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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))+ j=1 N g ij Ax j (t),i=1,,N,

where x i (t) 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 ,,x N )|x 1 =x 2 ==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.

34D05Asymptotic stability of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
34D23Global stability of ODE