Computer assisted proofs for transverse heteroclinics by the parameterization method

TL;DR

This paper introduces a rigorous computational framework using Fourier-Taylor and Chebyshev series to prove the existence of transverse heteroclinic orbits between hyperbolic periodic solutions in differential equations.

Contribution

It develops a functional analytic approach for computer-assisted proofs of heteroclinic connections, combining local manifold approximations with a posteriori error bounds.

Findings

Validated heteroclinic orbits in Lorenz system

Validated heteroclinic orbits in Hill Restricted Four Body Problem

Provided rigorous bounds on approximation errors

Abstract

This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a Fourier-Taylor approximation of the local stable/unstable manifold of the periodic orbit, combined with a numerical method for solving two point boundary value problems via Chebyshev series approximations. The a-posteriori analysis developed provides mathematically rigorous bounds on all approximation errors, providing both abstract existence results and quantitative information about the true heteroclinic solution. Example calculations are given for both the dissipative Lorenz system and the Hamiltonian Hill Restricted Four Body Problem.