We consider a two-dimensional weakly dissipative dynamical system withtime-periodic coefficients. Their time average is governed by a degenerateHamiltonian whose set of critical points has an interior. The dynamics ofthe system is studied in the presence of three time scales. Using themartingale problem approach and separating the involved time scales, weaverage the system to show convergence to a Markov process on a stratifiedspace. The corresponding strata of the reduced space are a two-sphere, apoint, and a line segment. Special attention is given to the domain of thelimiting generator, including the analysis of the gluing conditions at thepoint where the strata meet. The gluing conditions resulting from thehierarchy of the time scales are similar to the conditions on the domain ofskew Brownian motion.