Splitting of Separatrices in Hamiltonian Systems and Symplectic Maps

A century ago, the phenomenon of the splitting of separatrices was discovered by Henri Poincare in his celebrated memoir on the three-body problem [39]. While trying to integrate the problem of the three bodies, expanding the solutions with respect to a small parameter, Poincare noticed that the main obstruction was due to the possibility of transversal intersection of invariant manifolds that were coincident (separatrices) for the unperturbed integrable problem. To measure the size of such splitting, he developed a perturbative method in the parameter of perturbation, say ɛ. and he was confronted with a singular separatrix splitting problem, in the sense that the separatrices of the unperturbed problem depended on e in an essential way. He already noticed that the size of the splitting of the separatrices predicted by his perturbative method was exponentially small with respect to ɛ [39, page 223], a fact which prevented him to provide rigorous results, since the remainder of his perturbative expansion was, in principle, O(ɛ2).

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