Detectable canard cycles with singular slow dynamics of any order at the turning point

P. De Maesschalck; Freddy Dumortier

doi:10.3934/dcds.2011.29.109

Article Contents

2011, Volume 29, Issue 1: 109-140. Doi: 10.3934/dcds.2011.29.109

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Detectable canard cycles with singular slow dynamics of any order at the turning point

P. De Maesschalck^1, and
Freddy Dumortier^2,

1.
Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek
2.
Hasselt University, Campus Diepenbeek, Agoralaan gebouw D, B-3590 Diepenbeek

Received: December 31, 2009

Revised: March 31, 2010

Published: December 2010

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Abstract

This paper deals with the study of limit cycles that appear in a class of planar slow-fast systems, near a "canard'' limit periodic set of FSTS-type. Limit periodic sets of FSTS-type are closed orbits, composed of a Fast branch, an attracting Slow branch, a Turning point, and a repelling Slow branch. Techniques to bound the number of limit cycles near a FSTS-l.p.s. are based on the study of the so-called slow divergence integral, calculated along the slow branches. In this paper, we extend the technique to the case where the slow dynamics has singularities of any (finite) order that accumulate to the turning point, and in which case the slow divergence integral becomes unbounded. Bounds on the number of limit cycles near the FSTS-l.p.s. are derived by examining appropriate derivatives of the slow divergence integral.

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Mathematics Subject Classification: Primary: 34E15, 34C26; Secondary: 34E20, 37G15.

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