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 synchronizati on manifolds, and treat many ideas found in the physics and engineering lit erature analytically. In the first part of this work we introduce a modific ation of Fenichel theory which applies to chaotic synchronization and discu ss the Lyapunov-exponent-like quantities used to determine the transverse s tability of synchronization manifolds. The second part deals with the diffe rent methods for detecting synchrony: symmetry considerations, geometric si ngular perturbation theory and, in the case of uniformly asymptotically sta ble extensions, graph transforms. We also consider the case for which an ex tension of a system is only locally uniformly asymptotically stable and sho w that in such cases n : 1 synchrony occurs.