Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits

We consider perturbations of integrable systems which may be non-Hamiltonian and give sufficient conditions for them to be not meromorphically integrable near resonant tori such that the first integrals and commutative vector fields also depend meromorphically on the small parameter.Our proof of the result is based on generalized versions due to Ayoul and Zung of the Morales-Ramis theory and its extension, the Morales-Ramis-Simo theory, which enable us to show the nonintegrability of general differential equations using differential Galois groups of their variational and higher-order variational equations, respectively, along nonconstant particular solutions. Moreover, we discuss a relationship of our theory with the subharmonic Melnikov method for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems. We illustrate the theory for three examples: the periodically forced Duffing oscillator, second-order coupled oscillators and a two-dimensional system of pendulum-type subjected to a constant torque. In a companion paper, the theory is applied to prove the nonintegrability of the restricted three-body problem.