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

2014/07/25 by Xiuli Cen, Cen, Xiuli, Yulin Zhao +3
Mathematics · #34A34 #34C05 #37C07 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1407.7070

openalex publication_date 2014/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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≤ u, v≤ 3, u+v≤5 are realizable.

Related

• Bifurcation of limit cycles for a class of cubic Hamiltonian systems with nesting period annuli
• Bifurcation of limit cycles from a cubic reversible isochrone
• Limit cycles appearing from the perturbation of a cubic isochronous center
• Limit cycles bifurcating from a degenerate center
• Limit cycles bifurcating from the quasi-homogeneous polynomial centers of weight-degree 2 under non-smooth perturbations