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Birth of canard cycles. (English) Zbl 1186.34080

Summary: 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.
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.

MSC:

34E17 Canard solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C26 Relaxation oscillations for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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