2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67417It is well known that a generic small perturbation of a Liouville-integrable Hamiltonian system causes breakup of resonant and near-resonant invariant tori. A general approach to the simple resonance case in the convex real-analytic setting is developed, based on a new technique for solving the Hamilton-Jacobi equation. It is shown that a generic perturbation creates in the core of a resonance a partially hyperbolic lower-dimensional invariant torus, whose Lagrangian stable and unstable manifolds, described as global solutions of the Hamilton-Jacobi equation, split away from this torus at exponentially small angles. Optimal upper bounds with best constants are obtained for exponentially small splitting in the general case.40 pagesDynamical SystemsMathematical Physics37J40; 37J45text