Perturbation theory of the quadratic Lotka-Volterra double center

TL;DR

This paper analyzes how quadratic perturbations affect the double center in a Lotka-Volterra system, showing at most two limit cycles can bifurcate, using bifurcation functions and iterated path integrals.

Contribution

It introduces a novel perturbation approach for the quadratic Lotka-Volterra system with a double center, establishing an upper limit of two limit cycles.

Findings

Maximum of two limit cycles can bifurcate from the double center.

The distribution of limit cycles is constrained to at most two on the plane.

Bifurcation functions are expressed via iterated path integrals of length two.

Abstract

We revisit the bifurcation theory of the Lotka-Volterra quadratic system \begin{eqnarray} X_0 :\left\{\begin{aligned} \dot{x}=& - y -x^2+y^2 ,\\ \dot{y}= &\;\;\;\;x - 2xy \end{aligned} \right. \end{eqnarray} with respect to arbitrary quadratic deformations. The system $X_0$ has a double center, which is moreover isochronous. We show that the deformed system $X_0$ can have at most two limit cycles on the finite plane, with possible distribution $(i,j)$, where $i+j\leq2$. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two.