Oriol Castejón (Universitat Politècnica de Catalunya): Breakdown of Heteroclinic Orbits for Analytic Unfoldings of the Hopf-Zero Singularity. The Singular Case.

We study the exponentially small splitting of a
heteroclinic connection in a two-parameter family of analytic
vector fields in R3, which arises from analytic unfoldings of
the Hopf-zero singularity. Previous work showed that, in a
conservative setting, this heteroclinic connection is destroyed
if one considers perturbations of a higher order (the so-called
regular case), and an asymptotic formula of the distance between
the stable and unstable manifolds when they meet at the plane
z=0 was given. Moreover, its main term was a suitable version
of the Melnikov integral. Here, we study the singular case in
both the conservative and dissipative settings, and we show that
Melnikov theory is no longer valid. We give an asymptotic
expression of the splitting distance, which is exponentially
small with respect to one of the perturbation parameters.

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.

This is a joint work with I. Baldomà and T. M-Seara.