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Synchronization of complex dynamical networks with nonlinear inner-coupling functions and time delays. (English) Zbl 1094.34056

The subject of the paper is a network of coupled systems with many delays \[ \dot x_i = f(x_i) +\varepsilon \sum_{j=1}^{N} c_{ij} A(x_{j1}(t-\tau),x_{j2}(t-\tau_2),\dots,x_{jn}(t-\tau_n))^T, \] where \(x_i=(x_{i1},\dots,x_{in})^T\in \mathbb R^n\), \(A\) is an \(n\times n\) coupling matrix, \(\varepsilon\) and \(c_{ij}\) are the coupling parameters. The authors establish sufficient conditions for synchronization, i.e., the stability of a synchronous solution \(x(t)=(s(t),\dots,s(t))^T\). They use linear stability analysis and a Lyapunov-Krasovskii functional.

MSC:

34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K19 Invariant manifolds of functional-differential equations
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