Computer assisted proof of drift orbits along normally hyperbolic manifolds

Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds II: Application to the Restricted Three Body Problem

“…In this section we recall the results which we use for our diffusion mechanism. We follow the setup from [6], which is based on the scattering map theory described in [2], and the diffusion mechanism from [4], which is the main tool for our proof. Definition 2.…”

“…The strategy for proving diffusion using this scheme is described in [2,3,4,5]. For our proof we will use the results from [6], which provides a formulation for these methods, that can be implemented to obtain a computer assisted proof. The main feature of [6] is that the assumptions needed for the diffusion mechanism can be validated in a finite number of steps by checking various bounds on the properties of the system.…”

“…We also validate twist conditions, which ensure that the NHIM after perturbation contains a Cantor set of KAM tori. Then we can apply the theorems from [6] to prove diffusion. All the above steps can be established with the assistance of rigorous, interval arithmetic based estimates, performed with the aid of a computer.…”

“…This is based on a computer assisted construction of 'correctly aligned windows'. The difference is that in the current paper we only need to establish the intersections of the manifolds for the unperturbed system, and check some explicit conditions from [6]. The shadowing of orbits is then automatically taken care of by [4], and we do not need to carry out the explicit construction of windows as in [8].…”

“…To establish bounds on the stable and unstable manifolds and the transversality of their intersections needed to ensure that the scattering map is well defined we use cones. The assumptions of theorems [6] leading to diffusion can then be checked by computing sums along finite fragments of homoclinic orbits.…”