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 $( ilde{x}(t,ep,la), ilde{y}(t,ep,la))$ 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 $( ilde{x}(t,ep,la), ilde{y}(t,ep,la))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.
Bifurcation diagrams for singularly perturbed system / FRANCA, Matteo. - In: ELECTRONIC JOURNAL ON THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS. - ISSN 1417-3875. - ELETTRONICO. - 78:(2012), pp. 1-23. [10.14232/ejqtde.2012.1.78]
Bifurcation diagrams for singularly perturbed system
FRANCA, Matteo
Membro del Collaboration Group
2012
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 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 $( ilde{x}(t,ep,la), ilde{y}(t,ep,la))$ 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 $( ilde{x}(t,ep,la), ilde{y}(t,ep,la))$ is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.File | Dimensione | Formato | |
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