TITLE: Breakdown of Heteroclinic Orbits for Some Analytic
Unfoldings of the Hopf-Zero Singularity


AUTHORS:
Inmaculada Baldoma

Departament de Matematica Aplicada i Analisi, Universitat de Barcelona,
Gran Via 587, 08007 Barcelona, Spain

T.M. Seara

Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya,
Diagonal 647, 08028 Barcelona, Spain

E-mails: immaculada.baldoma@upc.edu, Tere.M-Seara@upc.edu

ABSTRACT:

In this paper we study the exponentially small splitting of a heteroclinic
connection in a one-parameter family of analytic vector fields in R^3. This
family arises from the conservative analytic unfoldings of the so-called Hopf
zero singularity (central singularity). The family under consideration can be
seen as a small perturbation of an integrable vector field having a
heteroclinic orbit between two critical points along the z axis. We prove that,
generically, when the whole family is considered, this heteroclinic connection
is destroyed. Moreover, we give an asymptotic formula of the distance between
the stable and unstable manifolds when they meet the plane z = 0. This distance
is exponentially small with respect to the unfolding parameter, and the main
term is a suitable version of the Melnikov integral given in terms of the Borel
transform of some function depending on the higher-order terms of the family.
The results are obtained in a perturbative setting that does not cover the
generic unfoldings of the Hopf singularity, which can be obtained as a singular
limit of the considered family. To deal with this singular case, other
techniques are needed. The reason to study the breakdown of the heteroclinic
orbit is that it can lead to the birth of some homoclinic connection to one of
the critical points in the unfoldings of the Hopf-zero singularity, producing
what is known as a Shilnikov bifurcation.