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Canard solutions at non-generic turning points

Author(s): Peter De Maesschalck; Freddy Dumortier
Journal: Trans. Amer. Math. Soc. 358 (2006), 2291-2334.
MSC (2000): Primary 34E15, 34E20, 34C26, 34A12; Secondary 34D15, 37G15, 34B99
Posted: December 20, 2005
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Abstract: This paper deals with singular perturbation problems for vector fields on $ 2$-dimensional manifolds. ``Canard solutions'' are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a ``turning point'' and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the ``control curve''.

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Additional Information:

Peter De Maesschalck
Affiliation: Departement Wiskunde, Natuurkunde, Informatica, Dynamical Systems, Hasselt University, Agoralaan, Gebouw D, B-3590 Diepenbeek, Belgium
Email: peter.demaesschalck@uhasselt.be

Freddy Dumortier
Affiliation: Departement Wiskunde, Natuurkunde, Informatica, Dynamical Systems, Hasselt University, Agoralaan, Gebouw D, B-3590 Diepenbeek, Belgium
Email: freddy.dumortier@uhasselt.be

DOI: 10.1090/S0002-9947-05-03839-0
PII: S 0002-9947(05)03839-0
Keywords: Singular perturbations, canard solutions, degenerate turning point, center manifolds, normal forms, blow up of families
Received by editor(s): May 19, 2003
Received by editor(s) in revised form: August 30, 2004
Posted: December 20, 2005
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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