Slow-fast cycles with points of self-intersections

Abstract

For a class of slow-fast systems in the form ̇ 𝑥=(𝐹(𝑥,𝜆)−𝑦)𝑥𝑛, ̇ 𝑦=𝜖𝑔(𝑥,𝑦,𝜆) for (𝑥,𝑦,𝜆)∈ℝ× ℝ×ℝ𝑝, we study the cyclicity of slow-fast cycles with a more degenerate point of self-intersection and a generic contact point. By using entry-exit functions and slow divergence integrals, we show that the cyclicity of slow-fast cycles with points of self-intersection is at most one under some non-degenerate conditions for 𝑛≥1. Finally, we apply our theory to the classical Holling-Tanner model.