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In this paper we consider general nearly integrable analytic Hamiltoniansystems of one and a half degrees of freedom which are a trigonometricpolynomial in the angular state variable. In the resonances of these systemsgenerically appear hyperbolic periodic orbits. We study the possibletransversal intersections of their invariant manifolds, which is exponentiallysmall, and we give an asymptotic formula for the measure of the splitting. Wesee that its asymptotic first order is of the form $K \varepsilon^{\beta}\text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in termsof the system features. We compare our results with the classical MelnikovTheory and we show that, tipically, in the resonances of nearly integrablesystems Melnikov Theory fails to predict correctly the constants $K$ and$\beta$ involved in the formula.

Splitting of separatrices in the resonances of nearly integrable Hamiltonian Systems of one and a half degrees of freedom