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We consider a dynamical system, possibly infinite dimensional ornon-autonomous, with fast and slow time scales which is oscillatory with highfrequencies in the fast directions. We first derive and justify the limitsystem of the slow variables. Assuming a steady state persists, we constructthe stable, unstable, center-stable, center-unstable, and center manifolds ofthe steady state of a size of order O(1) and give their leading orderapproximations. Finally, using these tools, we study the persistence ofhomoclinic solutions in this type of normally elliptic singular perturbationproblems.

Normally Elliptic Singular Perturbations and Persistence of Homoclinic Orbits