We consider Power Core Combiners 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 Yoga Leggings - AOP paper where the slow-time system is $1$-dimensional.
We prove the existence of a unique trajectory $(reve{x}(t, arepsilon,lambda),reve{y}(t, arepsilon,lambda))$ 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 $(reve{x}(t, arepsilon,lambda),reve{y}(t, arepsilon,lambda))$ is either homoclinic, heteroclinic, or unbounded.We derive explicit formulas for the tangents of these curves.
The results are illustrated by some examples.