Contraction principle and its applications in synchronization of nonlinearly coupled systems

Contraction principle is introduced in this paper. Based on such a principle, a rigorous criterion for stability of the synchronous manifold in nonlinearly-coupled identical systems for both chaotic and nonchaotic cases is derived. Such a criterion can be straightforwardly applied to arbitrarily nonlinearly coupled systems (whether regular or random, low or high dimensional, directed or undirected). Moreover, the synchrony guaranteed by the criterion is global and can be against the effect of noise.