Canard solutions at non-generic turning points
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- by Peter De Maesschalck and Freddy Dumortier PDF
- Trans. Amer. Math. Soc. 358 (2006), 2291-2334 Request permission
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”.References
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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
- Received by editor(s): May 19, 2003
- Received by editor(s) in revised form: August 30, 2004
- Published electronically: December 20, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 2291-2334
- MSC (2000): Primary 34E15, 34E20, 34C26, 34A12; Secondary 34D15, 37G15, 34B99
- DOI: https://doi.org/10.1090/S0002-9947-05-03839-0
- MathSciNet review: 2197445