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 Poincaré-Melnikov function.

For a positive integer n and R>0, we set $$B_R^n=\left\{x\in {ℝ}^n|{∥x∥}_{\infty }<R\right\}$$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $$h\left(r\right)=\frac{1}{2}{r}_1^2+...\frac{1}{2}{r}_{n-1}^2+r_n$$ on $${𝕋}^n\times B_R^n$$ , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of $${𝕋}^n\times B_R^n$$ , and setting $${\epsilon }_j:={∥h-H_j∥}_{C^0\left(V\right)}$$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface,...

We give sufficient conditions for the strong asymptotic stability of the distributions of dynamical systems with multiplicative perturbations. We apply our results to iterated function systems.