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Zbl 0867.93042
Wu, Chai Wah; Chua, Leon O.
Synchronization in an array of linearly coupled dynamical systems.
(English)
[J] IEEE Trans. Circuits Syst., I, Fundam. Theory Appl. 42, No.8, 430-447 (1995). ISSN 1057-7122

Consider a system of {\it mn} ODEs $${\dot{\bold x}_i}={{\bold f}_i}({{\bold x}_i},t)+ \sum_{j=1}^m{{\bold D}_{ij}{\bold x}_j}, \quad \text {for } i=1,...,n,$$ where ${{\bold x}_i}=(x_{i1}, ...,x_{in})^T, {{\bold x}_i}\in R^n, {{\bold f}'_i}$s are uniformly Lipschitz continuous, ${{\bold D}_{ij}}$ are $n \times n$ matrices. Various notions of synchronization are defined, which means that the trajectories of all the cells approach each other. Sufficient conditions for asymptotic synchronization of linearly coupled identical dynamic systems are proved and discussed for various coupling configurations using the main results of Lyapunov's direct method. The considered coupling configurations are described by symmetric matrices, irreducible matrices, normal matrices, circulant matrices, nonnegative matrices including their perturbations. Sufficient conditions for additive nonlinear coupling to make the synchronized state asymptotically stable are given.
[L.Bakule (Praha)]
MSC 2000:
*93C15 Control systems governed by ODE
93A15 Large scale systems
93D20 Asymptotic stability of control systems

Keywords: asymptotic stability; synchronization; coupling

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