Transition Tori in the Planar Restricted Elliptic Three Body Problem

TL;DR

This paper studies how small eccentricities in the elliptic three body problem perturb Lyapounov orbits into quasi-periodic tori, revealing chaotic dynamics through transversal intersections of invariant manifolds.

Contribution

It demonstrates the persistence of Lyapounov orbits as invariant tori under small elliptic perturbations and identifies conditions for chaotic dynamics via manifold intersections.

Findings

Existence of a Cantor set of Lyapounov orbits surviving perturbation.

Persistence of quasi-periodic invariant tori near the libration point L2.

Chaotic dynamics with diffusion over energy levels due to manifold intersections.

Abstract

We consider the elliptic three body problem as a perturbation of the circular problem. We show that for sufficiently small eccentricities of the elliptic problem, and for energies sufficiently close to the energy of the libration point L2, a Cantor set of Lyapounov orbits survives the perturbation. The orbits are perturbed to quasi-periodic invariant tori. We show that for a certain family of masses of the primaries, for such tori we have transversal intersections of stable and unstable manifolds, which lead to chaotic dynamics involving diffusion over a short range of energy levels. Some parts of our argument are nonrigorous, but are strongly backed by numerical computations.