Homoclinic saddle-node bifurcations in singularly perturbed systems

Abstract

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called “localized structures” in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O(e) perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, Ws(Γ) and Wu(Γ), the stable and unstable manifolds of a slow manifold Γ. Homoclinic orbits to Γ correspond to intersections Ws(Γ)∩Wu(Γ); Ws(Γ)∩Wu(Γ)=∅ for a a*. The bifurcation at a=a* is followed by a sequence of nearby, O(e2(loge)2) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on Γ. The structure of Ws(Γ)∩Wu(Γ) becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.