A planar model system for the saddle–node Hopf bifurcation with global reinjection

Abstract

A simultaneous saddle–node and Hopf bifurcation is a basic codimension-two local bifurcation of vector fields with a phase space of dimension at least three. Its local unfoldings are now well known and it has been found as an organizing centre of the dynamics in many vector fields arising in applications. Here we study this very bifurcation but in the presence of a reinjection mechanism that causes trajectories to return to the relevant local neighbourhood in phase space. This happens in applications, for example, in a semiconductor laser subject to optical injection.We propose and study a -symmetric planar vector field with an additional 2π periodicity as a global model for the planar vector field reduction near a saddle–node Hopf (SNH) bifurcation with global reinjection. The phase space of this model vector field is a half-cylinder and the SNH bifurcation reduces to a saddle–node pitchfork (SNP) bifurcation with global reinjection.Two-parameter unfoldings are presented for the different cases of the SNP bifurcation in the presence of global reinjection. New phenomena that we find are periodic, homoclinic and families of heteroclinic orbits that wind around the cylinder. The topological unfoldings are developed hand-in-hand with careful numerical investigations of these global bifurcations. We demonstrate how our results can be applied to a planar model of a semiconductor laser with optical injection. This is followed by a discussion of how to interpret the presented unfoldings in terms of the full three-dimensional dynamics near a SNH bifurcation with global reinjection.