  Important models for immortal solutions of Ricci flow that collapse with
bounded curvature come from locally G-invariant solutions on principal bundles,
where G is a nilpotent Lie group. In this paper, we establish convergence and
asymptotic stability, modulo smooth finite-dimensional center manifolds, of
certain R^{N}-invariant solutions. When the dimension of the total space is
three, these results are relevant to work of Lott classifying the asymptotic
behavior of all 3-dimensional Ricci flow solutions whose sectional curvatures
and diameters are respectively O(t^{-1}) and O(t^{1/2}).
