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

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.