Approximate invariant manifolds up to exponentially small terms

Abstract

This paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspaces E 0 ( dim m 0 ), E 1 ( dim m 1 ). Under light Diophantine conditions on the linear part, we prove that there is a polynomial change of coordinate in E 1 allowing to eliminate, in the E 1 component of the vector field, all terms depending only on the coordinate u 0 ∈ E 0 , up to an exponentially small remainder. This main result enables to prove the existence of analytic center manifolds up to exponentially small terms and extends to infinite-dimensional vector fields. In the elliptic case, our results also proves, with very light assumptions on the linear part in E 1 , that for initial data very close to a certain analytic manifold, the solution stays very close to this manifold for a very long time, which means that the modes in E 1 stay very small.