Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

TL;DR

This paper demonstrates the existence of chaotic motions in the planar restricted four body problem by applying a topological forcing theorem, verified through rigorous computer-assisted methods, revealing complex dynamics beyond integrability.

Contribution

It introduces a constructive, computer-assisted approach to prove chaos in the four body problem without relying on perturbation theory, applicable to non-equal masses.

Findings

Existence of chaotic motions in the four body problem.

Verification of transverse intersections of invariant manifolds.

Quantitative bounds on transport times and manifold locations.

Abstract

We prove the existence of chaotic motions in a planar restricted four body problem, establishing that the system is not integrable. The idea of the proof is to verify the hypotheses of a topological forcing theorem. The forcing theorem applies to two freedom Hamiltonian systems where the stable and unstable manifolds of a saddle-focus equilibrium intersect transversally in the energy level set of the equilibrium. We develop a mathematically rigorous computer assisted argument which verifies the hypotheses of the forcing theorem, and we implement our approach for the restricted four body problem. The method is constructive and works far from any perturbative regime and for non-equal masses. Being constructive, our argument results in useful byproducts like information about the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and upper/lower bounds on transport times.