Exponential synchronization of stochastic delayed discrete-time complex networks. (English) Zbl 1172.92002

Summary: This paper is concerned with the problem of synchronization for stochastic discrete-time driven-response networks with time-varying delays. By employing the Lyapunov functional method combined with stochastic analysis as well as feedback control techniques, several sufficient conditions are established that guarantee the exponentially mean-square synchronization of two identical delayed networks with stochastic disturbances. These sufficient conditions, which are expressed in terms of linear matrix inequalities (LMIs), can be solved efficiently by the LMI toolbox in Matlab.
A particular feature of the LMI-based synchronization criteria is that they are dependent not only on the connection matrices in the drive networks and the feedback gains in the response networks, but also on the lower and upper bounds of the time-varying delay, and are therefore less conservative than the delay-independent ones. Two numerical examples are exploited to demonstrate the feasibility and applicability of the proposed synchronization approaches.


93D15 Stabilization of systems by feedback
90B15 Stochastic network models in operations research
93E15 Stochastic stability in control theory


Matlab; LMI toolbox
Full Text: DOI


[1] Boukas, E.K., Al-Muthairi, N.F.: Delay-dependent stabilization of singular linear systems with delays. Int. J. Innov. Comput. Inf. Control 2(2), 283–291 (2006)
[2] Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) · Zbl 0816.93004
[3] Cao, J., Lu, J.: Adaptive synchronization of neural networks with or without time-varying delays. Chaos 16, 013133 (2006) · Zbl 1144.37331
[4] Cao, J., Li, P., Wang, W.W.: Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 353(4), 318–325 (2006)
[5] Chen, B., Lam, J., Xu, S.: Memory state feedback guaranteed cost control for neutral delay systems. Int. J. Innov. Comput. Inf. Control 2(2), 293–303 (2006)
[6] EI Ghaoui, L., Oustry, F., Ait Rami, M.: A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans. Autom. Control 42(8), 1171–1176 (1997) · Zbl 0887.93017
[7] Gao, H., Lam, J., Chen, G.: New criteria for synchronization stability of general complex dynamical networks with coupling delays. Phys. Lett. A. 360, 263–273 (2006) · Zbl 1236.34069
[8] Gao, H., Lam, J., Wang, Z.: Discrete bilinear stochastic systems with time-varying delay: stability analysis and control synthesis. Chaos Solitons Fractals 34, 394–404 (2007) · Zbl 1134.93413
[9] Li, C.G., Chen, G.: Synchronization in general complex dynamical networks with coupling delays. Physica A 343, 263–278 (2004)
[10] Li, Z., Chen, G.: Global synchronization and asymptotic stability of complex dynamical networks. IEEE Trans. Circuits Syst.-II 53(1), 28–33 (2006)
[11] Huang, X., Cao, J.: Generalized synchronization for delayed chaotic neural networks: a novel coupling scheme. Nonlinearity 19, 2797–2811 (2006) · Zbl 1111.37022
[12] Jost, J., Joy, M.: Special properties and synchronization in coupled map lattices. Phys. Rev. E 65, 061201 (2002)
[13] Khasminskii, R.Z.: Stochastic Stability of Differential Equations. Alphen aan den Rijn, Sijthoffand Noor, Khasminskiidhoff (1980)
[14] Leibfritz, F.: An LMI-based algorithm for designing suboptimal static H 2/H output feedback controllers. SIAM J. Control Optim. 57, 1711–1735 (2001) · Zbl 0997.93032
[15] Liu, Y., Wang, Z., Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 19(5), 667–675 (2006) · Zbl 1102.68569
[16] Liu, Y., Wang, Z., Liu, X.: Design of exponential state estimates for neural networks with mixed time delays. Phys. Lett. A 364, 401–412 (2007) · Zbl 05839158
[17] Lu, J.Q., Cao, J.: Synchronization-based approach for parameters identification in delayed chaotic neural networks. Physica A 382, 672–682 (2007)
[18] Lu, W.L., Chen, T.P.: Synchronization of coupled connected neural networks with delays. IEEE Trans. Circuits Syst.-I 51(12), 2491–2503 (2004) · Zbl 1371.34118
[19] Lü, J.H., Chen, G.: A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans. Autom. Control 50, 841–846 (2005) · Zbl 1365.93406
[20] Lü, J.H., Yu, X.H., Chen, G., Cheng, D.Z.: Characterizing the synchronizability of small-world dynamical networks. IEEE Trans. Circuits Syst.-I 51, 787–796 (2004) · Zbl 1374.34220
[21] Michael, B., Perez, J., Martinez-Zuniga, R.: Optimal filtering for nonlinear polynomial systems over linear observations with delay. Int. J. Innov. Comput. Inf. Control 2(4), 863–874 (2006)
[22] Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990) · Zbl 0938.37019
[23] Perez-Munuzuri, V., Perez-Villar, V., Chua, L.O.: Autowaves for image processing on a two-dimensional CNN array of excitable nonlinear circuits: flat and Wrinkled labyrinths. IEEE Trans. Circuits Syst.-I 40, 174–181 (1993) · Zbl 0800.92038
[24] Toroczkai, Z.: Complex networks: the challenge of interaction topology. Los Alamos Sci. (29) 94–109 (2005)
[25] Wang, X.-F., Chen, G.: Synchronization in small-world dynamical networks. Int. J. Bifurc. Chaos 12(1), 187–192 (2002)
[26] Wang, X.-F., Chen, G.: Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans. Circuits Syst.-I 49(1), 54–62 (2002) · Zbl 1368.93576
[27] Wang, Z., Liu, Y., Li, M., Liu, X.: Stability analysis for stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans. Neural Netw. 17(3), 814–820 (2006)
[28] Wang, Z., Liu, Y., Fraser, K., Liu, X.: Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays. Phys. Lett. A 354(4), 288–297 (2006) · Zbl 1181.93068
[29] Wu, C.W.: Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling. IEEE Trans. Circuits Syst.-I 50(2), 294–297 (2003) · Zbl 1368.34051
[30] Wu, C.W.: Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling. IEEE Trans. Circuits Syst.-II 52(5), 282–286 (2005)
[31] Zheleznyak, A., Chua, L.O.: Coexistence of low- and high-dimensional spatio-temporal chaos in a chain of dissipatively coupled Chua’s circuits. Int. J. Bifurc. Chaos 4(3), 639–674 (1994) · Zbl 0900.92015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.