Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity

TL;DR

This paper proves the existence of exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity, providing a formula for the distance between manifolds that occurs beyond all orders in the normal form.

Contribution

It establishes the breakdown of heteroclinic connections in the Hopf-Zero singularity and derives an explicit exponentially small formula for the manifold distance.

Findings

Heteroclinic connection breaks down in the Hopf-Zero singularity.

The manifold distance is exponentially small in the perturbation parameter.

Results apply to both conservative and dissipative unfoldings.

Abstract

In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity in an open set of the parameter space. This heteroclinic orbit appears at any order if one performs the normal form around the origin, therefore it is a phenomenon "beyond all orders". In this paper we provide a formula for the distance between the corresponding stable and unstable one dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative and dissipative unfoldings.