Skip to main content
SpringerLink
Log in

Exponential synchronization of stochastic delayed discrete-time complex networks

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper is concerned with the problem of synchronization for stochastic discrete-time drive-response networks with time-varying delay. By employing the Lyapunov functional method combined with the stochastic analysis as well as the feedback control technique, 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Google Scholar 

  2. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

    MATH  Google Scholar 

  3. Cao, J., Lu, J.: Adaptive synchronization of neural networks with or without time-varying delays. Chaos 16, 013133 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, C.G., Chen, G.: Synchronization in general complex dynamical networks with coupling delays. Physica A 343, 263–278 (2004)

    Article  MathSciNet  Google Scholar 

  10. Li, Z., Chen, G.: Global synchronization and asymptotic stability of complex dynamical networks. IEEE Trans. Circuits Syst.-II 53(1), 28–33 (2006)

    Article  Google Scholar 

  11. Huang, X., Cao, J.: Generalized synchronization for delayed chaotic neural networks: a novel coupling scheme. Nonlinearity 19, 2797–2811 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jost, J., Joy, M.: Special properties and synchronization in coupled map lattices. Phys. Rev. E 65, 061201 (2002)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  17. Lu, J.Q., Cao, J.: Synchronization-based approach for parameters identification in delayed chaotic neural networks. Physica A 382, 672–682 (2007)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Google Scholar 

  22. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  24. Toroczkai, Z.: Complex networks: the challenge of interaction topology. Los Alamos Sci. (29) 94–109 (2005)

    Google Scholar 

  25. Wang, X.-F., Chen, G.: Synchronization in small-world dynamical networks. Int. J. Bifurc. Chaos 12(1), 187–192 (2002)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  29. Wu, C.W.: Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling. IEEE Trans. Circuits Syst.-I 50(2), 294–297 (2003)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Jinling Liang

  2. Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

    Jinling Liang, Zidong Wang & Xiaohui Liu

Authors
  1. Jinling Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zidong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaohui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zidong Wang.

Additional information

This work was supported by the Royal Society Sino-British Fellowship Trust Award of the United Kingdom.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liang, J., Wang, Z. & Liu, X. Exponential synchronization of stochastic delayed discrete-time complex networks. Nonlinear Dyn 53, 153–165 (2008). https://doi.org/10.1007/s11071-007-9303-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-007-9303-5

Keywords

Search

Navigation