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We show that many coupled oscillator array configurations considered in the literature can be put into a simple form so that determining the stability of the synchronous state can be done by a master stability function, which can be tailored to one's choice of stability requirement. This solves, once and for all, the problem of synchronous stability for any… (More)

- J J Hopfield, A V M Herz, J F Heagy, L M Pecora, T L Caroll, T Yang +4 others
- 2000

That is, the system has two symmetric periodic attractors, one of which is shown in Fig. 2(c). In this lemma, we can see an essential function of the ICC that makes stable dynamics by averaging two expanding maps with opposite slopes (d=d x)f (x; 1) > 1, (d=d x)f (x; 01) < 01, and 1=2j(d=d x)f (x; 1) + (d=d x)f (x; 01)j < 1 for x a < j xj < x b. Then Lemma… (More)

453 tion) was-9 times larger than the average count for LMFE based upon the standard matrix inversion. This is a conservative estimate, as the computational load of LMFE is amenable to further improvement using inversion algorithms that exploit the centrosymmetric character [8] of the regularized modified covariance matrix.

- Louis M. Pecora, Thomas L. Carroll, Gregg A. Johnson, Douglas J. Mar, James F. Heagy
- 1997

The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and ''cottage industries'' have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic… (More)

Master-stability functions (MSFs) are fundamental to the study of synchronization in complex dynamical systems. For example, for a coupled oscillator network, a necessary condition for synchronization to occur is that the MSF at the corresponding normalized coupling parameters be negative. To understand the typical behaviors of the MSF for various chaotic… (More)

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has… (More)

We introduce the theory of identical or complete synchronization of identical oscillators in arbitrary networks. In addition, we introduce several graph theory concepts and results that augment the synchronization theory and tie is closely to random, semirandom, and regular networks. We then use the combined theories to explore and compare three types of… (More)

A definition of synchronization of coupled dynamical systems is provided. We discuss how such a definition allows one to identify a unifying framework for synchronization of dynamical systems, and show how to encompass some of the different phenomena described so far in the context of synchronization of chaotic systems.

Synchronization is of central importance in power distribution, telecommunication, neuronal and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form. Here we present a new framework and develop techniques for the… (More)