Bifurcation diagrams for singularly perturbed system.

We consider a singularly perturbed system where the fast dynamic of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system is $1$-dimensional and it admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. In this setting Battelli and Palmer proved the existence of a unique trajectory $(\tilde{x}(t,\ep,\la),\tilde{y}(t,\ep,\la))$ homoclinic to the slow manifold. The purpose of this paper is to construct curves which divide the $2$-dimensional parameters space in different areas where $(\tilde{x}(t,\ep,\la),\tilde{y}(t,\ep,\la))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.