Skip to main content
Springer Nature Link
Log in

On Synchronization of Coupled Delayed Neural Networks

  • Chapter
Recent Advances in Nonlinear Dynamics and Synchronization

Part of the book series: Studies in Computational Intelligence ((SCI,volume 254))

Abstract

In this chapter, the global synchronization problem is investigated for both continuous- and discrete-time coupled neural networks. The neural networks appear in the form of coupled arrays, where both the linear and nonlinear couplings are taken into account. The activation functions include both the Lipchitz-type and the sector-type ones. Due to the high dimension of the system under consideration, the Kronecker product is utilized to facilitate the derivation and simplify the presentation. By resorting to the matrix functional method, we aim to establish sufficient conditions under which the considered array of neural networks is globally synchronized. It is shown that the globally exponential synchronization can be achieved by suitably designing the coupling matrix, the inner linking matrix and some free matrices representing the relationships between the system matrices. The sufficient conditions obtained that guarantee the synchronization are directly related to several matrix quantities describing the coupling topology. Furthermore, these conditions are expressed in terms of several linear matrix inequalities (LMIs) which can be easily verified by utilizing the numerically efficient Matlab LMI toolbox. Several illustrative examples are given to show the feasibility and applicability of the proposed synchronization scheme.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

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

    MATH  Google Scholar 

  2. Cao, J., Li, P., Wang, W.W.: Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 353, 318–325 (2006)

    Article  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., Yuan, K., Ho, D.W.C., Lam, J.: Global point dissipativity of neural networks with mixed time-varying delays. Chaos 16(1) 013105 (2006)

    Google Scholar 

  5. Cao, J., Chen, G., Li, P.: Global synchronization in an array of delayed neural networks with hybrid coupling. IEEE Trans. Syst. Man Cybern. B 38(2), 488–498 (2008)

    Article  MathSciNet  Google Scholar 

  6. Chen, Y.: Global stability of neural networks with distributed delays. Neural Networks 15, 867–871 (2002)

    Article  Google Scholar 

  7. Chen, J.L., Chen, X.H.: Special Matrices. Tsinghua University Press, China (2001)

    Google Scholar 

  8. Chen, G., Zhou, J., Liu, Z.R.: Global synchronization of coupled delayed neural networks and applications to chaotic CNN models. Int. J. Bifurc. Chaos 14(7), 2229–2240 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, M., Zhou, D.: Synchronization in uncertain complex networks. Chaos 16, 013101 (2006)

    Article  MathSciNet  Google Scholar 

  10. Chua, L.O.: CNN: a Paradigm for Complexity. World Scientific, Singapore (1998)

    MATH  Google Scholar 

  11. Gao, H., Chen, T.: New results on stability of discrete-time systems with time-varying state delay. IEEE Tran. Autom. Control 52(2), 328–334 (2007)

    Article  MathSciNet  Google Scholar 

  12. 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 

  13. Gilli, M.: Strange attractors in delayed cellular neural networks. IEEE Trans. Circuits Syst. -I 40(11), 849–853 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gopalsamy, K., He, X.: Stability in asymmetric Hopfield nets with transmission delays. Physica D 76, 344–358 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gu, K.Q., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhauser, Boston (2003)

    MATH  Google Scholar 

  16. He, Y., Liu, G., Rees, D.: New delay-dependent stability criteria for neural networks with time-varying delay. IEEE Trans. Neural Networks 18(1), 310–314 (2007)

    Article  MathSciNet  Google Scholar 

  17. He, Y., Wu, M., She, J.H.: Delay-dependent exponential stability of delayed neural networks with time-varying delay. IEEE Trans. Circuits Syst.-II 53(7), 553–557 (2006)

    Article  Google Scholar 

  18. Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-stage neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)

    Article  Google Scholar 

  19. Hoppensteadt, F.C., Izhikevich, E.M.: Pattern recognition via synchronization in phase locked loop neural networks. IEEE Trans. Neural Networks 11(3), 734–738 (2000)

    Article  Google Scholar 

  20. Huang, X., Cao, J.: Generalized synchronization for delayed chaotic neural networks: a novel coupling scheme. Nonlinearity 19(12), 2797–2811 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. 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 

  23. Liang, J., Cao, J., Lam, J.: Convergence of discrete-time recurrent neural networks with variable delay. Int. J. Birfurc. Chaos 15(2), 581–595 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Liu, Y., Wang, Z., Liu, X.H.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Networks 19(5), 667–675 (2006)

    Article  MATH  Google Scholar 

  25. Liu, Y., Wang, Z., Serrano, A., Liu, X.: Discrete-time recurrent neural networks with time-varying delays: exponential stability analysis. Phys. Lett. A 362, 480–488 (2007)

    Article  Google Scholar 

  26. Lu, H.T.: Chaotic attractors in delayed neural networks. Phys. Lett. A 298, 109–116 (2002)

    Article  MATH  Google Scholar 

  27. 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 

  28. Mohamad, S., Gopalsamy, K.: Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Appl. Math. Comput. 135(1), 17–38 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  29. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. 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 

  32. Song, Q., Wang, Z.: A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays. Phys. Lett. A (2007)

    Google Scholar 

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

    Article  Google Scholar 

  34. 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 

  35. Wang, Z., Liu, Y., Liu, X.: On global asymptotic stability of neural networks with discrete and distributed delays. Phys. Lett. A 345(4-6), 299–308 (2005)

    Article  MATH  Google Scholar 

  36. 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 

  37. Wang, Z., Liu, Y., Li, M., Liu, X.: Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays. IEEE Trans. Neural Networks 17(3), 814–820 (2006)

    Article  Google Scholar 

  38. Wang, Z., Liu, Y., Yu, L., Liu, X.: Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 356(4-5), 346–352 (2006)

    Article  MATH  Google Scholar 

  39. 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 

  40. 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 

  41. Wu, C.W.: Synchronization in networks of nonlinear dynamical systems coupled via a directed graph. Nonlinearity 18, 1057–1064 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wu, C.W., Chua, L.O.: Synchronization in an array of linearly coupled dynamical systems. IEEE Trans. Circuits Syst.-I 42(8), 430–447 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  43. Xiong, W., Cao, J.: Global exponential stability of discrete-time Cohen-Grossberg neural networks. Neurocomputing 64, 433–446 (2005)

    Article  Google Scholar 

  44. Zhang, J., Suda, Y., Iwasa, T.: Absolutely exponential stability of a class of neural networks with unbounded delay. Neural Networks 17(3), 391–397 (2004)

    Article  MATH  Google Scholar 

  45. Zhao, H.Y.: Global stability of neural networks with distributed delays. Phys. Rev. E 68, 051909 (2003)

    Article  Google Scholar 

  46. 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  MATH  MathSciNet  Google Scholar 

  47. Zhou, J., Chen, T.P.: Synchronization in general complex delayed dynamical networks. IEEE Trans. Circuits Syst.-I 53(3), 733–744 (2006)

    Article  MathSciNet  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, U.K

    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

Editor information

Editors and Affiliations

  1. Institute for Smart-Systems Technologies, Alpen-Adria Universität Klagenfurt , Universitätsstraße 65-67, 9020, Klagenfurt, Austria

    Kyandoghere Kyamakya  & Jean Chamberlain Chedjou  & 

  2. Lehrgebiet Informationstechnik, FernUniversität in Hagen , Universitätsstraße 27 - PRG, 58097, Hagen, Germany

    Wolfgang A. Halang  & Zhong Li  & 

  3. Lehrgebiet Kommunikationsnetze, FernUniversität in Hagen , Universitätsstraße 27 - PRG, 58097, Hagen, Germany

    Herwig Unger

  4. Institute for Nonlinear Science, University of California, San Diego , 9500 Gilman Drive, 92093-0402, La Jolla, CA, USA

    Nikolai F. Rulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Liang, J., Wang, Z., Liu, X. (2009). On Synchronization of Coupled Delayed Neural Networks. In: Kyamakya, K., Halang, W.A., Unger, H., Chedjou, J.C., Rulkov, N.F., Li, Z. (eds) Recent Advances in Nonlinear Dynamics and Synchronization. Studies in Computational Intelligence, vol 254. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04227-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04227-0_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04226-3

  • Online ISBN: 978-3-642-04227-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics