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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

dx i (t) dt=f(x i (t))+c j=1,ji m a ij Γ[x j (t-τ)-x i (t)],

where i=1,,m, x i (t) n denotes the state variable of the ith node, Γ= diag {γ 1 ,,γ n } is the inner connection matrix with γ j 0 and a ij 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 |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.

34K25Asymptotic theory of functional-differential equations
34K19Invariant manifolds (functional-differential equations)
34K23Complex (chaotic) behavior of solutions of functional-differential equations