TITLE:
Periodic perturbation of a 3D conservative flow with a heteroclinic connection
to saddle-foci

AUTHORS:
Ainoa Murillo^(1), Arturo Vieiro^(1,2),
   
(1) Department de Matematiques i Informatica,
    Universitat de Barcelona (UB), Barcelona, Spain.
(2) Centre de Recerca Matematica, Edifici C,
    Campus Bellaterra, 08193 Bellaterra, Spain.

ABSTRACT:

The 2-jet normal form of the elliptic volume-preserving Hopf-zero bifurcation
provides a one-parameter family of volume-preserving vector fields with a pair
of saddle-foci points whose 2-dimensional invariant manifolds form a 2-sphere
of spiralling heteroclinic orbits. We study the effect of an external periodic
forcing on the splitting of these 2-dimensional invariant manifolds. The
internal frequency (related to the foci and already presented in the
unperturbed system) interacts with an external one (coming from the periodic
forcing). If both frequencies are incommensurable, this interaction leads to
quasi-periodicity in the splitting behaviour, which is exponentially small in
(a suitable function of) the unfolding parameter of the Hopf-zero bifurcation.
The corresponding behaviour is described by a Melnikov function. The changes of
dominant harmonics correspond to primary quadratic tangencies between the
invariant manifolds. Combining analytical and numerical results, we provide a
detailed description of the asymptotic behaviour of the splitting under
concrete arithmetic properties of the frequencies.