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 center of the dynamics in many vector fields arising in applications.
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, where the saddle-node bifurcation takes place on an invariant circle.
We derive and study a Z2-symmetric planar vector with an additional 2Pi periodicity as a global model for the planar vector field reduction near a saddle-node Hopf bifurcation with global reinjection. This vector field, whose phase space is a half-cylinder, is as simple as possible and acts as a model vector field for this global bifurcation.
We present two-parameter unfoldings for the different cases of the saddle-node Hopf 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 theoretical unfoldings are confirmed with careful numerical investigations of these global bifurcations. We demonstrate how our results can be applied to the semiconductor laser with optical injection.
Additional information: Later published by Institute of Physics, (2004) Nonlinearity, 17(4), pp. 1119-1151, ISSN 0951-7715
Sponsorship: We are grateful to Cathryn Crooks for providing the original data on bifurcation diagrams presented in figure 17, and to Alan Champneys, Hank Broer and Renato Vitolo for helpful discussions. The research of B.K. was supported by an EPSRC Advanced Fellowship grant and that of B.O. by EPSRC grant GR/R2395/01.