Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation

Abstract

The saddle-node Hopf bifurcation (SNH) is a generic codimension-two bifurcation of equilibria of vector fields in dimension at least three. It has been identified as an organizing centre in numerous vector field models arising in applications. We consider here the case that there is a global reinjection mechanism, because the centre manifold of the zero eigenvalue returns to a neighbourhood of the equilibrium. Such a SNH bifurcation with global reinjection occurs naturally in applications, most notably in models of semiconductor lasers.We construct a three-dimensional model vector field that allows us to study the possible dynamics near a SNH bifurcation with global reinjection. This model follows on from our earlier results on a planar (averaged) vector field model, and it allows us to find periodic and homoclinic orbits with global excursions out of and back into a neighbourhood of the SNH point. Specifically, we use numerical continuation techniques to find a two-parameter bifurcation diagram for a well-known and complicated case of a SNH bifurcation that involves the break-up of an invariant sphere. As a particular feature we find a concrete example of a phenomenon that was studied theoretically by Rademacher: a curve of homoclinic orbits that accumulates on a segment in parameter space while the homoclinic orbit itself approaches a saddle-periodic orbit.