Nonintegrability of time-periodic perturbations of analytically integrable systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of analytically integrable systems in the sense of Bogoyavlenskij and study their \emph{real-meromorphic} nonintegrability, using a generalized version due to Ayoul and Zung of the Morales-Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not \emph{real-meromorphically} integrable near homo- and heteroclinic orbits. We illustrate the theory for rotational motions of a periodically forced rigid body, which provides a mathematical model of a quadrotor helicopter.