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Generalized connection graph method for synchronization in asymmetrical networks. (English) Zbl 1118.34044

The paper considers the network of identical oscillators

x i ' =F(x i )+ k=1 n d ik (t)Px k ,i=1,,n,

where x i l , F: l l , P is a projection matrix that selects the components of x i that are involved in the coupling. The connection matrix D={d ij (t)} is assumed to satisfy

k d ik =0,d ii =- ki d ik ,i=1,,n·

The main result provides conditions for global complete synchronization of this network, i.e. x i (t)-x j (t)0 for all i,j and t. These conditions extend the results of the authors in [Physica D 195, No. 1–2, 159–187 (2004; Zbl 1098.82622)] to the case of arbitrary asymmetrical connection matrix D.

34D05Asymptotic stability of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C28Complex behavior, chaotic systems (ODE)
34C30Manifolds of solutions of ODE (MSC2000)
34C60Qualitative investigation and simulation of models (ODE)
34D20Stability of ODE
34C14Symmetries, invariants (ODE)