Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay. (English) Zbl 1111.34056

The authors consider the linearly coupled system of delay-differential equations \[ \frac{dx_i(t)}{dt} = f(x_i(t)) + c \sum_{j=1,j\neq i}^{m}a_{ij} \Gamma [x_j(t-\tau)-x_i(t)], \] where \(i=1,\dots,m\), \(x_i(t)\in \mathbb{R}^n\) denotes the state variable of the \(i\)th node, \(\Gamma=\mathrm{diag} \{ \gamma_1,\dots,\gamma_n\}\) is the inner connection matrix with \(\gamma_j\geq 0\) and \(a_{ij}\geq 0\) for all \(i\) and \(j\).
Main results of the paper concern the conditions of complete synchronization, i.e., conditions for the following asymptotic behavior: \(\lim_{t\to \infty} | x_j(t)-x_i(t)| =0\) for all \(i\) and \(j\). In particular, the authors extend the master stability function methodology due to L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar and J. F. Heagy [Chaos 7, 520–543 (1997; Zbl 0933.37030)] and the methodology used by W. Lu and T. Chen [Physica D 213, 214–230 (2006; Zbl 1105.34031)] to delay systems.


34K25 Asymptotic theory of functional-differential equations
34K19 Invariant manifolds of functional-differential equations
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
Full Text: DOI


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