A New Approach to Synchronization Analysis of Linearly Coupled Map Lattices*

In this paper, a new approach to analyze synchronization of linearly coupled map lattices (LCMLs) is presented. A reference vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ifmmode\expandafter\hat\else\expandafter\^\fi{x} $$\end{document}(t) is introduced as the projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating the difference between the trajectory and the projection. By this method, some criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to the eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for the coupled system. Moreover, it is revealed that the stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense. That is, the solution of the coupled system does not converge to a certain knowable s(t) satisfying s(t+1) = f(s(t)) but to the reference vector on the synchronization manifold, which in fact is a certain weighted average of each xi(t) for i = 1, ⋯ ,m, but not a solution s(t) satisfying s(t + 1) = f(s(t)).