  This paper contains theory on two related topics relevant to manifolds of
normally hyperbolic singularities. First, theorems on the formal and $ C^k $
normal forms for these objects are proved. Then, the theorems are applied to
give asymptotic properties of the transition map between sections transverse to
the centre-stable and centre-unstable manifolds of some normally hyperbolic
manifolds. A method is given for explicitly computing these so called Dulac
maps. The Dulac map is revealed to have similar asymptotic structures as in the
case of a saddle singularity in the plane.
