Exponential synchronization for arrays of coupled neural networks with time-delay couplings

  • Tao LiEmail author
  • Ting Wang
  • Ai-guo Song
  • Shu-min Fei
Technical Notes and Correspondence


This paper deals with global exponential synchronization in arrays of coupled delayed neural networks with both delayed coupling and one single delayed one. Through employing Kronecker product and convex combination technique, two novel synchronization criteria are presented in terms of linear matrix inequalities (LMIs), and these conditions are dependent on the bounds of both time-delay and its derivative. Through employing Matlab LMI Toolbox and adjusting some matrix parameters in the derived results, we can realize the design and applications of the addressed systems, which shows that our methods improve and extend those reported methods. The efficiency and applicability of the proposed results can be demonstrated by three numerical examples with simulations.


Coupled neural networks exponential synchronization LMI approach Lyapunov-Krasovskii functional time-varying delay 


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  1. [1]
    L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, No. 8, pp. 821–824, 1990.CrossRefMathSciNetGoogle Scholar
  2. [2]
    T. L. Carroll and L. M. Pecora, “Synchronization chaotic circuits,” IEEE Trans. on Circuits and Systems I, vol. 38, no. 4, pp. 453–456, 1991.CrossRefGoogle Scholar
  3. [3]
    C. W. Wu and L. O. Chua, “Application of graph theory to the synchronization in an array of coupled nonlinear oscillators,” IEEE Trans. on Circuits and Systems I, vol. 42, no. 8, pp. 494–497, 1995.CrossRefMathSciNetGoogle Scholar
  4. [4]
    L. Zheleznyak and L. O. Chua, “Coexistence of low-and high dimensional spatiotemporal chaos in a chain of dissipatively coupled Chua’s circuits,” Int. J. of Bifurcations and Chaos, vol. 4, no. 3, pp. 639–674, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    V. Perez-Munuzuri and V. Perez-Villar, “Spiral waves on a 2-D array of nonlinear circuits,” IEEE Trans. on Circuits Systems I, vol. 40, no. 11, pp. 872–877, 1993.zbMATHCrossRefGoogle Scholar
  6. [6]
    D. V. Senthilkumar, J. Kurths, and M. Lakshmanan, “Stability of synchronization in coupled time-delay systems using Krasovskii-Lyapunov theory,” Physical Review E, vol. 79, 066208 1–4 2009.CrossRefMathSciNetGoogle Scholar
  7. [7]
    W. L. Guo, F. Austin, and S. H. Chen, “Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling,” Commun. in Nonlinear Sci. and Numerical Simulation, Available online 2009.Google Scholar
  8. [8]
    P. Li and Z. Yi, “Synchronization analysis of delayed complex networks with time-varying couplings,” Physica A, vol. 387, no. 14, pp. 3729–3737, 2008.CrossRefGoogle Scholar
  9. [9]
    W. Wu and T. P. Chen, “Global synchronization criteria of linearly coupled neural network systems with time-varying coupling,” IEEE Trans. on Neural Networks, vol. 19, no. 2, pp. 319–332, 2008.CrossRefGoogle Scholar
  10. [10]
    Y. P. Zhang and J. T. Sun, “Robust synchronization of coupled delayed neural networks under general impulsive control,” Chaos, Solitons and Fractals, vol. 41, no. 3, pp. 1476–1480, 2009.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Y. H. Xia, Z. J. Yang, and M. A. Han, “Synchronization schemes for coupled identical Yang-Yang type fuzzy cellular neural networks,” Commun. in Nonlinear Sci. and Numerical Simulation, vol. 14, no. 9–10, pp. 3645–3659, 2009.CrossRefMathSciNetGoogle Scholar
  12. [12]
    X. Y. Lou and B. T. Cui, “Synchronization of neural networks based on parameter identification and via output or state coupling,” Journal of Computational and Applied Mathematics, vol. 222, no. 2, pp. 440–457, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    S. Y. Xu and Y. Yang, “Synchronization for a class of complex dynamical networks with time-delay,” Commun. in Nonlinear Sci. and Numerical Simulation, vol. 14, no. 8, pp. 3230–3238, 2009.CrossRefMathSciNetGoogle Scholar
  14. [14]
    Q. K. Song, “Synchronization analysis of coupled connected neural networks with mixed time delays,” Neurocomputing, vol. 72, no. 16–18, pp. 3907–3914 2009.CrossRefGoogle Scholar
  15. [15]
    K. Yuan, Robust “Synchronization in arrays of coupled networks with delay and mixed coupling,” Neurocomputing, vol. 72, no. 4–6, pp. 1026–1031 2009.CrossRefGoogle Scholar
  16. [16]
    J. D. Cao, G. R. Chen, and P. Li, “Global synchronization in an array of delayed neural networks with Hybrid coupling,” IEEE Trans. on Systems, Man, and Cybernetics-Part B, vol. 38, No. 2, pp. 488–498 2008.CrossRefMathSciNetGoogle Scholar
  17. [17]
    W. W. Yu, J. D. Cao, and J. H. Li, “Global synchronization of linearly hybrid coupled networks with time-varying delay,” SIAM Journal of Applied dynamical Systems, vol. 7, no. 1, 108–133 2008.zbMATHCrossRefGoogle Scholar
  18. [18]
    Z. Y. Fei, H. J. Gao, and W. X. Zheng, “New synchronization stability of complex networks with an interval time-varying coupling delay,” IEEE Trans. on Circuits and Systems-II, vol. 56, no. 6, pp. 499–503 2009.CrossRefGoogle Scholar
  19. [19]
    W. L. He and J. D. Cao, “Global synchronization in arrays of coupled networks with one single timevarying delay coupling,” Physics Letters A, vol. 373, no. 31, pp. 2682–2694 2009.CrossRefMathSciNetGoogle Scholar
  20. [20]
    J. D. Cao and L. L. Li, “Cluster synchronization in an array of hybrid coupled neural networks with delay,” Neural Networks, vol. 22, no. 4, pp. 335–342 2009.CrossRefMathSciNetGoogle Scholar
  21. [21]
    J. L. Liang, Z. D. Wang, and Y. R. Liu, “Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks,” IEEE Trans. on Neural Networks, vol. 19, no. 11, pp. 1910–1921, 2008.CrossRefGoogle Scholar
  22. [22]
    H. Y. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    T. Li, A. G. Song, and S. M. Fei, “Novel stability criteria on discrete-time neural networks with timevarying and distributed delays,” International Journal of Neural Systems, vol. 19, no. 4, pp. 269–283, 2009.CrossRefGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Electronic Engineering and AutomationHenan Polytechnic UniversityJiaozuoHenan, China
  2. 2.School of Instrument Science and EngineeringSoutheast UniversityNanjingJiangsu, China
  3. 3.Key Laboratory of Measurement and Control of CSE (School of Automation, Southeast University)Ministry of EducationNanjingJiangsu, China

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