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Title: Birth of canard cycles
Authors: DUMORTIER, Freddy 
Roussarie, Robert
Issue Date: 2009
Source: Discrete and Continuous Dynamical Systems. Series S, 4. p. 723-781
Series/Report no.: 2
Abstract: In this paper we consider singular perturbation problems occuring in planar slow-fast systems ((x) over dot = y - F(x, lambda), (y) over dot = -epsilon G(x, lambda)) where F and are smooth or even real analytic for some results, A is a multiparameter and epsilon is a small parameter. We deal with turning points that are limiting situations of (generalized) Hopf bifurcations and that we call slow-fast Hopf points. We investigate the number of limit cycles that can appear near a slow-fast Hopf point and this under very general conditions. One of the results states that for any analytic family of planar systems, depending on a finite number of parameters, there is a finite upperbound for the number of limit cycles that can bifurcate from a slow-fast Hopf point.The most difficult problem to deal with concerns the uniform treatment of the evolution that a limit cycle undergoes when it grows from a small limit cycle near the singular point to a canard cycle of detectable size. This explains the title of the paper. The treatment, is based on blow-up, good normal forms and appropriate Chebyshev systems. In the paper we also relate the slow-divergence integral as it is used in singular perturbation theory to Abelian integrals that have to be used in studying limit cycles close to the singular point.
Keywords: Slow-fast system;singular perturbation;turning point;Hopf bifurcation;canard cycle;Lienard equation
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ISSN: 1937-1632
e-ISSN: 1937-1179
DOI: 10.3934/dcdss.2009.2.723
ISI #: 000498207000002
Category: A1
Type: Journal Contribution
Validations: ecoom 2021
Appears in Collections:Research publications

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