Subharmonic Melnikov functions and nonintegrability for autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems near periodic orbits

Abstract

We study autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems and give sufficient conditions for their real-analytic non-integrability near periodic orbits of the unperturbed systems such that the first integrals and commutative vector fields depend analytically on the small parameter by using the subharmonic Melnikov functions. Moreover, we show that autonomous dissipative perturbations prevent real-analytic integrability of these systems. Our results reveal that the perturbed systems can be real-analytically non-integrable even if there is no homoclinic/heteroclinic orbit in the unperturbed systems. We illustrate our theory with a periodically forced duffing equation and a damped Morse oscillator.