Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case

Abstract

In this paper we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singulariry. The method consists in obtaining an asymptotic formula for the difference between this manifolds which turns to be exponentially small respect to the unfolding parameter. The formula obtained is explicit but depends on the so-called Stokes constants, which arise in the study of original vector field and which corresponds to the so called inner equation in singular perturbation theory.