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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

À propos de canards (Apropos canards)


Author: Martin Wechselberger
Journal: Trans. Amer. Math. Soc. 364 (2012), 3289-3309
MSC (2010): Primary 34E15, 34C40, 34D35; Secondary 37C50, 58K45
Published electronically: January 20, 2012
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Abstract: We extend canard theory of singularly perturbed systems to the general case of $ k$ slow and $ m$ fast dimensions, with $ k\ge 2$ and $ m\ge 1$ arbitrary. A folded critical manifold of a singularly perturbed system, a generic requirement for canards to exist, implies that there exists a local $ (k+1)$-dimensional center manifold spanned by the $ k$ slow variables and the critical eigendirection of the fast variables. If one further assumes that the $ m-1$ nonzero eigenvalues of the $ m\times m$ Jacobian matrix of the fast equation have all negative real part, then the $ (k+m)$-dimensional singularly perturbed problem is locally governed by the flow on the $ (k+1)$-dimensional center manifold. By using the blow-up technique (a desingularization procedure for folded singularities) we then show that the local flow near a folded singularity of a $ k$-dimensional folded critical manifold is, to leading order, governed by a three-dimensional canonical system for any $ k\ge 2$. Consequently, results on generic canards from the well-known case $ k=2$ can be extended to the general case $ k\ge 2$.


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Additional Information

Martin Wechselberger
Affiliation: School of Mathematics & Statistics, University of Sydney, Sydney NSW 2006, Australia
Email: wm@maths.usyd.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05575-9
PII: S 0002-9947(2012)05575-9
Keywords: Singularly perturbed systems, differential-algebraic equations, folded singularities, canards
Received by editor(s): May 20, 2010
Received by editor(s) in revised form: January 18, 2011
Published electronically: January 20, 2012
Additional Notes: This work was supported by Marsden Fund in NZ. The author would like to thank Vivien Kirk for carefully reading the manuscript.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.