A CRITERION FOR NONINTEGRABILITY BASED ON POINCARES THEOREM

The non-integrability of a two-degrees-of-freedom Hamiltonian is inves tigated by a method based on a well-known theorem of Poincare. It is p roved that the perturbed system is non-integrable for values of the pe rturbative parameter in an open interval around zero, if, on a dense s et of resonant tori of the unperturbed system, the average value of th e perturbative function, evaluated along the periodic orbits on each t orus, depends on the particular orbit. An application to the separable quartic oscillator with a quadratic perturbation is made and it is sh own that if the perturbation is non-separable in the same coordinates, the system is non-integrable.