Breakdown of heteroclinic connections in the analytic Hopf-Zero singularity: Rigorous computation of the Stokes constant

TL;DR

This paper develops a rigorous computational method to verify the non-vanishing of the Stokes constant in analytic unfoldings of the Hopf-Zero singularity, confirming the existence of heteroclinic connections.

Contribution

It introduces a computer-assisted approach to rigorously compute bounds for the Stokes constant in the Hopf-Zero singularity, a task previously lacking analytic techniques.

Findings

Confirmed the non-zero value of the Stokes constant in two specific unfoldings

Provided rigorous bounds for the Stokes constant using computer assistance

Validated the existence of heteroclinic connections in the studied singularities

Abstract

Consider analytic generic unfoldings of the three dimensional conservative Hopf-Zero singularity. Under open conditions on the parameters determining the singularity, the unfolding possesses two saddle-foci when the unfolding parameter is small enough. One of them has one dimensional stable manifold and two dimensional unstable manifold whereas the other one has one dimensional unstable manifold and two dimensional stable manifold. Baldom\'a, Castej\'on and Seara [BCS13] gave an asymptotic formula for the distance between the one dimensional invariant manifolds in a suitable transverse section. This distance is exponentially small with respect to the perturbative parameter, and it depends on what is usually called a Stokes constant. The non-vanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. However, up to now there do not exist analytic techniques to check that condition. In this paper we provide a method for obtaining accurate rigorous computer assisted bounds for the Stokes constant. We apply it to two concrete unfoldings of the Hopf-Zero singularity, obtaining a computer assisted proof that the constant is non-zero.