Homoclinic saddle-node bifurcations in singularly perturbed systems

Abstract

In this paper we study the creation of homoclinic orbits by saddlenode bifurcations Inspired on similar phenomena appearing in the analysis of socalled localized structures in modulation or amplitude equations we consider a family of nearly integrable singularly perturbed three dimensional vector elds with two bifurcation parameters a and b The O perturbation destroys a manifold consisting of a family of integrable homoclinic orbits it breaks open into two manifolds Ws and Wu the stable and unstable manifolds of a slow manifold Homoclinic orbits to correspond to intersections Ws Wu Ws Wu for a a a pair of pulse homoclinic orbits emerges as rst intersection of Ws and Wu as a a The bifurcation at a a is followed by a sequence of nearby Olog homoclinic saddlenode bifurcations at which pairs of Npulse homoclinic orbits are created these orbits make N circuits through the fast eld The second parameter b distinguishes between two signicantly dierent cases in the cooperating respectively counteracting case the averaged eect of the fast eld is in the same respectively opposite direction as the slow ow on The structure of Ws Wu becomes highly complicated in the counteracting case we show the existence of many new types of sometimes exponentially close homoclinic saddlenode bifurcations