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Synchronization of weighted networks and complex synchronized regions. (English) Zbl 1220.34074
Summary: Since the Laplacian matrices of weighted networks usually have complex eigenvalues, the problem of complex synchronized regions should be investigated carefully. The present Letter addresses this important problem by converting it to a matrix stability problem with respect to a complex parameter, which gives rise to several types of complex synchronized regions, including bounded, unbounded, disconnected, and empty regions. Because of the existence of disconnected synchronized regions, the convexity characteristic of stability for matrix pencils is further discussed. Then, some efficient methods for designing local feedback controllers and inner-linking matrices to enlarge the synchronized regions are developed and analyzed. Finally, a weighted network of smooth Chua’s circuits is presented as an example for illustration.
MSC:
34D06Synchronization
05C82Small world graphs, complex networks (graph theory)
94B10Convolutional codes
94C15Applications of graph theory to circuits and networks
93B52Feedback control