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- W2077367792 abstract "In this paper we consider singular perturbation problems occuring in planar slow-fast systems $(dot x=y-F(x,lambda),dot y=-varepsilon G(x,lambda))$ where $F$ and $G$ are smooth or even real analytic for some results, $lambda$ is a multiparameter and $varepsilon$ 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.<br> 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." @default.
- W2077367792 created "2016-06-24" @default.
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- W2077367792 date "2009-01-01" @default.
- W2077367792 modified "2023-10-16" @default.
- W2077367792 title "Birth of canard cycles" @default.
- W2077367792 doi "https://doi.org/10.3934/dcdss.2009.2.723" @default.
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