  We consider a dynamical system, possibly infinite dimensional or
non-autonomous, with fast and slow time scales which is oscillatory with high
frequencies in the fast directions. We first derive and justify the limit
system of the slow variables. Assuming a steady state persists, we construct
the stable, unstable, center-stable, center-unstable, and center manifolds of
the steady state of a size of order O(1) and give their leading order
approximations. Finally, using these tools, we study the persistence of
homoclinic solutions in this type of normally elliptic singular perturbation
problems.
