Abstract:

We consider two-dimensional slow-fast systems which canard cycles occuring in the layer equation obtained for $varepsilon = 0$. The canard cycles under consideration may be broken by two independent mechanisms: either a turning point or jump between two contact points. Each of these mechanisms is associated to a parameter permitting a generic breaking of the canard cycle. For this reason the canard cycles under consideration are called canard cycles with two breaking parameters. They also pass through two independent horizontal levels parametrized by $u$, $v$. Then, for a given system we have a whole 2-parameter family of canard cycles $Gamma_{u,v}$. The properties of the system depend on four slow-fast divergence integrals, which are functions of $u$, $v$ and also of a parameter $lambda$. In this talk I give some results about the limit cycles which bifurcate from the canard cycles. Ideas coming from Khovanskii can be apply to reduce the problem to the consideration of this system of slow divergence integrals. We can deduce from this system the number of bifurcating limit cycles and the bifurcation diagram of limit cycles.

These results appear in a paper written by Lylia Mamouhdi and myself in which we generalize at any finite codimension $c$ the case $c = 2$ which was studied by Freddy Dumortier and myself in a previous work.