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

Lu, Jianquan; Cao, Jinde

Nonlinear Anal., Real World Appl. 8, No. 4, 1252-1260 (2007).

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 \(f(x_i)=(f_1(x_i),\dots,f_n(x_i))^T\): \(\mathbb R^n\to \mathbb R^n\), \(x_i=(x_{i1},\dots,x_{in})\in R^n\) 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. \(\| x_i(t)-x_j(t)\| \to 0\) for \(t\to \infty\), any \(i,j\) and all initial conditions.

Reviewer: Sergiy Yanchuk (Berlin)

MSC:

34D05	Asymptotic properties of solutions to ordinary differential equations
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
34H05	Control problems involving ordinary differential equations
34D23	Global stability of solutions to ordinary differential equations

Keywords:

synchronization; thee-like network; adaptive control

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