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Synchronization in complex dynamical networks with nonsymmetric coupling. (English) Zbl 1178.34056

Subject of the paper are networks of coupled systems of the form \[ x_i'(t)=f(x_i(t))+\sum_{j=1}^N g_{ij}Ax_j(t),\quad i=1,\dots,N, \] where \(x_i(t)\in\mathbb{R}^n\) is the state variable of one node of the network, \(G=(g_{ij})\) is the coupling matrix. The authors extend the master stability method [L. Pecora, T. Carroll, G. Johnson, D. Mar, and J. Heagy, Chaos 7, 520–543 (1997; Zbl 0933.37030)] to obtain criteria for global synchronization, i.e. global stability of the invariant subspace \(\{ (x_1,\dots,x_N)|\;x_1=x_2=\cdots=x_N \}\). Using the technique of T. Nishikawa and A. E. Motter [Physica D 224, No. 1–2, 77–89 (2006; Zbl 1117.34048)], the authors include also the case, when the coupling matrix is nondiagonalizable.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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