Abstract

We consider the Hamiltonian polynomial function H of degree fourth given by either H ( x, y, p(x,) p(y)) = 1/2 ( p(x)(2) + p(y)(2)) + 1/2 ( x(2) + y(2)) + V-3 ( x, y) + V-4 ( x, y), or H ( x, y, p(x), p(y)) = 1/2 ( p(x)(2) + p(y)(2)) + 1/2 (-x(2) + y(2)) + V-3 ( x; y) + V-4 ( x, y), where V-3 ( x, y) and V-4 ( x, y) are homogeneous polynomials of degree three and four, respectively. Our main objective is to prove the existence and stability of periodic solutions associated to H using the classical averaging method.