Homoclinic saddle-node bifurcations in singularly perturbed systems

摘要

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 budThe O perturbation destroys a manifold consisting of a family of integrable homoclinic orbits it breaks open into two manifolds Wsud and Wuud the stable and unstable manifolds of a slow manifold ud Homoclinic orbits to ud correspond to intersections Wsud Wuud Wsud Wuud for a a a pair of udpulse homoclinic orbits emerges as rst intersection of Wsud and Wuud 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 eldudThe 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 ud The structure of Wsud Wuud becomes highly complicated in the counteracting case we show the existence of many new types of sometimes exponentially close homoclinic saddlenode bifurcations