Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers

Linping Peng; Zhaosheng Feng; Changjian Liu

doi:10.3934/dcds.2014.34.4807

Article Contents

2014, Volume 34, Issue 11: 4807-4826. Doi: 10.3934/dcds.2014.34.4807

This issue Previous Article Self-intersections of trajectories of the Lorentz process Next Article Linearised higher variational equations

Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers

1.
School of Mathematics and System Sciences, Beijing University of Aeronautics and Astronautics, LIMB of the Ministry of education, Beijing, 100191
2.
Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539
3.
School of Mathematics, Soochow University, Suzhou, Jiangsu 215006

Received: November 30, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

This paper is concerned with the bifurcation of limit cycles from a quadratic reversible Lotka-Volterra system with two centers of genus one under small quadratic perturbations. It shows that the cyclicities of each period annulus and two period annuli of the considered system under small quadratic perturbations are two, respectively. This not only gives at least partially a positive answer to an open conjecture, but also improves the corresponding results in the literature. In addition, we present the configurations of limit cycles of the perturbed system as (2, 0), (1, 1), (1, 0), (0, 2), (0, 1) and (0, 0), where $(i,\, j)$ indicates that the perturbed system has $i$ limit cycles surrounding the positive singularity while it has $j$ limit cycles surrounding the negative one.

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Mathematics Subject Classification: Primary: 34C07, 37G15; Secondary: 34C05.

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