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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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À propos de canards (Apropos canards)
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by Martin Wechselberger PDF
Trans. Amer. Math. Soc. 364 (2012), 3289-3309 Request permission

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
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 3289-3309
  • MSC (2010): Primary 34E15, 34C40, 34D35; Secondary 37C50, 58K45
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05575-9
  • MathSciNet review: 2888246