Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials

TL;DR

This paper demonstrates that quadratic near-integrable systems perturbed by quadratic polynomials can produce at least four limit cycles, answering an open question in the field.

Contribution

It provides a proof that quadratic systems can have four limit cycles with a (3,1) distribution using Melnikov function analysis.

Findings

Quadratic systems can generate at least four limit cycles.

The (3,1) distribution of limit cycles is achievable.

Melnikov function method is effective for this analysis.

Abstract

In this paper, we give a positive answer to the open question: Can there exist 4 limit cycles in quadratic near-integrable polynomial systems? It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least 4 limit cycles with (3,1) distribution. The method of Melnikov function is used.