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Discrete Dynamics in Nature and Society
Volume 4 (2000), Issue 3, Pages 245-256

Invariant manifolds and cluster synchronization in a family of locally coupled map lattices

1Department of Mathematics, Volga State Academy, 5, Nesterov Street, Nizhny Novgorod 603 600, Russia
2Department of Differential Equations, Institute of Applied Mathematics, 10, Ul'yanov Street, Nizhny Novgorod 603 005, Russia
3Department of Physics, Technical University of Denmark, Lyngby 2800, Denmark

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents an analysis of the invariant manifolds for a general family of locally coupled map lattices. These manifolds define the different types of full, partial, and anti-phase chaotic synchronization that can arise in discrete dynamical systems. Existence of various invariant manifolds, self-similarity as well as orderings and embeddings of the manifolds of a coupled map array are established. A general variational equation for the stability analysis of invariant manifolds is derived, and stability conditions for full and partial chaotic synchronization of concrete coupled maps are obtained. The general results are illustrated through examples of three coupled two-dimensional standard maps with damping.