Skip to main content
SpringerLink
Log in

Lagrange \(\alpha \)-Exponential Synchronization of Non-identical Fractional-Order Complex-Valued Neural Networks

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this article, Lagrange \(\alpha \)-exponential synchronization of non-identical fractional-order complex-valued neural networks (FOCVNNs) is studied. Numerous favorable conditions for achieving Lagrange \(\alpha \)-exponential synchronization and \(\alpha \)-exponential convergence of the descriptive networks are constructed using additional inequalities and the Lyapunov method. Furthermore, the structure of the \(\alpha \)-exponential convergence ball, in which the rate of convergence is linked with the system’s characteristics and order of differential, has also been demonstrated. These findings, which do not require consideration of the existence and uniqueness of equilibrium points, help to generalize and improve previous works and may be used to mono-stable and multi-stable of the FOCVNNs. The salient feature of the article is the graphical exhibition of the effectiveness of the proposed method by using numerical simulation for synchronization of a particular case of the considered fractional-order drive and response systems.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. N. Aguila-Camacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951–2957 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. I. Aizenberg, Complex-Valued Neural Networks with Multi-valued Neurons, vol. 353 (Springer, Berlin, 2011)

    MATH  Google Scholar 

  3. M.F. Amin, K. Murase, Single-layered complex-valued neural network for real-valued classification problems. Neurocomputing 72(4–6), 945–955 (2009)

    Article  Google Scholar 

  4. C. Aouiti, M. Bessifi, X. Li, Finite-time and fixed-time synchronization of complex-valued recurrent neural networks with discontinuous activations and time-varying delays. Circuits Syst. Signal Process. 39(11), 5406–5428 (2020)

    Article  Google Scholar 

  5. A. Arbi, Novel traveling waves solutions for nonlinear delayed dynamical neural networks with leakage term. Chaos Solitons Fractals 152, 111436 (2021)

    Article  MathSciNet  Google Scholar 

  6. A. Arbi, J. Cao, A. Alsaedi, Improved synchronization analysis of competitive neural networks with time-varying delays. Nonlinear Anal. Model. Control 23(1), 82–107 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Arbi, Y. Guo, J. Cao, Convergence analysis on time scales for Hobam neural networks in the Stepanov-like weighted pseudo almost automorphic space. Neural Comput. Appl. 33(8), 3567–3581 (2021)

    Article  Google Scholar 

  8. A. Arbi, N. Tahri, C. Jammazi, C. Huang, J. Cao, Almost anti-periodic solution of inertial neural networks with leakage and time-varying delays on timescales. Circuits Syst. Signal Process. 41(4), 1940–1956 (2022)

    Article  Google Scholar 

  9. H. Bao, J.H. Park, J. Cao, Synchronization of fractional-order complex-valued neural networks with time delay. Neural Netw. 81, 16–28 (2016)

    Article  MATH  Google Scholar 

  10. L. Chen, H. Yin, T. Huang, L. Yuan, S. Zheng, L. Yin, Chaos in fractional-order discrete neural networks with application to image encryption. Neural Netw. 125, 174–184 (2020). https://doi.org/10.1016/j.neunet.2020.02.008https://www.sciencedirect.com/science/article/pii/S0893608020300538

    Article  Google Scholar 

  11. S. Chen, L. Hanzo, S. Tan, Symmetric complex-valued RBF receiver for multiple-antenna-aided wireless systems. IEEE Trans. Neural Netw. 19(9), 1659–1665 (2008)

    Article  Google Scholar 

  12. M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 650–659 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Elfarhani, A. Jarraya, S. Abid, M. Haddar, Fractional derivative and hereditary combined model for memory effects on flexible polyurethane foam. Mech. Time-Depend. Mater. 20(2), 197–217 (2016)

    Article  Google Scholar 

  14. Y. Guo, S.S. Ge, A. Arbi, Stability of traveling waves solutions for nonlinear cellular neural networks with distributed delays. J. Syst. Sci. Complex. 35(1), 18–31 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. W. He, J. Cao, Adaptive synchronization of a class of chaotic neural networks with known or unknown parameters. Phys. Lett. A 372(4), 408–416 (2008)

    Article  MATH  Google Scholar 

  16. A. Hirose, Dynamics of fully complex-valued neural networks. Electron. Lett. 28(16), 1492–1494 (1992)

    Article  Google Scholar 

  17. A. Hirose, Complex-Valued Neural Networks, vol. 400 (Springer Science & Business Media, 2012).

  18. D.T. Hong, N.H. Sau, M.V. Thuan, New criteria for dissipativity analysis of fractional-order static neural networks. Circuits Syst. Signal Process. 41, 2221–2243 (2021)

    Article  Google Scholar 

  19. C. Huang, J. Cao, M. Xiao, A. Alsaedi, T. Hayat, Bifurcations in a delayed fractional complex-valued neural network. Appl. Math. Comput. 292, 210–227 (2017)

    MathSciNet  MATH  Google Scholar 

  20. J. Jian, P. Wan, Lagrange \(\alpha \)-exponential stability and \(\alpha \)-exponential convergence for fractional-order complex-valued neural networks. Neural Netw. 91, 1–10 (2017)

    Article  MATH  Google Scholar 

  21. J. Jian, B. Wang, Stability analysis in Lagrange sense for a class of bam neural networks of neutral type with multiple time-varying delays. Neurocomputing 149, 930–939 (2015)

    Article  Google Scholar 

  22. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 (Elseiver, Amsterdam, 2006)

    Book  MATH  Google Scholar 

  23. U. Kumar, S. Das, C. Huang, J. Cao, Fixed-time synchronization of quaternion-valued neural networks with time-varying delay. Proc. R. Soc. A 476(2241), 20200324 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Li, X. Liao, J. Yu, Complex-valued recurrent neural network with IIR neuron model: training and applications. Circuits Syst. Signal Process. 21(5), 461–471 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Li, Y. Kao, H.L. Li, Globally \(\beta \)-Mittag–Leffler stability and \(\beta \)-Mittag–Leffler convergence in Lagrange sense for impulsive fractional-order complex-valued neural networks. Chaos Solitons Fractals 148, 111061 (2021). https://doi.org/10.1016/j.chaos.2021.111061https://www.sciencedirect.com/science/article/pii/S096007792100415X

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Li, J. Jian, Exponential convergence and Lagrange stability for impulsive Cohen–Grossberg neural networks with time-varying delays. J. Comput. Appl. Math. 277, 23–35 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Li, Z. Wang, J. Lu, Y. Li, Adaptive synchronization of fractional-order complex-valued neural networks with discrete and distributed delays. Entropy 20(2), 124 (2018)

    Article  Google Scholar 

  28. X. Liao, Q. Luo, Z. Zeng, Y. Guo, Global exponential stability in Lagrange sense for recurrent neural networks with time delays. Nonlinear Anal. Real World Appl. 9(4), 1535–1557 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. N. Liu, J. Fang, W. Deng, Z.J. Wu, G.Q. Ding, Synchronization for a class of fractional-order linear complex networks via impulsive control. Int. J. Control Autom. Syst. 16(6), 2839–2844 (2018)

    Article  Google Scholar 

  30. K. Mathiyalagan, J.H. Park, R. Sakthivel, Exponential synchronization for fractional-order chaotic systems with mixed uncertainties. Complexity 21(1), 114–125 (2015a)

    Article  MathSciNet  Google Scholar 

  31. K. Mathiyalagan, J.H. Park, R. Sakthivel, Synchronization for delayed memristive BAM neural networks using impulsive control with random nonlinearities. Appl. Math. Comput. 259, 967–979 (2015b)

    MathSciNet  MATH  Google Scholar 

  32. S.M.A. Pahnehkolaei, A. Alfi, J.T. Machado, Delay-dependent stability analysis of the quad vector field fractional order quaternion-valued memristive uncertain neutral type leaky integrator echo state neural networks. Neural Netw. 117, 307–327 (2019a)

    Article  MATH  Google Scholar 

  33. S.M.A. Pahnehkolaei, A. Alfi, J.T. Machado, Delay independent robust stability analysis of delayed fractional quaternion-valued leaky integrator echo state neural networks with quad condition. Appl. Math. Comput. 359, 278–293 (2019b)

    MathSciNet  MATH  Google Scholar 

  34. S.M.A. Pahnehkolaei, A. Alfi, J.T. Machado, Stability analysis of fractional quaternion-valued leaky integrator echo state neural networks with multiple time-varying delays. Neurocomputing 331, 388–402 (2019c)

    Article  MATH  Google Scholar 

  35. J. Pan, X. Liu, W. Xie, Exponential stability of a class of complex-valued neural networks with time-varying delays. Neurocomputing 164, 293–299 (2015)

    Article  Google Scholar 

  36. R. Rakkiyappan, J. Cao, G. Velmurugan, Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans. Neural Netw. Learn. Syste. 26(1), 84–97 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Rakkiyappan, J. Cao, G. Velmurugan, 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). https://doi.org/10.1109/TNNLS.2014.2311099

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Rakkiyappan, R. Sivaranjani, G. Velmurugan, J. Cao, Analysis of global o (t- \(\alpha \)) stability and global asymptotical periodicity for a class of fractional-order complex-valued neural networks with time varying delays. Neural Netw. 77, 51–69 (2016)

    Article  MATH  Google Scholar 

  39. W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill Inc, New York, 1987)

    MATH  Google Scholar 

  40. H. Shu, Q. Song, J. Liang, Z. Zhao, Y. Liu, F.E. Alsaadi, Global exponential stability in Lagrange sense for quaternion-valued neural networks with leakage delay and mixed time-varying delays. Int. J. Syst. Sci. 50(4), 858–870 (2019). https://doi.org/10.1080/00207721.2019.1586001

    Article  MathSciNet  MATH  Google Scholar 

  41. E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity. Mol. Quantum Acoust. 23, 397–404 (2002)

    Google Scholar 

  42. G. Stamov, I. Stamova, Impulsive fractional-order neural networks with time-varying delays: almost periodic solutions. Neural Comput. Appl. 28(11), 3307–3316 (2017)

    Article  MATH  Google Scholar 

  43. G. Tanaka, K. Aihara, Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction. IEEE Trans. Neural Netw. 20(9), 1463–1473 (2009)

    Article  Google Scholar 

  44. P. Wan, J. Jian, J. Mei, Periodically intermittent control strategies for \(\alpha \)-exponential stabilization of fractional-order complex-valued delayed neural networks. Nonlinear Dyn. 92(2), 247–265 (2018a)

    Article  MATH  Google Scholar 

  45. P. Wan, J. Jian, J. Mei, Periodically intermittent control strategies for \(\alpha \)-exponential stabilization of fractional-order complex-valued delayed neural networks. Nonlinear Dyn. 92(2), 247–265 (2018b)

    Article  MATH  Google Scholar 

  46. H. Wei, R. Li, C. Chen, Z. Tu, Stability analysis of fractional order complex-valued memristive neural networks with time delays. Neural Process. Lett. 45(2), 379–399 (2017)

    Article  Google Scholar 

  47. A. Wu, Z. Zeng, Lagrange stability of memristive neural networks with discrete and distributed delays. IEEE Trans. Neural Netw. Learn. Syst. 25(4), 690–703 (2013)

    Article  Google Scholar 

  48. V.K. Yadav, V.K. Shukla, S. Das, Exponential synchronization of fractional-order complex chaotic systems and its application. Chaos Solitons Fractals 147, 110937 (2021). https://doi.org/10.1016/j.chaos.2021.110937https://www.sciencedirect.com/science/article/pii/S0960077921002915

    Article  MathSciNet  MATH  Google Scholar 

  49. X. Yang, C. Li, T. Huang, Q. Song, J. Huang, Synchronization of fractional-order memristor-based complex-valued neural networks with uncertain parameters and time delays. Chaos Solitons Fractals 110, 105–123 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  50. Y. Yang, J. Cao, Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects. Nonlinear Anal. Real World Appl. 11(3), 1650–1659 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. X. Yao, M. Tang, F. Wang, Z. Ye, X. Liu, New results on stability for a class of fractional-order static neural networks. Circuits Syst. Signal Process. 39, 5926–5950 (2020)

    Article  Google Scholar 

  52. C. Yin, X. Huang, Y. Chen, S. Dadras, Zhong Sm, Y. Cheng, Fractional-order exponential switching technique to enhance sliding mode control. Appl. Math. Model. 44, 705–726 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  53. F. Yu, H. Shen, Z. Zhang, Y. Huang, S. Cai, S. Du, Dynamics analysis, hardware implementation and engineering applications of novel multi-style attractors in a neural network under electromagnetic radiation. Chaos Solitons Fractals (2021). https://doi.org/10.1016/j.chaos.2021.111350

    Article  MathSciNet  Google Scholar 

  54. J. Yu, C. Hu, H. Jiang, \(\alpha \)-stability and \(\alpha \)-synchronization for fractional-order neural networks. Neural Netw. 35, 82–87 (2012)

    Article  MATH  Google Scholar 

  55. S. Zhang, M. Tang, X. Liu, Synchronization of a Riemann–Liouville fractional time-delayed neural network with two inertial terms. Circuits Syst. Signal Process. 40, 5280–5308 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are extending their heartfelt thanks to the revered reviewers for their valuable suggestions toward the improvement of the quality of the article.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Sapna Baluni & Subir Das

  2. Department of Mathematics, Amity School of Applied Sciences, Amity University, Gurugram, Haryana, 122413, India

    Vijay K. Yadav

  3. School of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  4. Yonsei Frontier Lab, Yonsei University, Seoul, 03722, South Korea

    Jinde Cao

Authors
  1. Sapna Baluni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Subir Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vijay K. Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jinde Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Subir Das.

Ethics declarations

Competing interests

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baluni, S., Das, S., Yadav, V.K. et al. Lagrange \(\alpha \)-Exponential Synchronization of Non-identical Fractional-Order Complex-Valued Neural Networks. Circuits Syst Signal Process 41, 5632–5652 (2022). https://doi.org/10.1007/s00034-022-02042-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-02042-2

Keywords

Search

Navigation