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.