We study a family of volume preserving diffeomorphisms that have a pair of hyperbolic fixed points. Using a geometric definition of the primary intersection, we study the topology of the heteroclinic intersection. We relate this to the set of zeroes of a Melnikov function defined on a saddle connection.
Numerical experiments show possible bifurcations in the primary intersections.
In addition, we develop a general Melnikov method for maps to detect the transverse intersection of stable and unstable manifolds. A review of the subject, for non specialists, will be given.