Quadratic perturbations of quadratic codimension-four centers

TL;DR

This paper investigates quadratic differential systems with a specific codimension-four center, analyzing how small perturbations can generate limit cycles, and establishes an upper bound of eight limit cycles using elliptic integrals and Picard-Fuchs equations.

Contribution

It provides a detailed analysis of the limit cycles in quadratic systems with a codimension-four center, employing elliptic integrals and Picard-Fuchs equations to bound their number.

Findings

Limit cycles are determined by zeros of a complete elliptic integral.

Unperturbed system orbits are elliptic curves.

Upper bound of eight limit cycles from the period annulus.

Abstract

We study the stratum in the set of all quadratic differential systems $\dot{x}=P_2(x,y), \dot{y}=Q_2(x,y)$ with a center, known as the codimension-four case $Q_4$. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincar\'e-Pontryagin-Melnikov integral $I$. We show that the orbits of the unperturbed system are elliptic curves, and $I$ is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center.