract

We present an unfolding of the codimension-two scenario of the simultaneous occurrence of a discontinuous bifurcation and an Andronov-Hopf bifurcation in a piecewise-smooth, continuous system of autonomous ordinary differential equations in the plane. We find the Hopf cycle undergoes a grazing bifurcation that may be very shortly followed by a saddle-node bifurcation of the orbit. We derive scaling laws for the bifurcation curves that emanate from the codimension-two bifurcation.