Canard Cycles of Finite Codimension with Two Breaking Parameters

Abstract

We consider two-dimensional slow-fast systems which canard cycles occurring 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 Γuv. The properties of the system depend on four slow–fast divergence integrals, which are functions of u, v and also of a parameter λ. First, in terms of these integrals we obtain an upper bound for the total number of limit cycles bifurcating from the region containing the canard cycles. Next, the codimension of each canard cycle is defined in terms of these integrals. The cyclicity of a canard cycle of finite codimension c is bounded by c + 1. We also give conditions on the four slow divergence integrals in order to have a versal c-unfolding. If c = 2 the versal parameters are just the two breaking parameters, but if c > 2, we need the (c − 2)-dimensional parameter λ to obtain a versal unfolding. Finally, for any finite c ≥ 2, we give an example of polynomial Lienard family exhibiting a versal unfolding of such a canard cycle of codimension c. These results generalize the case c = 2 which was studied in Dumortier and Roussarie (Discret Contin Dyn Sys 17(4):787–806, 2007).