Geometric Proof of Strong Stable/Unstable Manifolds, with Application to the Restricted Three Body Problem

TL;DR

This paper introduces a cone condition-based method for proving invariant manifolds near saddle--center fixed points, applicable to computer-assisted proofs, and demonstrates its use in establishing a homoclinic orbit in the restricted three body problem.

Contribution

The paper develops a new geometric approach for invariant manifold proofs that avoids rigorous integration, specifically tailored for computer-assisted validation in dynamical systems.

Findings

Existence of invariant manifolds proved without rigorous integration.

Application to the restricted three body problem confirms a homoclinic orbit.

Method facilitates computer-assisted proofs in celestial mechanics.

Abstract

We present a method for establishing invariant manifolds for saddle--center fixed points. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs, and does not require rigorous integration of the vector field in order to prove the existence of the invariant manifolds. We apply our method to the restricted three body problem and show that for a given choice of the mass parameter, there exists a homoclinic orbit to one of the libration points.