Limit cycles bifurcating from periodic orbits near a centre and a homoclinic loop with a nilpotent singularity of Hamiltonian systems

For a planar analytic near-Hamiltonian system, whose unperturbed system has a family of periodic orbits filling a period annulus with the inner boundary an elementary centre and the outer boundary a homoclinic loop through a nilpotent singularity of arbitrary order, we characterize the coefficients of the terms with degree greater than or equal to 2 in the expansion of the first order Melnikov function near the homoclinic loop. Based on these expression of the coefficients, we discuss the limit cycle bifurcations and obtain more number of limit cycles which bifurcate from the family of periodic orbits near the homoclinic loop and the centre. Finally, as an application of our main results we study limit cycle bifurcation of a (m + 1)th order Lienard system with an elliptic Hamiltonian function of degree 4, and improve the lower bound of the maximal number of the isolated zeros of the related Abelian integral for any m >= 4.