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

This paper contains two parts. In the first part, we shall study the Abelian integrals for Żoła̧dek's example [13], in which the existence of 11 small-amplitude limit cycles around a singular point in a particular cubic vector field is claimed. We will show that the bases 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.