Center manifolds for nonautonomous dynamics via evolution semigroups

Abstract

Using nonlinear evolution semigroups, we construct center-stable, center-unstable and center invariant manifolds for a nonautonomous dynamics in a simple manner. Our approach consists essentially of two steps: after introducing a nonlinear evolution semigroup for the original dynamics, we show that the problem of finding center-stable invariant manifolds is equivalent to a corresponding problem for the nonlinear semigroup. As a nontrivial consequence, we show that the regularity properties of the invariant manifolds for the evolution family and for the evolution semigroup are the same. The main novelty of our work is the method of proof, which leads to quite simple arguments for important results.