Regularity of invariant graphs for forced systems

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space; this is a special case of the well known concept of an inertial manifold for more general systems. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set is a normally hyperbolic manifold, and hence smooth. The aim of this paper is to extend this result in two directions: firstly, where we only have uniform contraction for a compact invariant set of input states, and secondly where the contraction rates are non-uniform (and hence defined by Liapunov exponents and analogous quantities). In both cases the invariant graph is only defined over closed subsets of the input space, and hence we need to define an appropriate notion of smoothness for such functions. This is done in terms of the Whitney extension theorem: a function is considered Whitney smooth if it satisfies the conditions of this theorem and hence can be extended to a smooth function of the whole input space.