Canard-cycle transition at a fast fast passage through a jump point

We consider transitory canard cycles that consist of a generic breaking mechanism, i.e. a Hopf or a jump breaking mechanism, in combination with a fast-fast passage through a jump point. Such cycle separates two types of canard cycles with a different shape. We obtain upper bounds on the number of periodic orbits that can appear near the canard cycle, and this under very general conditions. (C) 2013 Published by Elsevier Masson SAS on behalf of Academie des sciences.