Normally elliptic singular perturbations and persistence of homoclinic orbits

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.