Bifurcation diagrams for singularly perturbed system: the multi-dimensional case

We consider a singularly perturbed system where the fast dynamics of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. We generalize to the multi-dimensional case the results obtained in a previous paper where the slow-time system is $1$-dimensional. We prove the existence of a unique trajectory $(reve{x}(t,ep,la),reve{y}(t,ep,la))$ homoclinic to a centre manifold of the slow manifold. Then we construct curves in the $2$-dimensional parameters space, dividing it in different areas where $(reve{x}(t,ep,la),reve{y}(t,ep,la))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.