Cyclicity of canard cycles with hyperbolic saddles located away from the critical curve

Abstract

The goal of our paper is to study limit cycles in smooth planar slow-fast systems, Hausdorff close to canard cycles containing hyperbolic saddles located away from the critical (or slow) curve. Our focus is on a broad class of smooth slow-fast systems with a Hopf breaking mechanism (often called a generic turning point), a jump breaking mechanism or a non-generic turning point. Such canard cycles naturally occur in predator-prey systems with Holling type II and IV response functions and with a small predator's death rate. The study of canard cycles is also relevant for the Hilbert's 16th problem. We primarily focus on the canard cycles with one hyperbolic saddle (located away from the slow curve) and we allow isolated singularities in the slow dynamics.