Saddle-node bifurcation of homoclinic orbits in singular systems (Q1864090)

The author studies the following singularly perturbed system \[ \dot \xi=f_0(\xi)+\varepsilon f_1(\xi ,\eta ,\varepsilon),\quad \dot \eta =\varepsilon g(\xi ,\eta ,\varepsilon) ,\tag{1} \] where \(\xi \in \Omega \subset \mathbb{R}^n\), \(\eta \in \mathbb{R}\) and \(\varepsilon \in \mathbb{R}\) is a small parameter. It is assumed that the boundary layer problem associated to (1) of the form \(\dot \xi =f_0(\xi)\) has a nondegenerate heteroclinic solution \(\gamma (t)\) and the associated Melnikov function has a double zero. By using a functional analytic approach, it is shown that if a suitable second-order Melnikov function is not zero, then (1) has two heteroclinic orbits near \(\{\gamma (t)\}\times \mathbb{R}\) for \(\varepsilon \) on one side of \(\varepsilon =0\) and none on the other side. The transversality of those heteroclinic solutions is also investigated. The paper is finished with an example of a 3-dimensional differential system.