The number of limit cycles bifurcating from the period annulus of quasi-homogeneous Hamiltonian systems at any order

Abstract

A necessary and sufficient condition is given for quasi-homogeneous polynomial Hamiltonian systems having a center. Then it is shown that there exists a bound on the number of limit cycles bifurcating from the period annulus of quasi-homogeneous Hamiltonian systems at any order of Melnikov functions; and the explicit expression of this bound is given in terms of (n,k,s1,s2), where n is the degree of perturbation polynomials, k is the order of the first nonzero higher order Melnikov function, and (s1,s2) is the weight exponent of quasi-homogeneous Hamiltonian with center. This extends some known results and solves the Arnol'd-Hilbert's 16th problem for the perturbations of homogeneous or quasi-homogeneous polynomial Hamiltonian systems.