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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

Canard cycles in generic fast-slow systems on the torus


Author: I. V. Shchurov
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 71 (2010).
Journal: Trans. Moscow Math. Soc. 2010, 175-207
MSC (2010): Primary 34E17; Secondary 34E15, 37G15, 70K70
Posted: December 21, 2010
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Abstract: In generic fast-slow systems with a single parameter on the two-dimensional torus, for arbitrarily small values of this parameter there exist attracting canard cycles. This is a key distinction between the dynamics on the torus and the dynamics of similar systems on the plane. This has already been proved for systems with a convex slow curve. This paper looks at systems with a nonconvex slow curve. Upper and lower estimates for the number of canard cycles are obtained. An open set of systems having a preassigned number of attracting canard cycles is constructed.


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Additional Information

I. V. Shchurov
Affiliation: Moscow State University
Email: ilya@schurov.com

DOI: http://dx.doi.org/10.1090/S0077-1554-2010-00184-7
PII: S 0077-1554(2010)00184-7
Keywords: Fast-slow system, attracting canard cycle, slow curve, dynamical system
Posted: December 21, 2010
Additional Notes: This research was partially supported by the Russian Foundation for Basic Research (grant no. 10-01-00739-a).
Article copyright: © Copyright 2010 American Mathematical Society