Synchronization of chaotic systems and invariant manifolds

The theory of normally hyperbolic invariant manifolds (Fenichel theory) can be used to define strict chaotic synchronization in terms of synchronization manifolds, and treat many ideas found in the physics and engineering literature analytically. In the first part of this work we introduce a modification of Fenichel theory which applies to chaotic synchronization and discuss the Lyapunov-exponent-like quantities used to determine the transverse stability of synchronization manifolds. The second part deals with the different methods for detecting synchrony: symmetry considerations, geometric singular perturbation theory and, in the case of uniformly asymptotically stable extensions, graph transforms. We also consider the case for which an extension of a system is only locally uniformly asymptotically stable and show that in such cases n :1 synchrony occurs.