Breakdown of heteroclinic orbits for analytic unfoldings of the hopf-zero singularity. The singular case

Abstract

In this paper we prove the breakdown of an heteroclinic connection in the analytic versal unfoldings of the generic Hopf-Zero singularity. 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 correspond- ing stable and unstable one dimensional manifolds which is given by an exponentially small function in the perturbation parameter. Our result applies both for conservative unfoldings and for dissipative ones in some open regions of the parameter plane