We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in Gidea (2014 arXiv: 1405.0866 ). This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an ‘outer dynamics’, given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an ‘inner dynamics’, given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, Gidea (2014 arXiv: 1405.0866 ) does not rely on the KAM theory and/or Aubry–Mather theory to establish the existence of diffusion. Moreover, it does not require to check twist conditions or non-degeneracy conditions near resonances. The conditions are explicit and can be checked by finite precision calculations in concrete systems (roughly, they amount to checking that Melnikov-type integrals do not vanish and that some manifolds are transversal). As an application, we study the planar elliptic restricted three-body problem. We present a rigorous theorem that shows that if some concrete calculations yield a non zero value, then for any sufficiently small, positive value of the eccentricity of the orbits of the main bodies, there are orbits of the infinitesimal body that exhibit a change of energy that is bigger than some fixed number, which is independent of the eccentricity. We verify numerically these calculations for values of the masses close to that of the Jupiter/Sun system. The numerical calculations are not completely rigorous, because we ignore issues of round-off error and do not estimate the truncations, but they are not delicate at all by the standard of numerical analysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where 1 would be enough.) The code of these verifications is available. We hope that some full computer assisted proofs will be obtained in the near future since there are packages (CAPD) designed for problems of this type.

We consider a class of autonomous Hamiltonian systems subject to small, time‐periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non‐zero.We describe a topological method to establish orbits which diffuse in energy for every suitably small perturbation parameter . The method yields quantitative estimates: the existence of orbits along which the energy drifts by an amount independent of ε; the time required by such orbits to drift is ; the existence of orbits along which the energy makes chaotic excursions; explicit estimates for the Hausdorff dimension of the set of such chaotic orbits; the existence of orbits along which the time evolution of energy approaches a stopped diffusion process (Brownian motion with drift), as ε tends to 0. For each ε fixed, the set of initial conditions of the orbits that yield the diffusion process has positive Lebesgue measure, and in the limit the measure of these sets approaches 0. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion for appropriate sets of initial conditions. A key feature of our topological method is that it can be implemented in computer‐assisted proofs. We give an application to a concrete model of the planar elliptic restricted three‐body problem, on the motion of an infinitesimal body relative to the Neptune‐Triton system. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.