Applications of the Melnikov method to twist maps in higher dimensions using the variational approach

Abstract

We work with symplectic diffeomorphisms of the $n$-annulus ${\Bbb{A}}^n=T^*({\Bbb{R}}^n/{\Bbb{Z}}^n)$. Using the variational approach of Aubry and Mather, we are able to give a local description of the stable (and unstable) manifold for a hyperbolic fixed point. We use this in order to get a Melnikov-like formula for exact symplectic twist maps. This formula involves an infinite series that could be computed in some specific cases. We apply our formula to prove the existence of heteroclinic orbits for a family of twist maps in ${\Bbb{R}}^4$.