Advertisement

Anti-periodic Synchronization of Quaternion-valued Generalized Cellular Neural Networks with Time-varying Delays and Impulsive Effects

  • Yongkun LiEmail author
  • Yanchao Fang
  • Jiali Qin
Article
  • 5 Downloads

Abstract

In this paper, a class of quaternion-valued generalized cellular neural networks (QVGCNNs) with time-varying delays and impulsive effects is considered. Firstly, by constructing an appropriate Lyapunov function and applying inequality techniques, some sufficient conditions on the existence of anti-periodic solutions are established. Then, the global exponential anti-periodic synchronization of delayed QVGCNNs with impulsive effects and anti-periodic coefficients is investigated by designing a novel nonlinear state-feedback controller and constructing suitable Lyapunov functions. Our results are completely new even if the considered quaternion-valued systems degenerated into real-valued or complex-valued systems. Finally, two numerical examples are given to illustrate the feasibility and effectiveness of the obtained results.

Keywords

Anti-periodic solution generalized cellular neural network quaternion synchronization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Sudbery, “Quaternionic analysis,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 85, no. 2, pp. 199–225, 1979.MathSciNetzbMATHGoogle Scholar
  2. [2]
    S. Jankowski, A. Lozowski, and J. M. Zurada, “Complexvalued multistate neural associative memory,” IEEE Trans. on Neural Networks, vol. 7, no. 6, pp. 1491–1496, 1996.Google Scholar
  3. [3]
    H. Aoki and Y. Kosugi, “An image storage system using complex-valued associative memories,” Proc. of Int. Conf. on Pattern Recognition, IEEE, pp. 626–629, 2000.Google Scholar
  4. [4]
    H. Aoki, “A complex-valued neuron to transform gray level images to phase information,” Proc. of Int. Conference on Neural Information Processing, pp. 1084–1088, 2002.Google Scholar
  5. [5]
    G. Tanaka and K. Aihara, “Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction,” IEEE Transactions on Neural Networks, vol. 20, no. 9, pp. 1463–1473, 2009.Google Scholar
  6. [6]
    M. K. Muezzinoglu, C. Guzelis, and J. M. Zurada, “A new design method for the complex-valued multistate Hopfield associative memory,” IEEE Transactions on Neural Networks, vol. 14, no. 4, pp. 891–899, 2003.Google Scholar
  7. [7]
    P. Zheng, “Threshold complex-valued neural associative memory,” IEEE Transactions on Neural Networks & Learning Systems, vol. 25, no. 9, pp. 1714–1718, 2014.Google Scholar
  8. [8]
    M. Kobayashi, “Quaternionic Hopfield neural networks with twin-multistate activation function,” Neurocomputing, vol. 267, pp. 304–310, 2017.Google Scholar
  9. [9]
    N. Matsui, T. Isokawa, H. Kusamichi, F. Peper, and H. Nishimura, “Quaternion neural network with geometrical operators,” Journal of Intelligent & Fuzzy Systems, vol. 15, no. 3, 4, pp. 149–164, 2004.Google Scholar
  10. [10]
    X. Chen, Z. Li, Q. Song, J. Hu, and Y. Tan, “Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties,” Neural Networks, vol. 91, pp. 55–65, 2017.Google Scholar
  11. [11]
    D. Zhang, K. I. Kou, Y. Liu, and J. Cao, “Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in quaternion field,” Neural Networks, vol. 94, pp. 55–66, 2017.Google Scholar
  12. [12]
    Y. Liu, D. Zhang, J. Lu, and J. Cao, “Global m-stability criteria for quaternion-valued neural networks with unbounded time-varying delays,” Information Sciences, vol. 360, pp. 273–288, 2016.Google Scholar
  13. [13]
    Y. Li and J. Qin, “Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays,” Neurocomputing, vol. 292, pp. 91–103, 2018.Google Scholar
  14. [14]
    Y. Li and X. Meng, “Existence and global exponential stability of pseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on time scales,” Complexity, vol. 2017. Article ID 9878.69, 15 pages, 2017.Google Scholar
  15. [15]
    C. A. Popa and E. Kaslik, “Multistability and multiperiodicity in impulsive hybrid quaternion-valued neural networks with mixed delays,” Neural Networks, vol. 99, pp. 1–18, 2018.Google Scholar
  16. [16]
    Q. Song and X. Chen, “Multistability analysis of quaternion-valued neural networks with time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, pp. 5430–5440, 2018.MathSciNetGoogle Scholar
  17. [17]
    J. Hu, C. Zeng, and J. Tan, “Boundedness and periodicity for linear threshold discrete-time quaternion-valued neural network with time-delays,” Neurocomputing, vol. 267, pp. 417–425, 2017.Google Scholar
  18. [18]
    J. Zhu and J. Sun, “Stability of quaternion-valued neural networks with mixed delays,” Neural Processing Letters, May 2018. DOI: 10.1007/s11063-018-9849-xGoogle Scholar
  19. [19]
    Z. Cai, L. Huang, Z. Guo, L. Zhang, and X. Wan, “Periodic synchronization control of discontinuous delayed networks by using extended Filippov-framework,” Neural Networks, vol. 68, pp. 96–110, 2015.zbMATHGoogle Scholar
  20. [20]
    Y. Li, B. Li, S. Yao, and L. Xiong, “The global exponential pseudo almost periodic synchronization of quaternionvalued cellular neural networks with time-varying delays,” Neurocomputing, vol. 303, pp. 75–87, 2018.Google Scholar
  21. [21]
    Y. Li, H. Wang, and X. Meng, “Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with time-varying and distributed delays,” IMA Journal of Mathematical Control and Information, May 2018. DOI: 10.1093/imamci/dny015Google Scholar
  22. [22]
    W. Wu and T. Chen, “Global synchronization criteria of linearly coupled neural network systems with time-varying coupling,” IEEE Transactions on Neural Networks, vol. 19, no. 2, p. 319, 2008.Google Scholar
  23. [23]
    J. Chen, Z. Zeng, and P. Jiang, “Global Mittag-Leffler stability and synchronization of memristor-based fractionalorder neural networks,” Neural Networks the Official Journal of the International Neural Network Society, vol. 51, no. 3, pp. 1–8, 2014.zbMATHGoogle Scholar
  24. [24]
    L. Pan, X. Tang, and Y. Pan, “Generalized and exponential synchronization for a class of novel complex dynamic networks with hybrid time-varying delay via IPAPC,” International Journal of Control, Automation and Systems, vol. 16, no. 5, pp. 2501–2517, 2018.Google Scholar
  25. [25]
    W. K. Wong, W. Zhang, Y. Tang, and X. Wu, “Stochastic synchronization of complex networks with mixed impulses,” IEEE Transactions on Circuits & Systems I, vol. 60, no. 10, pp. 2657–2667, 2013.MathSciNetGoogle Scholar
  26. [26]
    H. B. Bao and J. Cao, “Projective synchronization of fractional-order memristor-based neural networks,” Neural Networks, vol. 63, pp. 1–9, 2015.zbMATHGoogle Scholar
  27. [27]
    A. Abdurahman, H. Jiang, and Z. Teng, “Function projective synchronization of impulsive neural networks with mixed time-varying delays,” Nonlinear Dynamics, vol. 78, no. 4, pp. 2627–2638, 2014.MathSciNetzbMATHGoogle Scholar
  28. [28]
    D. Tong, L. Zhang, W. Zhou, J. Zhou, and Y. Xu, “Asymptotical synchronization for delayed stochastic neural networks with uncertainty via adaptive control,” Int. J. of Control, Automation and Systems, vol. 14, no. 3, pp. 706–712, 2016.Google Scholar
  29. [29]
    M. Zarefard and S. Effati, “Adaptive synchronization between two non-identical BAM neural networks with unknown parameters and time-varying delays,” Int. J. of Control, Automation and Systems, vol. 15, no. 4, pp. 1877–1887, 2017.Google Scholar
  30. [30]
    R. Wei, J. Cao, and A. Alsaedi, “Fixed-time synchronization of memristive Cohen-Grossberg neural networks with impulsive effects,” Int. J. of Control, Automation and Systems, vol. 16, no. 5, 2214.2224, 2018.Google Scholar
  31. [31]
    J. A. Wang and X. Wen, “Pinning exponential synchronization of nonlinearly coupled Neural networks with mixed delays via intermittent control,” Int. J. of Control, Automation and Systems, vol. 16, no. 4, pp. 1558–1568, 2018.Google Scholar
  32. [32]
    H. Gu, H. Jiang, and Z. Teng, “On the dynamics in highorder cellular neural networks with time-varying delays,” Differential Equations & Dynamical Systems, vol. 19, no. 1–2, pp. 119–132, 2011.MathSciNetzbMATHGoogle Scholar
  33. [33]
    L. Zhou, “Dissipativity of a class of cellular neural networks with proportional delays,” Nonlinear Dynamics, vol. 73, no. 3, pp. 1895–1903, 2013.MathSciNetzbMATHGoogle Scholar
  34. [34]
    L. Zhou, “Delay-dependent exponential stability of cellular neural networks with multi-proportional delays,” Neural Processing Letters, vol. 38, no. 3, pp. 347–359, 2013.Google Scholar
  35. [35]
    M. Tan, “Global asymptotic stability of fuzzy cellular neural networks with unbounded distributed delays,” Neural Processing Letters, vol. 31, no. 2, pp. 147–157, 2010.Google Scholar
  36. [36]
    Y. Zhang and J. Sun, “Stability of impulsive neural networks with time delays,” Physics Letters A, vol. 348, no. 1, pp. 44–50, 2005.zbMATHGoogle Scholar
  37. [37]
    Q. Song and J. Zhang, “Global exponential stability of impulsive Cohen-Grossberg neural network with timevarying delays,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 500–510, 2008.MathSciNetzbMATHGoogle Scholar
  38. [38]
    X. Li and Z. Chen, “Stability properties for Hopfield neural networks with delays and impulsive perturbations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 3253–3265, 2009.MathSciNetzbMATHGoogle Scholar
  39. [39]
    C. Xu and Y. Wu, “Anti-periodic solutions for high-order cellular neural networks with mixed delays and impulses,” Advances in Difference Equations, vol. 2015. no. 1, p. 161, 2015.Google Scholar
  40. [40]
    A. Zhang, “Existence and exponential stability of antiperiodic solutions for HCNNs with time-varying leakage delays,” Advances in Difference Equations, vol. 2013, no. 1, pp. 1–14, 2013.MathSciNetGoogle Scholar
  41. [41]
    L. Pan and J. Cao, “Anti-periodic solution for delayed cellular neural networks with impulsive effects,” Nonlinear Analysis Real World Applications, vol. 12, no. 6, pp. 3014–3027, 2011.MathSciNetzbMATHGoogle Scholar
  42. [42]
    Y. Li, J. Qin, and B. Li, “Anti-periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays,” Neural Processing Letters, Jun 2018. DOI: 10.1007/s11063-018-9867-8Google Scholar
  43. [43]
    S. Shen, B. Li, and Y. Li, “Anti-periodic dynamics of quaternion-valued fuzzy cellular neural networks with time-varying delays on time scales,” Discrete Dynamics in Nature and Society, vol. 2018. Article ID 5290.86, 14 pages, 2018.Google Scholar
  44. [44]
    N. Huo and Y. Li, “Anti-periodic solutions for quaternionvalued shunting inhibitory cellular neural networks with distributed delays and impulses,” Complexity, vol. 2018. Article ID 6420.56, 12 pages, 2018.Google Scholar
  45. [45]
    Y. Li, J. Qin, and B. Li, “Existence and global exponential stability of anti-periodic solutions for delayed quaternionvalued cellular neural networks with impulsive effects,” Mathematical Methods in the Applied Sciences, vol. 42, no. 1, pp. 5–23, 2019.Google Scholar

Copyright information

© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.Department of MathematicsYunnan UniversityKunming, YunnanP. R. China

Personalised recommendations