Synchronization analysis of linearly coupled networks of discrete time systems

In this paper, we present a framework for analysis of synchronization of linearly coupled map lattices (LCMLs). The dynamics at each node of the coupled map lattices is described by a general difference equation, which can be of chaotic or others. As for coupling configuration of the linearly coupled map lattices, it is not assumed that the coupling matrix is symmetric and irreducible. Criteria are obtained for both local and global synchronization. These criteria reveal that two factors influence synchronization: dynamical behaviors at each node and coupling configuration. By these criteria, one can verify whether the LCML is synchronized, which is important and useful for both understanding and interpreting synchronization phenomena and designing coupling configuration. A scale “d” is presented to measure the synchronization capability of a linearly coupled map lattice. As we show, “d” can be regarded as a function of the smallest and the largest eigenvalues of the coupling matrix. Various kinds of coupling schemes, including complex networks such as small-world (SW) lattice and scale-free network, are studied. The quantity “d” is carefully analyzed and calculated for each case.