POINCAR E MELNIKOV ARNOLD METHOD FOR TWIST MAPS AMADEU DELSHAMS AND RAFAEL RAM REZ ROS Introduction A general theory for perturbations of an integrable planar map with a separatrix

A general theory for perturbations of an integrable planar map with a separatrix to a hyperbolic xed point has been developed in a previous lecture The splitting of the perturbed invariant curves was measured in rst order with respect to the parameter of perturbation by means of a periodic Melnikov function M de ned on the unperturbed separatrix In the case of planar twist maps M has zero mean and therefore there exists a periodic function L called the Melnikov potential such that M L Consequently if L is not identically constant respectively has non degenerate critical points the separatrix splits respectively the perturbed curves cross transversely The aim of this lecture is to present a similar theory for more dimensions The natural frame is to consider twist maps on cotangent bundles Once the suitable de nition of unperturbed separatrix has been introduced a non trivial problem in the high dimensional case a scalar function L can be de ned on it in such a way that L veri es the same properties than in the planar case The derivation of L is easily related to variational principles and the property of being a scalar function instead of a vectorial function like the classical Melnikov function makes it more useful for computations and geometrical understanding Even more it allows the application of Morse theory to establish the minimal number of transverse homoclinic orbits The results to be presented in this lecture are valid for exact symplectic maps on arbitrary exact symplectic manifolds that is the twist character is not essential We have restricted ourselves to twist maps only for simplicity Full details of the ideas presented here are contained in where another more general situation i e the exact symplectic case is studied Related ideas can be found in