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Zbl 1111.34056
Lu, Wenlian; Chen, Tianping; Chen, Guanron
Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay. (English)
[J] Physica D 221, No. 2, 118-134 (2006). ISSN 0167-2789

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

MSC 2000:: *34K25 Asymptotic theory of functional-differential equations
34K19 Invariant manifolds
34K23 Complex (chaotic) behavior of solutions of FDE

Keywords: synchronization; delay; linearly coupled; master stability function

Citations: Zbl 0933.37030; Zbl 1105.34031

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