Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka-Volterra type with a rational first integral of degree 2

TL;DR

This paper investigates the maximum number of limit cycles bifurcating from cubic polynomial Lotka-Volterra systems with a rational first integral of degree 2, using Abelian integrals and Chebyshev criterion.

Contribution

It establishes a sharp upper bound of 3 for the zeros of Abelian integrals in this context and explores bifurcation configurations under polynomial perturbations.

Findings

Maximum of 3 limit cycles for degree 3 perturbations.

All configurations with certain parameters are realizable.

Analysis of bifurcation and distribution of limit cycles.

Abstract

In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using \emph{Chebyshev criterion}, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for $n=3$. The simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered. All configurations $(u,v)$ with $0\leq u, v\leq 3, u+v\leq5$ are realizable.