À propos de canards (Apropos canards)
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- 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 $k\ge 2$.References
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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