Homoclinic dynamics in a restricted four body problem

TL;DR

This paper introduces a method to compute and analyze invariant manifolds in the planar restricted four body problem, revealing complex homoclinic dynamics and connecting orbits through advanced numerical techniques.

Contribution

The paper develops a novel computational approach for atlases of invariant manifolds and applies it to the four body problem, including symmetry breaking and continuation methods.

Findings

Computed atlases for stable and unstable manifolds at symmetric configurations.

Identified approximate intersections of invariant manifolds.

Found connecting orbits by breaking symmetries and applying continuation methods.

Abstract

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point, and implement the method for the saddle-focus libration points of the planar equilateral restricted four body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable and unstable manifolds. The resulting atlases are comprised of thousands of individual chart maps, with each chart represented by a two variable Taylor polynomial. Post-processing the atlas data yields approximate intersections of the invariant manifolds, which we refine via a shooting method for an appropriate two point boundary value problem. Finally we apply numerical continuation to the BVPs. This breaks the symmetries and leads to connecting orbits for some non-equal values of the primary masses.