Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification

TL;DR

This paper introduces a novel diffusion mechanism for Hamiltonian systems that avoids traditional KAM and Aubry-Mather theory, and applies it to demonstrate energy changes in the planar elliptic restricted three-body problem with numerical verification.

Contribution

It presents a new diffusion mechanism based on shadowing pseudo-orbits, applicable without twist or non-degeneracy conditions, and applies it to celestial mechanics with numerical validation.

Findings

Existence of orbits with significant energy change for small eccentricities

Numerical verification near Jupiter/Sun system parameters

Mechanism does not rely on KAM or Aubry-Mather theory

Abstract

We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in [25]. 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, [25] 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. 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 this verifications is available. We hope that some full computer assisted proofs will be obtained in a near future since there are packages (CAPD) designed for problems of this type.