Splitting of separatrices in Hamiltonian systems and symplectic maps

Poincar\'e, Melnikov and Arnol'd introduced the standard method for measuring the splitting of separatrices of Hamiltonian systems. It is based on the study of the zeros of the so-called Melnikov integral, a vectorial function for three or more degrees of freedom, that gives the first-order behavior. In the most interesting cases, it turns out that the splitting is exponentially small with respect of the parameter of the perturbation, and that means that the remainder has to be bounded very carefully. The mechanism for obtaining rigorously this exponentially small splitting for the one and a half degrees of freedom Hamiltonians is reviewed, and the main ideas for its generalization to more degrees of freedom are presented. Concerning symplectic maps, the Melnikov function is not an integral anymore, but an infinite sum. Nevertheless, for meromorphic perturbations of $2D$-area preserving maps, the Melnikov function turns out to be an elliptic function, and moreover can be evaluated via residues. Furthermore, general results on non-integrability can be provided. For instance, the elliptic billiard turns out to be non-integrable when perturbed by any non-trivial entire perturbation. For more degrees of freedom, using variational arguments, the Melnikov vectorial function for a symplectic map can be deduced from a scalar function (the Melnikov potential), and the splitting of separatrices associated to hyperbolic points can also be easily detected in several situations, for instance for generalized standard maps.