SPLITTING OF SEPARATRICES IN THE RESONANCES OF NEARLY INTEGRABLE HAMILTONIAN SYSTEMS OF ONE AND A HALF DEGREES OF FREEDOM

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form K epsilon(beta)c(-alpha/epsilon) and we identify the constants K, beta, alpha in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, typically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants K and beta involved in the formula.