  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.
