Global Synchronization In Coupled Map Lattices

This paper presents a global synchronization theorem for coupled map lattices. Roughly speaking the theorem states that the coupled map lattice x_{n+1} = AF(x_n) synchronizes if A has an eigenvalue 1 of multiplicity 1 corresponding to the synchronization manifold and the other eigenvalues of A are close to zero. Examples of coupled map lattices of logistic maps are used to illustrate the result. In particular, we give global results regarding synchronization in coupled map lattices for which previously only numerical evidence and local results were available. We show that (1) globally coupled maps synchronize if the coupling is large enough, (2) randomly coupled maps are synchronized if the number of couplings for each map is large enough and (3) coupled maps connected on a graph will synchronize if the ratio between the largest and the smallest nonzero eigenvalue of the Laplacian matrix of the graph is small.