Skip to main content
SpringerLink
Log in

Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays

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

Abstract

In this paper, we employ a novel method for solving the problem of the global exponential stability of quaternion-valued recurrent neural networks (QVNNs) with time-varying delays. Theoretically, a QVNN can be separated into four real-valued systems, forming an equivalent real-valued system. From the view of matrix measure, based on Halanay inequality instead of Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability for QVNNs. Moreover, the activation functions are not assumed to be derivative any more, which makes the analytical procedure compact. Finally, a numerical example is provided to validate the advantage of the proposed method and to show the effectiveness of the main results.

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

Fig. 1

Similar content being viewed by others

References

  1. Wu, B., Liu, Y., Lu, J.: New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays. Math. Comput. Model. 55(3), 837–843 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zhang, L., Zhu, Y., Zheng, W.X.: Energy-to-peak state estimation for Markov jump RNNS with time-varying delays via nonsynchronous filter with nonstationary mode transitions. IEEE Trans. Neural Netw. Learn. Syst. 26(10), 2346–2356 (2015)

    Article  MathSciNet  Google Scholar 

  3. Yang, R., Wu, B., Liu, Y.: A Halanay-type inequality approach to the stability analysis of discrete-time neural networks with delays. Appl. Math. Comput. 265, 696–707 (2015)

    MathSciNet  Google Scholar 

  4. Wang, Y., Wu, H.: Adaptive robust backstepping control for a class of uncertain dynamical systems using neural networks. Nonlinear Dyn. 81(4), 1597–1610 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Liu, Y., Zhang, D., Lu, J., Cao, J.: Global \(\mu \)-stability criteria for quaternion-valued neural networks with unbounded time-varying delays. Inf. Sci. 360, 273–288 (2016)

    Article  Google Scholar 

  6. Falahian, R., Dastjerdi, M.M., Molaie, M., Jafari, S., Gharibzadeh, S.: Artificial neural network-based modeling of brain response to flicker light. Nonlinear Dyn. 81(4), 1951–1967 (2015)

    Article  MathSciNet  Google Scholar 

  7. Rakkiyappan, R., Cao, J., Velmurugan, G.: Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syst. 26(1), 84–97 (2015)

    Article  MathSciNet  Google Scholar 

  8. Zhang, W., Li, C., Huang, T.: Global robust stability of complex-valued recurrent neural networks with time-delays and uncertainties. Int. J. Biomath. 7(02), 145006 (2014)

    MathSciNet  Google Scholar 

  9. Liu, Y., Xu, P., Lu, J., Liang, J.: Global stability of Clifford-valued recurrent neural networks with time delays. Nonlinear Dyn. 84(2), 767–777 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Isokawa, T., Matsui, N., Nishimura, H.: Quaternionic neural networks: fundamental properties and applications. In: Nitta, T. (ed.) Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters, chap. XVI, pp. 411–439. Information Science Reference, Hershey, New York (2009)

  11. Isokawa, T., Kusakabe, T., Matsui, N., Peper, F.: Quaternion neural network and its application. In: Palade, V., Howlett, R.J., Jain, L.C. (eds.) Proceedings of Knowledge-Based Intelligent Information and Engineering Systems (KES2003). Lecture Notes in Artificial Intelligence, vol. 2774, pp. 318–324. Springer (2003)

  12. Gupta, S.: Linear quaternion equations with application to spacecraft attitude propagation. In: 1998 IEEE Aerospace Conference, pp. 69–76 (1998)

  13. Kusamichi, H., Isokawa, T., Matsui, N., Ogawa, Y., Maeda, K.: A new scheme for color night vision by quaternion neural network. In: Proceedings of the 2nd International Conference on Autonomous Robots and Agents, pp. 101–106 (2004)

  14. Matsui, N., Isokawa, T., Kusamichi, H., Peper, F., Nishimura, H.: Quaternion neural network with geometrical operators. J. Intell. Fuzzy Syst. Appl. Eng. Technol. 15(3–4), 149–164 (2004)

    MATH  Google Scholar 

  15. Yoshida, M., Kuroe, Y., Mori, T.: Models of Hopfield-type quaternion neural networks and their energy functions. Int. J. Neural Syst. 15(01–02), 129–135 (2005)

    Article  Google Scholar 

  16. Sahoo, A., Xu, H., Jagannathan, S.: Neural network-based event-triggered state feedback control of nonlinear continuous-time systems. IEEE Trans. Neural Netw. Learn. Syst. 27(3), 497–509 (2016)

    Article  MathSciNet  Google Scholar 

  17. Michel, A.N., Farrell, J., Sun, H.F., et al.: Analysis and synthesis techniques for Hopfield type synchronous discrete time neural networks with application to associative memory. IEEE Trans. Circuits Syst. 37(11), 1356–1366 (1990)

    Article  MathSciNet  Google Scholar 

  18. Velmurugan, G., Rakkiyappan, R., Cao, J.: Further analysis of global \(\mu \)-stability of complex-valued neural networks with unbounded time-varying delays. Neural Netw. 67, 14–27 (2015)

    Article  Google Scholar 

  19. Wu, Z.G., Shi, P., Su, H., Chu, J.: Delay-dependent stability analysis for switched neural networks with time-varying delay. IEEE Trans. Syst. Man Cybern. Part B Cybern. 41(6), 1522–1530 (2011)

    Article  MathSciNet  Google Scholar 

  20. Wu, Z.G., Shi, P., Su, H., Chu, J.: Exponential synchronization of neural networks with discrete and distributed delays under time-varying sampling. IEEE Trans. Neural Netw. Learn. Syst. 23(9), 1368–1376 (2012)

    Article  Google Scholar 

  21. Ma, S., Lu, Q., Wang, Q., Feng, Z.: Effects of time delay on two neurons interaction Morris–Lecar model. Int. J. Biomath. 1(2), 161–170 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tang, Y., Xing, X., Karimi, H.R., Kocarev, L., Kurths, J.: Tracking control of networked multi-agent systems under new characterizations of impulses and its applications in robotic systems. IEEE Trans. Ind. Electron. 63(2), 1299–1307 (2016)

    Article  Google Scholar 

  23. Khajanchi, S., Banerjee, S.: Stability and bifurcation analysis of delay induced tumor immune interaction model. Appl. Math. Comput. 248, 652–671 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Liu, Y., Lu, J., Wu, B.: Some necessary and sufficient conditions for the output controllability of temporal Boolean control networks. ESAIM Control Optim. Calc. Var. 20(1), 158–173 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, J., Zhong, J., Ho, D.W., Tang, Y., Cao, J.: On controllability of delayed Boolean control networks. SIAM J. Control Optim. 54(2), 475–494 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wu, X., Tang, Y., Zhang, W.: Input-to-state stability of impulsive stochastic delayed systems under linear assumptions. Automatica 66(4), 195–204 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tang, Y., Gao, H., Zhang, W., Kurths, J.: Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses. Automatica 53(1), 346–354 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  29. Zhang, Z., Lin, C., Chen, B.: Global stability criterion for delayed complex-valued recurrent neural networks. IEEE Trans. Neural Netw. Learn. Syst. 25(9), 1704–1708 (2014)

    Article  Google Scholar 

  30. Chen, F., Jiang, R., Wen, C., Su, R.: Self-repairing control of a helicopter with input time delay via adaptive global sliding mode control and quantum logic. Inf. Sci. 316, 123–131 (2015)

    Article  Google Scholar 

  31. Kou, K.I., Liu, Y., Zhang, D., Tu, Y.: Ensemble control of linear systems with parameter uncertainties. Int. J. Control 89(7), 1495–1508 (2016)

    Article  MathSciNet  Google Scholar 

  32. Vidyasagar, M.: Nonlinear Systems Analysis, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ (2002)

  33. Cao, J., Wang, J.: Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50(1), 34–44 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  35. Bohner, M., Rao, V.S.H., Sanyal, S.: Global stability of complex-valued neural networks on time scales. Differ. Equ. Dyn. Syst. 19(1–2), 3–11 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu, X., Chen, T.: Global exponential stability for complex-valued recurrent neural networks with asynchronous time delays. IEEE Trans. Neural Netw. Learn. Syst. 27(3), 593–606 (2016)

    Article  MathSciNet  Google Scholar 

  37. Liu, X., Chen, T.: Robust \(\mu \)-stability for uncertain stochastic neural networks with unbounded time-varying delays. Phys. A Stat. Mech. Appl. 387(12), 2952–2962 (2008)

    Article  MathSciNet  Google Scholar 

  38. Senthilraj, S., Raja, R., Zhu, Q., Samidurai, R., Yao, Z.: New delay-interval-dependent stability criteria for static neural networks with time-varying delays. Neurocomputing 186, 1–7 (2016)

    Article  MATH  Google Scholar 

  39. Liu, D., Wang, L., Pan, Y., Ma, H.: Mean square exponential stability for discrete-time stochastic fuzzy neural networks with mixed time-varying delay. Neurocomputing 171, 1622–1628 (2016)

    Article  Google Scholar 

  40. Song, Q., Zhao, Z.: Stability criterion of complex-valued neural networks with both leakage delay and time-varying delays on time scales. Neurocomputing 171, 179–184 (2016)

    Article  Google Scholar 

  41. Kundu, A., Das, P., Roy, A.: Stability, bifurcations and synchronization in a delayed neural network model of \(n\)-identical neurons. Math. Comput. Simul. 121, 12–33 (2016)

    Article  MathSciNet  Google Scholar 

  42. He, Y., Ji, M.D., Zhang, C.K., Wu, M.: Global exponential stability of neural networks with time-varying delay based on free-matrix-based integral inequality. Neural Netw. 77, 80–82 (2016)

    Article  Google Scholar 

  43. Tang, Y., Qian, F., Gao, H., Kurths, J.: Synchronization in complex networks and its application—a survey of recent advances and challenges. Annu. Rev. Control 38(2), 184–198 (2014)

    Article  Google Scholar 

Download references

Acknowledgments

The authors wish to thank the editors and the anonymous reviewers for a number of constructive comments and suggestions that have improved the quality of the paper. This work was partially supported by Zhejiang Provincial Natural Science Foundation of China under Grant LY14A010008, the China Postdoctoral Science Foundation under Grants 2016T90406, 2015M580378, 2014M560377, 2015T80483 and the National Natural Science Foundation of China under Grants 11671361, 61573102.

Author information

Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, China

    Yang Liu & Dandan Zhang

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

    Yang Liu & Jianquan Lu

Authors
  1. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dandan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianquan Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Y., Zhang, D. & Lu, J. Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear Dyn 87, 553–565 (2017). https://doi.org/10.1007/s11071-016-3060-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3060-2

Keywords

Search

Navigation