Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps

Abstract

We study families of volume preserving diffeomorphisms in ℝ3 that have a pair of hyperbolic fixed points with intersecting codimension one stable and unstable manifolds. Our goal is to elucidate the topology of the intersections and how it changes with the parameters of the system. We show that the "primary intersection" of the stable and unstable manifolds is genetically a neat submanifold of a "fundamental domain." We compute the intersections perturbatively using a codimension one Melnikov function. Numerical experiments show various bifurcations in the homotopy class of the primary intersections. © 2000 American Institute of Physics.