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:
| [1] | DOI: 10.1016/S0167-2789(03)00174-X · Zbl 1039.94016 · doi:10.1016/S0167-2789(03)00174-X |
| [2] | DOI: 10.1080/00207160412331284105 · Zbl 1066.65160 · doi:10.1080/00207160412331284105 |
| [3] | DOI: 10.1016/j.physleta.2005.12.092 · doi:10.1016/j.physleta.2005.12.092 |
| [4] | DOI: 10.1109/TCSII.2006.882363 · doi:10.1109/TCSII.2006.882363 |
| [5] | DOI: 10.1016/j.physa.2004.01.056 · doi:10.1016/j.physa.2004.01.056 |
| [6] | DOI: 10.1109/TAC.2006.890320 · Zbl 1366.39011 · doi:10.1109/TAC.2006.890320 |
| [7] | DOI: 10.1016/j.physleta.2006.08.033 · Zbl 1236.34069 · doi:10.1016/j.physleta.2006.08.033 |
| [8] | DOI: 10.1109/TSP.2004.827188 · Zbl 1369.93175 · doi:10.1109/TSP.2004.827188 |
| [9] | DOI: 10.1007/978-1-4612-0039-0 · doi:10.1007/978-1-4612-0039-0 |
| [10] | DOI: 10.1109/TAC.2006.887907 · Zbl 1366.34097 · doi:10.1109/TAC.2006.887907 |
| [11] | DOI: 10.1109/TCSI.2006.883876 · Zbl 1374.37128 · doi:10.1109/TCSI.2006.883876 |
| [12] | Kolmanovskii V., Introduction to the Theory and Applications of Functional Differential Equations (1999) · Zbl 0917.34001 |
| [13] | DOI: 10.1016/j.physa.2004.05.058 · doi:10.1016/j.physa.2004.05.058 |
| [14] | DOI: 10.1109/TCSI.2003.818611 · Zbl 1368.37087 · doi:10.1109/TCSI.2003.818611 |
| [15] | DOI: 10.1109/TCSII.2005.854315 · doi:10.1109/TCSII.2005.854315 |
| [16] | DOI: 10.1109/TAC.2005.849233 · Zbl 1365.93406 · doi:10.1109/TAC.2005.849233 |
| [17] | DOI: 10.1103/PhysRevE.68.045104 · doi:10.1103/PhysRevE.68.045104 |
| [18] | DOI: 10.1140/epjb/e2004-00112-3 · doi:10.1140/epjb/e2004-00112-3 |
| [19] | DOI: 10.1109/81.974874 · Zbl 1368.93576 · doi:10.1109/81.974874 |
| [20] | DOI: 10.1109/MCAS.2003.1228503 · doi:10.1109/MCAS.2003.1228503 |
| [21] | DOI: 10.1109/78.912924 · doi:10.1109/78.912924 |
| [22] | DOI: 10.1109/TNN.2006.872355 · doi:10.1109/TNN.2006.872355 |
| [23] | DOI: 10.1016/j.physleta.2005.07.025 · Zbl 1345.92017 · doi:10.1016/j.physleta.2005.07.025 |
| [24] | DOI: 10.1109/TAC.2006.880794 · Zbl 1366.90065 · doi:10.1109/TAC.2006.880794 |
| [25] | DOI: 10.1016/j.nonrwa.2005.10.004 · Zbl 1122.34065 · doi:10.1016/j.nonrwa.2005.10.004 |
| [26] | DOI: 10.1080/00207179608921866 · Zbl 0841.93014 · doi:10.1080/00207179608921866 |
| [27] | DOI: 10.1109/TAC.2004.841935 · Zbl 1365.93377 · doi:10.1109/TAC.2004.841935 |
| [28] | DOI: 10.1109/TSP.2004.836535 · Zbl 1370.93111 · doi:10.1109/TSP.2004.836535 |
| [29] | DOI: 10.1109/TCSI.2005.859050 · Zbl 1374.37056 · doi:10.1109/TCSI.2005.859050 |
| [30] | DOI: 10.1109/TAC.2006.872760 · Zbl 1366.93544 · doi:10.1109/TAC.2006.872760 |
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