General Decay Synchronization for Recurrent Neural Networks with Mixed Time Delays


This paper studies the general decay synchronization (GDS) of a class of recurrent neural networks (RNNs) with general activation functions and mixed time delays. By constructing suitable Lyapunov-Krasovskii functionals and employing useful inequality techniques, some sufficient conditions on the GDS of considered RNNs are established via a type of nonlinear control. In addition, one example with numerical simulations is presented to illustrate the obtained theoretical results.

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  1. [1]

    Chua L O and Yang L, Cellular neural networks: Applications, IEEE Trans. Circuits Syst., 1988, 35: 1273–1290.

    MathSciNet  Article  Google Scholar 

  2. [2]

    Haykin S, Neural Networks, Prentice-Hall, New Jersey, 1994.

    Google Scholar 

  3. [3]

    Stamov T G, Impulsive cellular neural networks and almost periodicity, Proc. Jpn. Acad., 2004, 80(10): 198–203.

    MathSciNet  Article  Google Scholar 

  4. [4]

    Gopalsamy K, Stability of artificial neural networks with impulses, Appl. Math. Comput., 2004, 154: 783–813.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    Zeng Z G, Wang J, and Liao X X, Global exponential stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circuits Syst. I, 2003, 50(10): 1353–1358.

    MathSciNet  Article  Google Scholar 

  6. [6]

    Cao J D and Wang J, Absolute exponential stability of recurrent neural networks with Lipschitzcontinuous activation functions and time delays, Neural Netw., 2004, 17: 379–390.

    Article  Google Scholar 

  7. [7]

    Huang X, Cao J D, and Ho D W C, Existence and attractivity of almost periodic solution for recurrent neural networks with unbounded delays and variable coefficients, Nonlinear Dyn., 2006, 45(3-4): 337–351.

    MathSciNet  Article  Google Scholar 

  8. [8]

    Zhang H G, Wang Z S, and Liu D R, Global asymptotic stability of recurrent neural networks with multiple time-varying delays, IEEE Trans. Neural Netw., 2008, 19(5): 855–873.

    Article  Google Scholar 

  9. [9]

    Hu J and Wang J, Global stability of complex-valued recurrent neural networks with time-delays, IEEE Trans. Neural Netw. Learn. Syst., 2012, 23(6): 853–865.

    Article  Google Scholar 

  10. [10]

    Wen S P, Zeng Z G, Huang T W, et al., Passivity analysis of memristor-based recurrent neural networks with time-varying delays, J. Frankl. Inst., 2013, 350: 2354–2370.

    MathSciNet  Article  Google Scholar 

  11. [11]

    Zhou L Q and Zhang Y Y, Global exponential periodicity and stability of recurrent neural networks with multi-proportional delays, ISA Trans., 2016, 60: 89–95.

    Article  Google Scholar 

  12. [12]

    Li T, Fei S M, and Zhang K J, Synchronization control of recurrent neural networks with distributed delays, Physica A, 2008, 387: 982–996.

    Article  Google Scholar 

  13. [13]

    Wu A L, Zeng Z G, Zhu X S, et al., Exponential synchronization of memristor-based recurrent neural networks with time delays, Neurocomputing, 2011, 74: 3043–3050.

    Article  Google Scholar 

  14. [14]

    Wu A L, Wen S P, and Zeng Z G, Synchronization control of a class of memristor-based recurrent neural networks, Inf. Sci., 2012, 183: 106–116.

    MathSciNet  Article  Google Scholar 

  15. [15]

    Jiang M H, Wang S T, Mei J, et al., Finite-time synchronization control of a class of memristorbased recurrent neural networks, Neural Netw., 2015, 63: 133–140.

    Article  Google Scholar 

  16. [16]

    Zhang Z Q, Li A L, and Yu S H, Finite-time synchronization for delayed complex-valued neural networks via integrating inequality method, Neurocomputing, 2018, 318: 248–260.

    Article  Google Scholar 

  17. [17]

    Liu C, Li C D, and Li C J, Quasi-synchronization of delayed chaotic systems with parameters mismatch and stochastic perturbation, Commun. Nonlinear Sci. Numer. Simulat., 2011, 16: 4108–4119.

    MathSciNet  Article  Google Scholar 

  18. [18]

    Abdurahman A, Jiang H J, and Teng Z D, Function projective synchronization of impulsive neural networks with mixed time-varying delays, Nonlinear Dyn., 2014, 78: 2627–2638.

    MathSciNet  Article  Google Scholar 

  19. [19]

    Muhammadhaji A, Abdurahman A, and Jiang H J, Finite-time synchronization of complex dynamical networks with time-varying delays and nonidentical nodes, J. Ctrl. Sci. Eng., 2017, 2017: 1–13.

    MATH  Google Scholar 

  20. [20]

    Abdurahman A, Jiang H J, and Hu C, General decay synchronization of memristor-based Cohen-Grossberg neural networks with mixed time-delays and discontinuous activations, J. Frankl. Inst., 2017, 354: 7028–7052.

    MathSciNet  Article  Google Scholar 

  21. [21]

    Hu M F and Xu Z Y, Adaptive feedback controller for projective synchronization, Nonlinear Anal. RWA, 2008, 9: 1253–1260.

    MathSciNet  Article  Google Scholar 

  22. [22]

    Zhang Z Q and Ren L, New sufficient conditions on global asymptotic synchronization of inertial delayed neural networks by using integrating inequality techniques, Nonlinear Dyn., 2018,

    Google Scholar 

  23. [23]

    Li Y and Li C D, Matrix measure strategies for stabilization and synchronization of delayed BAM neural networks, Nonlinear Dyn., 2016, 84(3): 1759–1770.

    MathSciNet  Article  Google Scholar 

  24. [24]

    Cao J D and Wan Y, Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays, Neural Netw., 2014, 53: 165–172.

    Article  Google Scholar 

  25. [25]

    Xiao J Y, Zhong S M, Li Y T, et al., Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays, Neurocomputing, 2017, 219: 431–439.

    Article  Google Scholar 

  26. [26]

    Wang D S, Huang L H, and Tang L K, Dissipativity and synchronization of generalized BAM neural networks with multivariate discontinuous activations, IEEE Trans. Neural Netw. Learn. Syst., 2018, 29(8): 3815–3827.

    MathSciNet  Article  Google Scholar 

  27. [27]

    Chen C, Li L X, Peng H P, et al., Fixed-time synchronization of memristor-based BAM neural networks with time-varying discrete delay, Neural Netw., 2017, 96: 47–54.

    Article  Google Scholar 

  28. [28]

    Wang L M, Shen Y, and Zhang G D, Synchronization of a class of switched neural networks with time-varying delays via nonlinear feedback control, IEEE Trans. Cyber., 2016, 46(10): 2300–2310.

    Article  Google Scholar 

  29. [29]

    Wang L M, Shen Y, and Zhang G D, General decay synchronization stability for a class of delayed chaotic neural networks with discontinuous activations, Neurocomputing, 2016, 179: 169–175.

    Article  Google Scholar 

  30. [30]

    Wang J, Shi K B, Huang Q Z, et al., Stochastic switched sampled-data control for synchronization of delayed chaotic neural networks with packet dropout, Appl. Math. Comput., 2018, 335: 211–230.

    MathSciNet  MATH  Google Scholar 

  31. [31]

    Shi K B, Tang Y Y, Liu X Z, et al., Non-fragile sampled-data robust synchronization of uncertain delayed chaotic Lurie systems with randomly occurring controller gain fluctuation, ISA Trans., 2017, 66: 185–199.

    Article  Google Scholar 

  32. [32]

    Shi K B, Tang Y Y, Zhong S M, et al., Recursive filtering for state-saturated systems with randomly occurring nonlinearities and missing measurements, Int. J. Robust Nonliner Ctrl., 2018, 28(5): 1693–1714.

    MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Ahmadjan Muhammadhaji.

Additional information

This research was supported by the National Natural Science Foundation of Xinjiang under Grant No. 2016D01C075.

This paper was recommended for publication by Editor SUN Jian.

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Muhammadhaji, A., Teng, Z. General Decay Synchronization for Recurrent Neural Networks with Mixed Time Delays. J Syst Sci Complex 33, 672–684 (2020).

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  • General activation functions
  • general decay synchronization
  • mixed time delay
  • recurrent neural network