Homoclinic saddle-node bifurcations in singularly perturbed systems

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(ε) 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 W s(Γ) ∩ Wu(Γ); Ws(Γ) ∩ Wu(Γ) = Ø for a <*, a pair of 1-pulse homoclinic orbits emerges as first intersection of Ws(Γ) and Wu(Γ) and as a > a*. The bifurcation at a = a* is followed by a sequence of nearby, O(ε2(log ε) 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. © 2000 Plenum Publishing Corporation.