Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

Abstract

In this work the existence of periodic solutions is studied for the Hamiltonian functions (Formula presented.) where the first term consist of a harmonic oscillator and the second term are homogeneous polynomials of degree 5 defined by two real parameters (Formula presented.) . Using the averaging method of second order we provide the sufficient conditions on the parameters to guarantee the existence of periodic solutions for positive energy and we study the stability of these periodic solutions.