  In this paper, we obtain the upper bound of the number of zeros of Abelian
integral for a class of cubic Hamiltonian systems with nesting period annuli
under perturbations of polynomials of degree n. Furthermore, we consider the
Hopf and homoclinic bifurcation when a=-1,b=-2,c=1 and n=3, and obtain 18
distributions in which system has at least 3 limit cycles for each case.
