Center manifolds for smooth invariant manifolds

Abstract

We study dynamics of flows generated by smooth vector fields in R n {\mathbb {R}}^n in the vicinity of an invariant and closed smooth manifold Y Y . By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of Y Y ) based on the information of the linearization along Y Y , which contains every locally bounded solution and is persistent under small perturbations.