Perturbations of quadratic centers of genus one

TL;DR

This paper develops a method to analyze the maximum number of limit cycles in quadratic systems with genus one centers by classifying systems, identifying key perturbations, and computing Melnikov functions.

Contribution

It introduces a classification of quadratic systems with genus one centers and computes Melnikov functions to determine cyclicity under perturbations.

Findings

Identified essential quadratic perturbations producing maximum limit cycles

Computed Melnikov functions for specific reversible systems

Determined cyclicity of period annuli in example systems

Abstract

We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems.