Saddle-node bifurcation of homoclinic orbits in singular systems

Abstract

We consider the singularly perturbed system $\dot\xi = f_0(\xi) + \varepsilon f_1(\xi,\eta,\varepsilon)$, $\dot\eta = \varepsilon g(\xi,\eta,\varepsilon )$ where $\xi\in\Omega\subset\mathbb R^n$, $\eta\in\mathbb R$ and $\varepsilon\in\mathbb R$ is a small real parameter. We assume that $\dot\xi = f_{0}(\xi)$ has a non degenerate heteroclinic solution $\g(t)$ and that the Melnikov function $\int_{-\infty}^{+\infty} \psi^{*}(t) f_{1}(\g(t),\alpha,0)\dt$ has a double zero at some point $\alpha_{0}$. Using a functional analytic approach we show that if a suitable second order Melnikov function is not zero, the above system has, in a neighborhood of $\{\gamma(t)\}\times\mathbb R$, two heteroclinic orbits for $\varepsilon$ on one side of $\varepsilon=0$ and none for $\varepsilon$ on the other side. We also study the transversality of the intersection of the center-stable and the center-unstable manifolds along these orbits.