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Synchronization analysis of linearly coupled networks of discrete time systems. (English) Zbl 1071.39011
The authors study coupled map networks of the form $x^i(t+1) = f(x^i(t)) + \sum_{j\neq i}^{m} b_{ij} [f(x^j(t))-f(x^i(t))], \quad i=1,\dots,m,$ where $$x^i(t)= (x_1^i(t),\dots,x_n^i(t))^T \in \mathbb{R}^n$$ is the state variable of the $$i$$-th node, $$t\in \mathbb{N}$$ is the discrete time, $$f:\mathbb{R}^n\to \mathbb{R}^n$$ is continuous, $$B=(b_{ij})$$ is the coupling matrix connecting the nodes, $$b_{ij}>0$$ for all $$i\neq j$$.
As the main result, the criteria are obtained for the synchronization of the subsystems, i.e. conditions when $$\| x^i(t)-x^j(t)\| \to 0$$ as $$t\to\infty$$ for some set of initial conditions. The global as well as local synchronization is considered. The obtained criteria reveal that two factors influence synchronization: dynamical behaviors at each node and coupling configuration.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 93C55 Discrete-time control/observation systems
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