Center manifolds in Banach spaces

Center manifold theorems are powerful tools in the analysis of the behavior of dynamical systems. For example, when the equation v´ = A(t)v has a (uniformly) partially hyperbolic behavior with no unstable directions, then under some mild additional assumptions all solutions of v´ = A(t)v + f(t, v) converge exponentially to the center manifold. Hence, the stability of the system is completely determined by the behavior on the center manifold. Therefore, one often considers a reduction to the center manifold. This has also the advantage of reducing the dimension of the system. Furthermore, since one often needs to approximate the center manifolds to sufficiently high order, it is also important to discuss their regularity. Our main goal is to establish the existence of smooth invariant center manifolds in the presence of nonuniformly partially hyperbolic behavior. The method of proof is inspired in the arguments of Chapter 6. In particular, the smoothness of the center manifolds is obtained with a single fixed point problem, instead of one for each additional derivative. We follow closely [15], now with arbitrary stable and unstable subspaces.