Limit cycles near a nilpotent centre and a homoclinic loop to a nilpotent singularity of Hamiltonian systems

For a planar analytic near-Hamiltonian system, whose unperturbed Hamiltonian system has a period annulus with its inner boundary an elementary centre and its outer boundary a homoclinic loop to a nilpotent singularity, it was characterized that the coefficients of the high order terms in the expansion of the first order Melnikov function near the loop. The present paper pursues this   characterization on the coefficients for a homoclinic loop to a nilpotent singularity, in virtual of a nilpotent centre of arbitrary order. Based on the properties of these coefficients,  we establish a bifurcation theory for finding more limit cycles.  Finally this theory is applied to an (m+1)th order generalized Lienard system, and to an mth order near-Hamiltonian system with a hyperelliptic Hamiltonian of degree 6.