Topological bifurcation of heteroclinic primary intersections for volume preserving maps and generalizations

Professor Hector Lomeli, Department of Mathematics, ITAM

We study a family volume preserving diffeomorphisms that have a pair of hyperbolic fixed points. We show that the corresponding codimension one stable and unstable manifolds of these points intersect transversally in a one dimensional manifold. 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. We classify these intersections in terms of the homotopy classes of a fundamental domain considered as a 2-torus. In addition, we develop a general Melnikov method for maps to detect the transverse intersection of stable and unstable manifolds. We relate this intersection to the set of zeroes of a geometric object that plays the role of a Melnikov function.