Bifurcation diagrams for singularly perturbed system

Abstract

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 to 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 $(\tilde{x}(t,\varepsilon,\lambda),\tilde{y}(t,\varepsilon,\lambda))$ 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 $(\tilde{x}(t,\varepsilon,\lambda),\tilde{y}(t,\varepsilon,\lambda))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.

Keywords

• singular perturbation
• homoclinic trajectory
• transcritical bifurcation
• saddle-node bifurcation