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Global Robust Synchronization of Fractional Order Complex Valued Neural Networks with Mixed Time Varying Delays and Impulses

  • Pratap Anbalagan
  • Raja Ramachandran
  • Jinde CaoEmail author
  • Grienggrai Rajchakit
  • Chee Peng Lim
Regular Papers Intelligent Control and Applications
  • 21 Downloads

Abstract

In this article, we explore the theoretical issues on the drive-response synchronization of a class of fractional order uncertain complex valued neural networks (FOUCNNs) with mixed time varying delays and impulses. Based upon the contraction mapping principle, robust analysis techniques, as well as Riemann-Liouville (R-L) derivative, we derive a new set of novel sufficient conditions for the existence and uniqueness of equilibrium point of such neural network system, while by applying the Lyapunov functional approach, the global stability of the equilibrium solutions are obtained. Furthermore, the synchronization criterion of FOUCNNs is also attracted by means of the adaptive error feedback control strategy. Finally, two examples are provided along with the simulation results to demonstrate the effectiveness of our main proofs.

Keywords

Adaptive synchronization asymptotic stability complex valued neural networks Riemann-Liouville derivative 

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References

  1. [1]
    H. Bao, J. H. Park, and J. Cao, “Synchronization of fractional–order complex–valued neural networks with time delay,” Neural Networks, vol. 81, no. 1, pp. 16–28, 2016.CrossRefGoogle Scholar
  2. [2]
    A. Carpinteri, P. Cornetti, and M. Kolwankar, “Calculation of the tensile and flexural strength of disordered materials using fractional calculus,” Chaos, Solitons and Fractals, vol. 21, no. 3, pp. 623–632, 2004.CrossRefzbMATHGoogle Scholar
  3. [3]
    Z. Ding and Y. Shen, “Projective synchronization of nonidentical fractional–order neural networks based on sliding mode controller,” Neural Networks, vol. 76, no. 1, pp. 97–105, 2016.CrossRefGoogle Scholar
  4. [4]
    Y. Gu, Y. Yu, and H. Wang, “Synchronization for fractional–order time–delayed memristor–based neural networks with parameter uncertainty,” Journal of the Franklin Institute, vol. 353, no. 15, pp. 3657–3684, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    E. Kaslik and S. Sivasundaram, “Dynamics of fractionalorder neural networks,” Proc. of the International Conf. Neural Network, California, pp. 611–618, 2011.Google Scholar
  6. [6]
    W. Li and J. J. E. Slotine, Applied Nonlinear Control, Englewood Cliffs, NJ, 1991.zbMATHGoogle Scholar
  7. [7]
    C. Huang, J. Cao, M. Xiao, A. Alsaedi, and T. Hayat, “Bifurcations in a delayed fractional complex–valued neural network,” Applied Mathematics and Computation, vol. 292, no. 1, pp. 210–227, 2017.MathSciNetCrossRefGoogle Scholar
  8. [8]
    R. Rakkiyappan, K. Udhayakumar, G. Velmurugan, J. Cao, and A. Alsaedi, “Stability and Hopf bifurcation analysis of fractional–order complex–valued neural networks with time delays,” Advances in Difference Equations, vol. 2017. no. 1, ID 225, 2017.Google Scholar
  9. [9]
    Y. Cao, R. Samidurai, and R. Sriraman, “Robust passivity analysis for uncertain neural networks with leakage delay and additive time–varying delays by using general activation function,” Mathematics and Computers in Simulation, vol. 155, pp. 57–77, 2019.MathSciNetCrossRefGoogle Scholar
  10. [10]
    L. Pecora and T. Carrol, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 3, pp. 821–824, 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Rakkiyappan, G. Velmurugan, and J. Cao, “Finitetime stability analysis of fractional–order complex–valued memristor–based neural networks with time delays,” Nonlinear Dynamics, vol. 78, no. 4, pp. 2823–2836, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    R. Rakkiyappan, G. Velmurugan, and J. Cao, “Stability analysis of fractional–order complex–valued neural networks with time delays,” Chaos, Solitons and Fractals, vol. 78, no. 1, pp. 297–316, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    X. Zhang, X. Lv, and X. Li, “Sampled–data based lag synchronization of chaotic delayed neural networks with impulsive control,” Nonlinear Dynamics, vol. 90, no. 3, pp. 2199–2207, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R. Rakkiyappan, S. Dharani, and Q. Zhu, “Synchronization of reaction–diffusion neural networks with time–varying delays via stochastic sampled–data controller,” Nonlinear Dynamics, vol. 79, no. 1, pp. 485–500, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    W. Rudin, Real and Complex Analysis, Mcgraw–Hill, Newyork, 1987.zbMATHGoogle Scholar
  16. [16]
    J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus, Springer, Dordrecht, 2007.CrossRefzbMATHGoogle Scholar
  17. [17]
    Q. Song, Z. Zhao, and Y. Liu, “Stability analysis of complex–valued neural networks with probabilistic timevarying delays,” Neurocomputing, vol. 159, no. 2, pp. 96–104, 2015.CrossRefGoogle Scholar
  18. [18]
    I. Stamova, “Global Mittag–Leffler stability and synchronization of impulsive fractional order neural networs with time–varying delays,” Nonlinear Dynamics, vol. 77, no. 4, pp. 1251–1260, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D.Tripathil, S. Pandey, and S. Das, “Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3645–3654, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    G. Velmurugan, R. Rakkiyappan, and J. Cao, “Finite–time synchronization of fractional–order memristor–based neural networks with time delays,” Neural Networks, vol. 7, no. 1, pp. 36–46, 2016.CrossRefzbMATHGoogle Scholar
  21. [21]
    F. Wang, Y. Q. Yang, and M. F. Hu, “Asymptotic stability of delayed fractional–order neural networks with impulsive effects,” Neurocomputing, vol.154, no. 22, pp. 239–244, 2015.CrossRefGoogle Scholar
  22. [22]
    X. Li and J. Cao, “An impulsive delay inequality involving unbounded time–varying delay and applications,” IEEE Transactions on Automatic Control, vol. 62, no. 7, pp. 3618–3625, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    X. Yang and J. Cao, “Adaptive pinning synchronization of coupled neural networks with mixed delays and vectorform stochastic perturbations,” Acta Matematica Scientia, vol. 32, no. 3, pp. 955–977, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    X. Yang and J. Cao, “Hybrid adaptive and impulsive synchronization of uncertain complex networks with delays and general uncertain perturbations,” Applied Mathematics and Computation, vol. 227, no. 1, pp. 480–493, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Z. Wang, J. Cao, Z. Guo, and L. Huang, “Generalized stability for discontinuous complex–valued Hopfield neural networks via differential inclusions,” Proceedings of the Royal Society A, vol. 474, no. 2220. art. no. 2018.507, 2018.Google Scholar
  26. [26]
    J. Xiao, S. Zhong, Y. Li, and F. Xu, “Finite–time Mittag–Leffler synchronization of fractional–order memristive BAM neural networks with time delays,” Neurocomputing, vol. 219, no. 1, pp. 431–439, 2016.Google Scholar
  27. [27]
    X. Yang, C. Li, T. Huang, Q. Song, and X. Chen, “Quasiuniform synchronization of fractional–order memristorbased neural networks with delay,” Neurocomputing, vol. 234, no. 1, pp. 205–215, 2017.CrossRefGoogle Scholar
  28. [28]
    X. Li and X. Fu, “Lag synchronization of chaotic delayed neural networks via impulsive control,” IMA Journal of Mathematical Control and Information, vol. 29, no. 1, pp. 133–145, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    X. Li and S. Song, “Stabilization of delay systems: delaydependent impulsive control,” IEEE Transactions on Automatic Control, vol. 62, no. 1, pp. 406–411, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    L. Zhang and Y. Wang, “Complex projective synchronization of complex–valued neural network with structure identifcation,” Journal of the Franklin Institute, vol. 354, no. 12, pp. 5011–5025, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    C. Zhou, W. Zhang, X. Yang, C. Xu, and J. Feng, “Finitetime synchronization of complex–valued neural networks with mixed delays and uncertain perturbations,” Neural Process Letters, vol. 46, no. 1, pp. 1–21, 2017.CrossRefGoogle Scholar
  32. [32]
    W. Zhou and J. M. Zurada, “Discrete–time recurrent neural networks with complex–valued linear threshold neurons,” IEEE Transactions on Circuits and Systems II, vol. 56, no. 8, pp. 669–673, 2009.CrossRefGoogle Scholar
  33. [33]
    Q. Zhu, J. Cao, and R. Rakiappan, “Exponential input–tostate stability of stochastic Cohen–Grossberg neural networks with mixed delays,” Nonlinear Dynamics, vol. 79, no. 2, pp. 1085–1098, 2015.MathSciNetCrossRefGoogle Scholar
  34. [34]
    Q. Zhu and X. Li, “Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen–Grossberg neural networks,” Fuzzy Sets and Systems, vol. 203, no. 1, pp. 74–94, 2012.MathSciNetCrossRefzbMATHGoogle 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

  • Pratap Anbalagan
    • 1
  • Raja Ramachandran
    • 2
  • Jinde Cao
    • 3
    • 4
    Email author
  • Grienggrai Rajchakit
    • 5
  • Chee Peng Lim
    • 6
  1. 1.Department of MathematicsAlagappa UniversityKaraikudiIndia
  2. 2.Ramanujan Centre for Higher MathematicsAlagappa UniversityKaraikudiIndia
  3. 3.School of MathematicsSoutheast UniversityNanjingChina
  4. 4.School of Mathematics and StatisticsShandong Normal UniversityJi’nanChina
  5. 5.Department of Mathematics, Faculty of ScienceMaejo UniversityChiang MaiThailand
  6. 6.Institute of Intelligent System Research and InnovationDeakin UniversityDeakinAustralia

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