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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 $$ \dot x_i = f(x_i) + c \sum_{j=1}^{N} a_{ij} \Gamma x_j, \quad i=1,\dots, N, $$ where $f(x_i)=(f_1(x_i),\dots,f_n(x_i))^T$: $\Bbb R^n\to \Bbb R^n$, $x_i=(x_{i1},\dots,x_{in})\in R^n$ are the state variables of the nodes, $c>0$ is the coupling strength, $\Gamma$ 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. $\Vert x_i(t)-x_j(t)\Vert \to 0$ for $t\to \infty$, any $i,j$ and all initial conditions.

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