Cyclicity of slow–fast cycles with two canard mechanisms

Abstract

In this paper, we study the cyclicity of some degenerate slow-fast cycles with two canard mechanisms in planar slow-fast systems. One canard mechanism originates from a slow-fast Hopf point and the other from a point of self-intersection where the so-called entry-exit relation can be used. By studying the difference map, we show that the cyclicity of such slow-fast cycles is at most two (the associated slow divergence integral is nonzero or vanishes). As an example, we apply this result to the modified Holling-Tanner model.