Smooth center manifolds for nonuniformly partially hyperbolic trajectories

We establish the existence of unique smooth center manifolds for ordinary differential equations v′=A(t)v+f(t,v) in Banach spaces, assuming that v′=A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.