Crossing limit cycles in piecewise smooth Kolmogorov systems: an application to Palomba's model

TL;DR

This paper investigates the maximum number of crossing limit cycles in piecewise smooth Kolmogorov systems, providing new lower bounds and applying the results to Palomba's economic model, advancing understanding of such systems.

Contribution

It establishes new lower bounds for crossing limit cycles in piecewise smooth Kolmogorov systems and applies these findings to a specific economic model.

Findings

At least one crossing limit cycle exists in Palomba's model.

Lower bounds for the number of crossing limit cycles: M_K^p(2)≥1, M_K^p(3)≥12, M_K^p(4)≥18.

These bounds are among the best known in the literature for such systems.

Abstract

In this paper, we study the number of isolated crossing periodic orbits, so-called crossing limit cycles, for a class of piecewise smooth Kolmogorov systems defined in two zones separated by a straight line. In particular, we study the number of crossing limit cycles of small amplitude. They are all nested and surround one equilibrium point or a sliding segment. We denote by $\mathcal M_{K}^{p}(n)$ the maximum number of crossing limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a piecewise smooth Kolmogorov systems of degree $n=m+1$. We make a progress towards the determination of the lower bounds $M_K^p(n)$ of crossing limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a piecewise smooth Kolmogorov system of degree $n$. Specifically, we shot that $M_{K}^{p}(2)\geq 1$, $M_{K}^{p}(3)\geq 12$, and $M_{K}^{p}(4)\geq 18$. In particular, we show at least one crossing limit cycle in Palomba's economics model, considering it from a piecewise smooth point of view. To our knowledge, these are the best quotes of limit cycles for piecewise smooth polynomial Kolmogorov systems in the literature.