2024/10/11 by Yagor Romano Carvalho, Carvalho, Yagor Romano, Luiz Fernando da Silva Gouveia +3 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2410.09281
openalex publication_date 2024/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 MKp(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 MKp(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 MKp(2)≥ 1, MKp(3)≥ 12, and MKp(4)≥ 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.