Bifurcation of ten small-amplitude limit cycles by perturbing a quadratic Hamiltonian system

TL;DR

This paper analyzes the bifurcation of small-amplitude limit cycles in quadratic Hamiltonian systems, correcting previous claims and demonstrating a combined method to identify up to 10 bifurcating cycles.

Contribution

It corrects prior assumptions about limit cycle existence and introduces a combined Melnikov and focus value method for better bifurcation analysis.

Findings

Corrects previous claim of 11 limit cycles in a cubic vector field

Demonstrates up to 10 small-amplitude limit cycles bifurcating from a center

Shows advantages of combining Melnikov function and normal form methods

Abstract

This paper contains two parts. In the first part, we shall study the Abelian integrals for Zoladek's example [13], in which it is claimed the existence integrals of 11 small-amplitude limit cycles around a singular point in a particular cubic vector filed. We will show that the basis chosen in the proof of [13] are not independent, which leads to failure in drawing the conclusion of the existence of 11 limit cycles in this example. In the second part, we present a good combination of Melnikov function method and focus value (or normal form) computation method to study bifurcation of limit cycles. An example by perturbing a quadratic Hamiltonian system with cubic polynomials is presented to demonstrate the advantages of both methods, and 10 small-amplitude limit cycles bifurcating from a center are obtained by using up to 5th-order Melnikov functions.