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

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

Blog Article

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 GREY CLAY 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 $(reve{x}(t,arepsilon,lambda),reve{y}(t,arepsilon,lambda))$ 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 Feminine Medicated Products for the tangents of these curves.The results are illustrated by some examples.