The Construction and Smoothness of Invariant Manifolds by the Deformation Method
Marsden, J. E. and J. Scheurle
SIAM J. Math. Anal., 18, 1261-1274
This paper proves optimal results for the invariant manifold theorems near a fixed point for a mapping (or a differential equation) by using the deformation, or Lie transform, method from singularity theory. The method was inspired by the difficulties encountered by the implicit function theorem technique in the case of the center manifold. The idea here is simply to deform the given system into its linearization and to track this deformation using the flow of a time-dependant vector field. Corresponding to the difficulties with the center manifold encountered by other techniques, we run into a "derivative loss" in this case as well, which is overcome by utilizing estimates on the differentiated equation. A survey of the other methods used in the literature is also presented.