Further improvement on synchronization stability of complex networks with coupling delays. (English) Zbl 1183.34116
The subject of the paper is the following system of coupled delay differential equations (dynamical network)
\[ \dot u_i = g(u_i) + c \sum_{j=1}^N G_{ij} Au_j(t-\tau),\quad i=1,2,\dots,N, \]
where \(u_i\in \mathbb{R}^n\) \(g:\mathbb{R}^n\to \mathbb{R}^n\) is a continuously differentiable function, \(\tau>0\) is the time delay, \(G=(G_{ij})_{N\times N}\) is the coupling matrix of the network, which contains only 1 or 0 as elements. In addition, the following condition is satisfied
\[ G_{ii}=-\sum_{j=1,j\neq i}^n G_{ij}=-\sum_{j=1,j\neq i}^n G_{ji}, \]
which implies the existence of an invariant subspace of synchronous solutions, on which \(u_1=u_2=\cdots=u_N\).
The main result of this paper gives new conditions for stability of the subspace of synchronous solutions. Methods of the proof are based on a delay fractioning approach and a Lyapunov-Krasovski functional.
\[ \dot u_i = g(u_i) + c \sum_{j=1}^N G_{ij} Au_j(t-\tau),\quad i=1,2,\dots,N, \]
where \(u_i\in \mathbb{R}^n\) \(g:\mathbb{R}^n\to \mathbb{R}^n\) is a continuously differentiable function, \(\tau>0\) is the time delay, \(G=(G_{ij})_{N\times N}\) is the coupling matrix of the network, which contains only 1 or 0 as elements. In addition, the following condition is satisfied
\[ G_{ii}=-\sum_{j=1,j\neq i}^n G_{ij}=-\sum_{j=1,j\neq i}^n G_{ji}, \]
which implies the existence of an invariant subspace of synchronous solutions, on which \(u_1=u_2=\cdots=u_N\).
The main result of this paper gives new conditions for stability of the subspace of synchronous solutions. Methods of the proof are based on a delay fractioning approach and a Lyapunov-Krasovski functional.
Reviewer: Sergiy Yanchuk (Berlin)
MSC:
34K20 | Stability theory of functional-differential equations |
34K06 | Linear functional-differential equations |
34K25 | Asymptotic theory of functional-differential equations |
34D06 | Synchronization of solutions to ordinary differential equations |
Keywords:
complex dynamical networks; delay fractioning; linear matrix inequality; synchronization stability; time-varying structured uncertainties; Lyapunov-Krasovskii functionalReferences:
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