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

Yagor Romano Carvalho, Luiz Fernando da Silva Gouveia, Oleg Makarenkov

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.

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

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 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 . We make a progress towards the determination of the lower bounds of crossing limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a piecewise smooth Kolmogorov system of degree . Specifically, we shot that , , and . 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.

Paper Structure

This paper contains 5 sections, 6 theorems, 25 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bifurcations of Small Limit Cycles
  4. An application to Palomba's model
  5. Bifurcation of limit cycles in Piecewise Smooth Kolmogorov systems

Key Result

Theorem 1.1

The lower bounds of small limit cycles $\mathcal{M}^{c}_{p_K}(n)$ for the piecewise smooth Kolmogorov systems eqkolpdesc for $n=2,3,$ and $4$ are, at least, $1$, $12$, and $18$, respectively (as illustrated in Table 1).

Figures (8)

  • Figure 1: Crossing, Sliding and escaping segments.
  • Figure 2: Crossing, Sliding and escaping segments.
  • Figure 3: Phase portrait of the Palomba's model considering $\eta_{1}=1$, $\gamma_{1}=2$, $\eta_{2}=3$, and $\gamma_{2}=5$.
  • Figure 4: Phase portrait of the system \ref{['Palombadisc1']}. The green segment represents the sliding region. In the picture a) we consider $\varepsilon=-0.5$ and the sliding segment is an attractor. In the picture b), $\varepsilon=0$, there is no sliding segment and the point $p$ is a center. In the picture c), $\varepsilon=0.5$ and the sliding segment is repulsor.
  • Figure 5: Phase portrait of the Kolmogorov system \ref{['komquar_cub']}.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3: Chr2006
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 1 more