Separatrix Splitting in 3D Volume-Preserving Maps

We construct a family of integrable volume-preserving maps in R-3 with a two-dimensional heteroclinic connection of spherical shape between two fixed points of saddle-focus type. In other contexts, such structures are called Hill's spherical vortices or spheromaks. We study the splitting of the separatrix under volume-preserving perturbations using a discrete version of the Melnikov method. First, we establish several properties under general perturbations. For instance, we bound the topological complexity of the primary heteroclinic set in terms of the degree of some polynomial perturbations. We also give a sufficient condition for the splitting of the separatrix under some entire perturbations. A broad range of polynomial perturbations verify this sufficient condition. Finally, we describe the shape and bifurcations of the primary heteroclinic set for a specific perturbation.