The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities

Abstract

An analysis is presented of the homoclinic bifurcations occurring in a generic unfolding of a saddle-node/Hopf singularity (also known as a Gavrilov–Guckenheimer point). Specifically, an explanation is given of previously numerically observed oscillations of loci of homoclinic orbits to two different saddle focus equilibria. These oscillations occur within an exponentially thin wedge of parameter space that emerges from the codimension-two point. The frequency of oscillation tends to zero as the codimension-two point is approached. Earlier theory by Gaspard showed that homoclinic orbits must exist inside the parameter wedge. This result is here extended to give the frequency and amplitude of the oscillations of the homoclinic loci within the wedge. It is also shown how the two loci are related to each other, and, in the case of only cubic perturbations of the normal form, that they are precisely out-of-phase. The analysis is shown to agree with numerical results on perturbed normal forms and in two model systems arising in applications to atmospheric dynamics and to calcium wave propagation.