Cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points*

In this paper, we study the cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points in planar slow–fast systems, where the nilpotent contact point is a jump point or a slow–fast Hopf point. These slow–fast cycles can be classified into three cases based on the two nilpotent contact points: (i) both are generic jump points, (ii) one is a generic jump point and the other is a slow–fast Hopf point, and (iii) both are slow–fast Hopf points. By using slow divergence integrals and entry–exit functions, we show that the cyclicity of slow–fast cycles with one self-intersection point and two generic jump points (or one generic jump point and one slow–fast Hopf point) is at most two, and the cyclicity of slow–fast cycles with one self-intersection point and two slow–fast Hopf points is at most three under some specific conditions. Finally, we apply the main results to two predator-prey models.