Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Birth of canard cycles

Pages: 723 - 781, Volume 2, Issue 4, December 2009      doi:10.3934/dcdss.2009.2.723

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Freddy Dumortier - Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek, Belgium (email)
Robert Roussarie - Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France (email)

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.
   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, Liénard equation.
Mathematics Subject Classification:  34C05, 34C07, 34C08, 34C23, 34C25, 34C26, 34E15, 34E20.

Received: September 2008;      Revised: March 2009;      Available Online: September 2009.