Computer assisted proof of Shil'nikov homoclinics: with application to the Lorenz-84 model

TL;DR

This paper introduces a computational method for rigorously proving the existence of Shil'nikov homoclinic intersections using geometric bounds and interval arithmetic, with applications to the Lorenz-84 model.

Contribution

It develops a novel computer-assisted approach for verifying homoclinic intersections, ensuring parameter uniqueness and providing sharp bounds in a complex atmospheric model.

Findings

Validated homoclinic intersection in Lorenz-84 model.

Established sharp bounds for the parameter values.

Ensured uniqueness of the homoclinic intersection parameter.

Abstract

We present a methodology for computer assisted proofs of Shil'nikov homoclinic intersections. It is based on geometric bounds on the invariant manifolds using rate conditions, and on propagating the bounds by an interval arithmetic integrator. Our method ensures uniqueness of the parameter for which the homoclinic takes place. We apply the method for the Lorenz-84 atmospheric circulation model, obtaining a sharp bound for the parameter, and also for where the homoclinic intersection of the stable/unstable manifolds takes place.