Diffusion time and splitting of the separatrices in nearly integrable isochronous Hamiltonian systems

We consider the problem of Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincar ́e-Melnikov function.