Neural Computing and Applications

, Volume 19, Issue 4, pp 543–548

Algebraic condition of synchronization for multiple time-delayed chaotic Hopfield neural networks

Original Article


In this paper, an easy and efficient method is brought forward to design the feedback control for the synchronization of two multiple time-delayed chaotic Hopfield neural networks, whose activation functions and delayed activation functions can have different forms of mapping. Without many complex restrictions and Lyapunov analytic process, the feedback control is given based on the M-matrix theory, the system parameters and the feedback section coefficients. All the results are simulated by Matlab and Simulink, which shows the simplicity and validity of the control. As shown in the simulation results, the error systems converge to zero rapidly.


Synchronization Chaos Hopfield neural network Time delays Algebraic condition 


  1. 1.
    Li R, Wang W, Tseng H (1999) Detection and location of objects from mobile mapping image sequences by Hopfield neural networks. Photogramm Eng Remote Sensing 65:1199–1205Google Scholar
  2. 2.
    Engin M, Demirağ Serdar, Engin EZ (2007) The classification of human tremor signals using artificial neural network. Expert Syst Appl 33(3):754–761CrossRefGoogle Scholar
  3. 3.
    Hsiao FH, Liang YW, Xu SD (2007) Decentralized stabilization of neural network linearly interconnected systems via t-s fuzzy control. J Dyn Syst Meas Control 129(3):343–351CrossRefGoogle Scholar
  4. 4.
    Kaslik E, Balint St (2006) Configurations of steady states for Hopfield-type neural networks. Appl Math Comput 82(1):934–946CrossRefMathSciNetGoogle Scholar
  5. 5.
    Liu Y, Wei ZJ, Chen JX (2005) Reaction condition optimization of catalytic hydrogenation of m-dinitrobenzene by bp neural network (in Chinese). Chin J Catal 26(1):20–24Google Scholar
  6. 6.
    Jiang HL, Sun YM (1997) Fault-tolerance analysis of neural network for high voltage transmission line fault diagnosis. Proceedings of the 4th International Conference on advance in power system control, operation and management, APSCOM-97, Hong Kong, pp 433–438Google Scholar
  7. 7.
    Wang SH (2005) Classification with incomplete survey data: a Hopfield neural network approach. Comput Oper Res 32(10):2583–2594MATHCrossRefGoogle Scholar
  8. 8.
    Liao XX, Liao Y (1997) Stability of Hopfield type neural networks (II). Sci China (Series A) 40(8):813–816MATHCrossRefGoogle Scholar
  9. 9.
    Lu JG (2007) Robust global exponential stability for interval reaction–diffusion Hopfield neural networks with distributed delays. IEEE Trans Circuits Syst II Express Briefs 54(12):1115–1119CrossRefGoogle Scholar
  10. 10.
    Song QK, Wang ZD (2008) Stability analysis of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. Physica A 387(13):3314–3326CrossRefGoogle Scholar
  11. 11.
    Cao JD, Tao Q (2001) Estimation on domain of attraction and convergence rate of Hopfield continuous feedback neural networks. J Comput Syst Sci 62(3):528–534MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cao JD (2004) An estimation of the domain of attraction and convergence rate for Hopfield continuous feedback neural networks. Phys Lett A 325(5–6):370–374MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    He HL, Wang ZS, Liao XX (2005) Stability analysis of uncertain neural networks with linear and nonlinear time delays. Lect Notes Comput Sci 3496:199–202CrossRefGoogle Scholar
  14. 14.
    Kaslik E, Balint S (2007) Bifurcation analysis for a two-dimensional delayed discrete-time Hopfield neural network. Chaos Solitons Fractals 34(4):1245–1253MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Yan H, Yang XS (2006) Hyperchaos and bifurcation in a new class of four-dimensional Hopfield neural networks. Neurocomputing 69(13–15):1787–1795Google Scholar
  16. 16.
    Aihara K, Takable T, Toyoda M (1990) Chaotic neural networks. Phys Lett A 144(7):333–340CrossRefMathSciNetGoogle Scholar
  17. 17.
    Milanovic V, Zaghloul ME (1996) Synchronization of chaotic neural networks for secure communications. Circuits and systems, 1996. ISCAS ‘96, ‘Connecting the World’. 1996 IEEE International Symposium 3:28–31Google Scholar
  18. 18.
    Cheng CJ, Liao TL, Hwang CC (2005) Exponential synchronization of a class of chaotic. Chaos Solitons Fractals 24:197–206MATHMathSciNetGoogle Scholar
  19. 19.
    Lou XY, Cui BT (2008) Anti-synchronization of chaotic delayed neural networks. Acta Physica Sinica 57(4):2060–2067MATHMathSciNetGoogle Scholar
  20. 20.
    Meng J, Wang XY (2008) Robust anti-synchronization of a class of delayed chaotic neural networks. Chin J Comput Phys 25(2):247–252Google Scholar
  21. 21.
    Li P, Cao JD, Wang ZD (2007) Robust impulsive synchronization of coupled delayed neural networks with uncertainties. Physica A 373(Complete):261–272CrossRefGoogle Scholar
  22. 22.
    Liao XX, Xiao DM (2000) Globally exponential stability of Hopfield neural networks with time-varying delays. Acta Electronica Sinica 28(4):87–90Google Scholar
  23. 23.
    Liao XX (2001) Mathematical theory and application of stability (in Chinese). Huazhong Normal University Press, WuhanGoogle Scholar
  24. 24.
    He HL (2007) Numeral analysis (in Chinese). Science Press, PekinGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  1. 1.College of ScienceNaval University of EngineeringWuhanChina

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