Birth of canard cycles

Abstract

In this paper we consider singular perturbation problems occuring in planar slow-fast systems (x = y-F(x, λ), y =-G(x, λ)) where F and G are smooth or even real analytic for some results, λ is a multiparameter and " 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 slowdivergence 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.