Local and global exponential synchronization of complex delayed dynamical networks with general topology
Jin-Liang Wang Zhi-Chun Yang Tingwen Huang Mingqing Xiao
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 393-408 doi: 10.3934/dcdsb.2011.16.393
In this paper, we consider a generalized complex network possessing general topology, in which the coupling may be nonlinear, time-varying, nonsymmetric and the elements of each node have different time-varying delays. Some criteria on local and global exponential synchronization are derived in form of linear matrix inequalities (LMIs) for the complex network by constructing suitable Lyapunov functionals. Our results show that the obtained sufficient conditions are less conservative than ones in previous publications. Finally, two numerical examples and their simulation results are given to illustrate the effectiveness of the derived results.
keywords: Complex networks time-varying delays exponential synchronization.
Realization of joint spectral radius via Ergodic theory
Xiongping Dai Yu Huang Mingqing Xiao
Electronic Research Announcements 2011, 18(0): 22-30 doi: 10.3934/era.2011.18.22
Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.
keywords: random product of matrices joint spectral radius. The finiteness conjecture

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