Global Synchronization in Finite-time of Fractional-order Complexvalued Delayed Hopfield Neural Networks

  • Xinxin Zhang
  • Peifeng NiuEmail author
  • Nan Liu
  • Guoqiang Li
Regular Papers Intelligent Control and Applications


This paper deals with the synchronization issue of fractional-order complex-valued Hopfield neural networks with time delay. In this paper, by means of properties of the fractional-order inequality, such as Hölder inequality and Gronwall inequality, sufficient conditions are presented to guarantee the finite-time synchronization of the fractional-order complex-valued delayed neural networks when 1/2 ≤ γ < 1 and 0 < γ < 1/2. Finally, two numerical simulations are provided to show the effectiveness of the obtained results.


Complex-valued neural networks finite-time synchronization fractional-order inequality time delay 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Zeng, R. Zhang, Y. Liu, and S. Zhong, “Sampleddata synchronization of chaotic Lur’e systems via inputdelay–dependent–free–matrix zero equality approach,” Applied Mathematics and Computation, vol. 315, pp. 34–46, December 2017.MathSciNetCrossRefGoogle Scholar
  2. [2]
    R. Zhang, X. Liu, D. Zeng, S. Zhong, and K. Shi, “A novel approach to stability and stabilization of fuzzy sampleddata Markovian chaotic systems,” Fuzzy Sets and Systems, vol. 344, pp. 108–128, 2017.CrossRefzbMATHGoogle Scholar
  3. [3]
    I. Podlubny. Fractional Differential Equations, Academic, New York, NY, USA, 1999.zbMATHGoogle Scholar
  4. [4]
    D. Cafagna, “Fractional calculus: A mathematical tool from the past for present engineers [Past and present],” Industrial Electronics Magazine IEEE, vol. 1, no. 2, pp. 35–40, July 2007.MathSciNetCrossRefGoogle Scholar
  5. [5]
    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elesvier, New York, NY, USA, 2006.zbMATHGoogle Scholar
  6. [6]
    X. Wu, J. Li, and G. Chen, “Chaos in the fractional order unified system and its synchronization,” Journal of the Franklin Institute, vol. 345, no. 4, pp. 392–401, July 2008.CrossRefzbMATHGoogle Scholar
  7. [7]
    Y. Li, Y. Q. Chen, and I. Podlubny, “Technical communique: Mittag–Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, August 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    X. Wu, H. Lu, and S. Shen, “Synchronization of a new fractional–order hyperchaotic system,” Physics Letters A, vol. 373, no. 27, pp. 2329–2337, June 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. J. Seow, V. K. Asari, and A. Livingston, “Learning as a nonlinear line of attraction in a recurrent neural network,” Neural Computing and Applications, vol. 19, no. 2, pp. 337–342, March 2010.CrossRefGoogle Scholar
  10. [10]
    J. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,” IEEE Transactions on Neural Networks, vol. 18, no. 2, pp. 416–430, March 2007.MathSciNetCrossRefGoogle Scholar
  11. [11]
    W. Yu, J. Cao, and G. Chen, “Stability and Hopf bifurcation of a general delayed recurrent neural network,” IEEE Trans Neural Networks, vol. 19, no. 5, pp. 845–854, May 2008.CrossRefGoogle Scholar
  12. [12]
    Q. Zhu and J. Cao, “Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays,” IEEE Transactions on Neural Networks Learning Systems, vol. 23, no. 3, pp. 467–479, March 2012.MathSciNetCrossRefGoogle Scholar
  13. [13]
    R. Zhang, D. Zeng, S. Zhong, and Y. Yu, “Event–triggered sampling control for stability and stabilization of memristive neural networks with communication delays,” Applied Mathematics and Computation, vol. 310, pp. 57–74, October 2017.MathSciNetCrossRefGoogle Scholar
  14. [14]
    P. Arena, R. Caponetto, L. Fortuna, and D. Porto, “Bifurcation and chaos in noninteger order cellular neural networks,” International Journal of Bifurcation and Chaos, vol. 8, no. 7, pp. 1527–1539, July 1998.CrossRefzbMATHGoogle Scholar
  15. [15]
    I. Petras, “A note on the fractional–order cellular neural networks,” Proc. of IEEE International Joint Conference on Neural Networks, pp. 1021–1024, July 2006.Google Scholar
  16. [16]
    A. Boroomand and M. B. Menhaj, “Fractional–order Hopfield neural networks,” Proceedings of the 15th international conference on Advances in neuro–information processing. pp. 883–890, 2008.Google Scholar
  17. [17]
    H. Huang, T. Huang, and X. Chen, “A mode–dependent approach to state estimation of recurrent neural networks with Markovian jumping parameters and mixed delays,” Neural Networks, vol. 46, no. 10, pp. 50–61, October 2013.CrossRefzbMATHGoogle Scholar
  18. [18]
    H. Bao, J. Cao, and J. Kurths, “State estimation of fractional–order delayed memristive neural networks,” Nonlinear Dynamics, vol. 94, no. 2, pp. 1215–1225, 2018.CrossRefGoogle Scholar
  19. [19]
    H. Wu, X. Zhang, S. Xue, L. Wang, and Y. Wang, “LMI conditions to global Mittag–Leffler stability of fractionalorder neural networks with impulses,” Neurocomputing, vol. 193, pp. 148–154, June 2016.CrossRefGoogle Scholar
  20. [20]
    J. Yu, C. Hu, and H. Jiang, “a–stability and a–synchronization for fractional–order neural networks,” Neural Networks, vol. 35, pp. 82–87, August 2012.CrossRefzbMATHGoogle Scholar
  21. [21]
    L. Chen, Y. Chai, R. Wu, “Letters: dynamic analysis of a class of fractional–order neural networks with delay,” Neurocomputing, vol. 111, no. 6, pp. 190–194, July 2013.CrossRefGoogle Scholar
  22. [22]
    F. Wang, Y. Yang, and M. Hu, “Asymptotic stability of delay fractioanal–order neural networks with impulsive effects,” Neurocomputing, vol. 154, pp. 239–244, April 2015.CrossRefGoogle Scholar
  23. [23]
    H. Wu, X. Zhang, S. Xue, and P. Niu, “Quasi–uniform stability of Caputo–type fractional–order neural networks with mixed delay,” International Journal of Machine Learning and Cybernetics, vol. 8, no. 5, pp. 1501–1511, October 2017.CrossRefGoogle Scholar
  24. [24]
    E. Kaslik and S. Sivasundaram, “Nonlinear dynamics and chaos in fractional–order neural networks,” Neural Networks the Official Journal of the International Neural Network Society, vol. 32, no. 1, pp. 245–256, February 2012.CrossRefzbMATHGoogle Scholar
  25. [25]
    H. Bao, J. H. Park, and J. Cao, “Adaptive synchronization of fractional–order memristor–based neural networks with time delay,” Nonlinear Dynamics, vol. 82, no. 3, pp. 1343–1354, November 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    H. Bao, J. H. Park, and J. Cao, “Synchronization of fractional–order delayed neural networks with hybrid coupling,” Complexity, vol. 21, pp. 106–112, 2016.MathSciNetCrossRefGoogle Scholar
  27. [27]
    S. Zhou, X. Lin, L. Zhang, and Y. Li, “Chaotic synchronization of a fractional neurons network system with two neurons,” Proc. of International Conference on Communications, Circuits and Systems, pp. 773–776, 2010.Google Scholar
  28. [28]
    M. Bohner, V. S. H. Rao, and S. Sanyal, “Global stability of complex–valued neural networks on time scales,” Differential Equations and Dynamical Systems, vol. 19, no. 1–2, pp. 3–11, January 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J. Hu and J. Wang, “Global stability of complex–valued recurrent neural networks with time–delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 6, pp. 853–865, May 2012.CrossRefGoogle Scholar
  30. [30]
    R. Rakkiyappan, J. Cao, and G. Velmurugan, “Existence and uniform stability analysis of fractional–order complexvalued neural networks with time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 1, pp. 84–97, January 2015.MathSciNetCrossRefGoogle Scholar
  31. [31]
    S. Tyagi, S. Abbas, and M. Hafayed, “Global Mittag–Leffler stability of complex valued fractional–order neural network with discrete and distributed delays,” Rendiconti del Circolo Matematico di Palermo Series 2, vol. 65, no. 3, pp. 485–505, December 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    R. Zhang, D. Zeng, J. Park, Y. Liu, and S. Zhong, “Nonfragile Sampled–Data Synchronization for Delayed Complex Dynamical NetworksWith Randomly Occurring Controller Gain Fluctuations,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 48, no. 12, pp. 2271–2281, Dec. 2017.CrossRefGoogle Scholar
  33. [33]
    H. Bao, H. P. Ju, and J. Cao, “Synchronization of fractional–order complex–valued neural networks with time delay,” Neural Networks, vol. 681, pp. 16–28, May 2016.CrossRefGoogle Scholar
  34. [34]
    D. S. Mitrinovic, Analytic Inequalities, Springer, New York, 1970.CrossRefzbMATHGoogle Scholar
  35. [35]
    M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality. Birkhauser, Switzerlang, 2009.CrossRefzbMATHGoogle Scholar
  36. [36]
    C. Corduneanu, Principle of Differential and Intergral Equations, Allyn and Bacon, USA, 1971.Google Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Xinxin Zhang
    • 1
  • Peifeng Niu
    • 1
    Email author
  • Nan Liu
    • 2
  • Guoqiang Li
    • 1
  1. 1.School of Electrical EngineeringYanshan UniversityQinhuangdaoChina
  2. 2.Yanshan UniversityQinhuangdaoChina

Personalised recommendations