Lipschitz stable manifolds

We want to construct stable and unstable invariant manifolds without assuming the existence of a uniform exponential dichotomy for the linear variational equation. Our main objective is to describe the weakest possible setting under which one can construct the invariant manifolds.We still require some amount of hyperbolicity. Namely, we show that under fairly general assumptions the generalized notion of nonuniform exponential dichotomy allows us to establish the existence of stable and unstable invariant manifolds. In this chapter we only consider “Lipschitz manifolds”, that is, graphs of Lipschitz functions. We refer to Chapters 5 and 6 for the existence of smooth invariant manifolds (respectively in Rn and in arbitrary Banach spaces), under slightly stronger assumptions. We follow closely [12], although now considering the general case when the stable and unstable subspaces may depend on the time t. Lipschitz center manifolds were obtained with a similar approach in [8]; we refer to Chapter 8 for the construction of smooth center manifolds.