Existence and Stability of Periodic Orbits for a Hamiltonian System with Homogeneous Potential of Degree Five

In this paper we consider the autonomous Hamiltonian system with two degrees of freedom associated to the function H = 1/2 (x(2) + y(2)) + 1/2 (p(x)(2) + p(y)(2)) + V-5(x, y), where V-5(x, y) = (A/5x(5) + Bx(3)y(2) + C/5xy(4)) which is related to a homogeneous potential of degree five. We prove the existence of different families of periodic orbits and the type of stability is analyzed through the averaging theory which guarantee the existence of such orbits on adequate sets defined by the parameters A, B, C.