Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four body problem

TL;DR

This paper develops computer-assisted methods to rigorously prove the existence of transverse homoclinic orbits in the spatial equilateral restricted four body problem, demonstrating complex chaotic dynamics.

Contribution

It introduces a novel combination of boundary value problem formulation, parameterization, and validated numerics to establish homoclinic chaos in a non-perturbative celestial mechanics setting.

Findings

Existence of transverse homoclinic orbits confirmed for specific parameters.

Validated computational framework for connecting orbits in celestial mechanics.

Proof of chaos via homoclinic intersections in the four body problem.

Abstract

We develop computer assisted arguments for proving the existence of transverse homoclinic connecting orbits, and apply these arguments for a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four body problem. The idea is to formulate the desired connecting orbits as solutions of certain two point boundary value problems for orbit segments which originate and terminate on the local stable/unstable manifolds attached to a periodic orbit. These boundary value problems are studied via a Newton-Kantorovich argument in an appropriate Cartesian product of Banach algebras of rapidly decaying sequences of Chebyshev coefficients. Perhaps the most delicate part of the problem is controlling the boundary conditions, which must lie on the local stable/unstable manifolds of the periodic orbit. For this portion of the problem we use a parameterization method to develop Fourier-Taylor approximations equipped with a-posteriori error bounds. This requires validated computation of a finite number of Fourier-Taylor coefficients via Newton-Kantorovich arguments in appropriate Cartesian product of rapidly decaying sequences of Fourier coefficients, followed by a fixed point argument to bound the tail terms of the Taylor expansion. Transversality follows as a consequence of the Newton-Kantorovich argument.