Canard cycles in the presence of slow dynamics with singularities

Abstract

We study the cyclicity of limit, periodic sets that occur in families of vector fields of slow-fast type. The limit; periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow-fast two-saddle cycle, formed by a, regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral.