Invariant Manifolds

Summary

This longest and technically most complicated chapter of the book is devoted to the theory of invariant manifolds of RDS. Our aim is to give an up-to-date picture which is as complete and as reliable as possible.

For the whole chapter we use a “dynamical” method which today is widely used in the deterministic case and was carried over to RDS by Wanner [340]. The idea is to work with a scale of Banach spaces of orbits with certain exponential growth rates by which invariant manifolds can be characterized. The advantage of this method is (besides yielding global invariant manifolds and their regularity) that it also gives the Hartman-Grobman theorem for RDS, thus technically unifying the whole chapter.

After explaining the problem (Sect. 7.1) and doing various preparations to simplify the main work (Sect. 7.2), the long Sect. 7.3 is devoted to the construction of global invariant manifolds (Theorem 7.3.1 for unstable manifolds, Theorem 7.3.10 for stable manifolds and Theorems 7.3.14 and 7.3.17 for center manifolds). The moral of the two key technical lemmas 7.3.3 and 7.3.6 is that all one has to do is solving affine difference equations by the variation of constants formula.

The outcome of these constructions are invariant manifolds which are just Lipschitz-continuous no matter how smooth the RDS under consideration is. We will improve the regularity by playing with the parameters in the scale of Banach spaces. To accomplish e. g. C ^k stable manifolds for a C ^k RDS one has to place two parameters a < b into the gap between the two Lyapunov exponents at which the spectrum is split such that also ka < b — which for k > 1 is not always possible and is the origin of the “gap conditions” (Theorem 7.3.19).

In Sect. 7.4 we complete the picture by showing that the splitting of the Lyapunov spectrum between any two exponents yields foliations of invariant manifolds (Theorem 7.4.1). These can be used as nonlinear coordinates to topologically decouple the nonlinear RDS into blocks according to the (linear) splitting of the MET (Theorem 7.4.4). There is just one more step to (topologically) linearize all blocks having non-zero Lyapunov exponents (Hartman-Grobman Theorem 7.4.12).

Due to very restrictive conditions, global invariant manifolds rarely exist. In contrast, local invariant manifolds usually do exist (Theorem 7.5.5), and some can be dynamically characterized and extended to global objects (Theorem 7.5.13).

We hope that after all this the reader will enjoy studying the examples in Sect. 7.6.