Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation (Q2888614)

The paper presents a study of the splitting of separatrices in a Hamiltonian system of one degree of freedom with a fast periodic or quasiperiodic perturbation that is meromorphic in the state variables. The obtained results are different from the previous ones in the literature, which mainly assume algebraic or trigonometric polynomial dependence on the state variables. As a model, the pendulum equation with several meromorphic perturbations is considered. The sensitivity of the size of splitting on the width of the analyticity strip of the perturbation with respect to the state variables is investigated. In particular, the size of the splitting is shown to be exponentially small if the strip of analyticity is wide enough. Furthermore, the splitting grows as the width of the analyticity strip shrinks, even becoming non-exponentially small for very narrow strips. The results obtained prevent the use of polynomial truncations of a meromorphic perturbation to compute the size of the splitting of separatrices.