  We analytically compute asymptotic expansions of a 1-dimensional sub-manifold
of stable and unstable manifolds in a 4-dimensional symplectic mapping by using
the method called asymptotic expansions beyond all orders. This method enables
us to capture exponentially small splitting of separatrices and also to obtain
explicit functional approximations of the sub-manifolds. In addition, we show
the condition with which homoclinic structure caused by crossing between the
stable and unstable sub-manifolds is regarded as a direct product of
2-dimensional mappings.
