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Adaptive synchronization in tree-like dynamical networks. (English) Zbl 1125.34031

The authors investigate the synchronization in three-like dynamical networks, which can be described by the following system of coupled ordinary differential equations

x ˙ i =f(x i )+c j=1 N a ij Γx j ,i=1,,N,

where f(x i )=(f 1 (x i ),,f n (x i )) T : n n , x i =(x i1 ,,x in )R n are the state variables of the nodes, c>0 is the coupling strength, Γ is a diagonal matrix. The structure of the network is described by the coupling matrix A=(a ij ).

The main result reports the possibility of finding a coupling c(x) such that the system will be completely synchronized, i.e. x i (t)-x j (t)0 for t, any i,j and all initial conditions.

34D05Asymptotic stability of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
34H05ODE in connection with control problems
34D23Global stability of ODE