Abstract
The synchronization of a Riemann–Liouville-type fractional inertial neural network with a time delay and two inertial terms is studied in this paper. Some new Lyapunov functions are constructed. Based on the properties of the Riemann–Liouville fractional derivative, two new synchronization criteria are given in terms of linear matrix inequalities (LMIs). Suitable controllers are designed to ensure that synchronization can be achieved between the master system and slave system. Four numerical examples are provided to show the effectiveness and superiority of the obtained criteria.
Similar content being viewed by others
References
S. Arik, Z. Orman, Global stability analysis of Cohen–Grossberg neural networks with time-varying delays. Phys. Lett. A. 341, 410–421 (2005)
K.L. Babcock, R.M. Westervelt, Stability and dynamics of simple electronic neural networks with added inertia. Phys. D. 23, 464–469 (1986)
J. Chen, Z. Zeng, P. Jiang, Global Mittag-Leffler stability and synchronization of memrister-based fractional-order neural networks. Neural Netw. 51, 1–8 (2014)
L. Chen, Y. Chai, R. Wu, T. Ma, H. Zhai, Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111, 190–194 (2013)
L. Chen, R. Wu, J. Cao, J.B. Liu, Stability and synchronization of memrister-based fractional-order delayed neural networks. Neural Netw. 71, 37–44 (2015)
S.A. David, D.D. Quintino, C.M.C. Inacio, J.A.T. Machado, Fractional dynamic behavior in ethanol prices series. J. Comput. Appl. Math. 339, 85–93 (2018)
K. Diethelm, The Analysis of Fractional Differential Equations (Springer, New York, 2010), pp. 1–30
S. Ding, Z. Wang, Event-triggered synchronization of discrete-time neural networks: a switching approach. Neural Netw. 125, 31–40 (2020)
S. Ding, Z. Wang, H. Zhang, Intermittent control for quasisynchronization of delayed discrete-time neural networks. IEEE Trans. Cybern. 99, 1–12 (2020)
Y. Gu, H. Wang, Y. Yu, Stability and synchronization for Riemman–Liouville fractional-order time-delayed inertial neural networks. Neurocomputing 340, 270–280 (2019)
Y. Gu, H. Wang, Y. Yu, Synchronization for incommensurate Riemann–Liouville fractional-order time-delayed competitive neural networks with different time scales and known or unknown parameters. J. Comput. Nonlinear Dyn. 14(051002), 1–11 (2019)
T. Hu, X. Zhang, S. Zhong, Global asymptotic synchronization of nonidentical fractional-order neural networks. Neurocomputing 313, 39–46 (2018)
C. Huang, B. Liu, New studies on dynamic analysis of inertial neural networks involving non-reduced order method. Neurocomputing 325, 283–287 (2019)
M.D. Ji, Y. He, M. Wu, C.K. Zhang, Further results on exponential stability of neural networks with time-varying delay. Appl. Math. Comput. 256, 175–182 (2015)
X.Y. Li, X.T. Li, C. Hu, Some new results on stability and synchronization for delayed inertial neural networks based on non-reduced order method. Neural Netw. 96, 91–100 (2017)
K. Liang, W. Li, Exponential synchronization in inertial Cohen–Grossberg neural networks with time delays. J. Frank. Inst. 356, 11285–11304 (2019)
P. Liu, Z. Zeng, J. Wang, Multiple mittag-leffler stability of fractional-order recurrent neural networks. IEEE Trans. Syst. Man Cybern. Syst. 47, 2279–2288 (2017)
S. Liu, X.F. Zhou, X. Li, W. Jiang, Asymptotical stability of Riemann–Liouville fractional singular systems with multiple time-varying delays. Appl. Math. Lett. 65, 32–39 (2017)
Y. Liu, Z. Wang, X. Liu, Asymptotic stability for neural networks with mixed time-delays: the discrete-time case. Neural Netw. 22, 67–74 (2009)
Y.S. Moon, P.Y. Park, W.H. Kwon, Y.S. Lee, Delay dependent robust stabilization of uncertain state-delayed systems. Int. J. Control 74, 1447–1455 (2004)
L.M. Pecora, T.L. Carroll, Synchronization in chaotic system. Phys. Rev. Lett. 64, 821–824 (1990)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999), pp. 62–74
A. Pratap, R. Raja, J. Cao, C.P. Lim, O. Bagdasar, Stability and pinning synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous activations. Appl. Math. Comput. 359, 241–260 (2019)
A. Pratap, R. Raja, J. Cao, G. Rajchakit, Habib M. Fardoun, Stability and synchronization criteria for fractional order competitive neural networks with time delays: an asymptotic expansion of Mittag Leffler function. J. Frank. Inst. 356, 2212–2239 (2019)
A. Pratap, R. Raja, J. Cao, Fathalla A. Rihan, Aly R. Seadawy, Quasi-pinning synchronization and stabilization of fractional order BAM neural networks with delays and discontinuous neuron activations. Chaos, Solitons Frac. 131(109491), 1–22 (2020)
S. Tyagi, S. Abbas, M. Kirane, Global asymptotic and exponential synchronization of ring neural network with reaction-diffusion term and unbounded delay. Neural Comput. Appl. 30, 487–501 (2018)
F. Wang, Y. Yang, M. Hu, X. Xu, Projective cluster synchronization of fractional-order coupled-delay complex network via adaptive pinning control. Phys. A. 434, 134–143 (2015)
F. Wang, X. Liu, M. Tang, L. Chen, Further results on stability and synchronization of fractional-order Hopfield neural networks. Neurocomputing 346, 12–19 (2019)
H. Wang, Y. Yu, G. Wen, Stability analysis of fractional-order Hopfield neural networks with time delays. Neural Netw. 55, 98–109 (2014)
S.P. Xiao, H.H. Lian, H.B. Zeng, G. Chen, W.H. Zheng, Analysis on robust passivity of uncertain neural networks with time-varying delays via free-matrix-based integral inequality. Int. J. Control Autom. 15, 2385–2394 (2017)
Y. Yang, Y. He, Y. Wang, M. Wu, Stability analysis of fractional-order neural networks: an LMI approach. Neurocomputing 285, 82–93 (2018)
X. Yao, M. Tang, F. Wang, Z. Ye, X. Liu, New results on stability for a class of fractional-order static neural networks. Circ. Syst. Signal Process. 39, 5926–5950 (2020)
J. Yu, C. Hu, H. Jiang, Z. Teng, Exponential synchronization of Cohen–Grossberg neural networks via periodically intermittent control. Neurocomputing 74, 1776–1782 (2011)
C. Zhang, F. Deng, Y. Peng, Adaptive synchronization of Cohen–Grossberg neural network with mixed time-varying delays and stochastic perturbation. Appl. Math. Comput. 269, 792–801 (2015)
X.M. Zhang, Q.L. Han, New Lyapunov–Krasovskii functionals for global asymptotic stability of delayed neural networks. IEEE Trans. Neural Netw. 20, 533–539 (2009)
X.M. Zhang, Q.L. Han, Global asymptotic stability for a class of generalized neural networks with interval time-varying delay. IEEE Trans. Neural Netw. 22, 1180–1192 (2011)
Y. Zhang, Y. Yu, X. Cui, Dynamical behaviors analysis of memristor-based fractional-order complex-valued neural networks with time delay. Appl. Math. Comput. 339, 242–258 (2018)
Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare that they have no conflicts of interest.
Data availability statements
All data generated or analyzed during this study are included in this published article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partly supported by the National Science Foundation of China under Grants 61773404 and 61271355.
Rights and permissions
About this article
Cite this article
Zhang, S., Tang, M. & Liu, X. Synchronization of a Riemann–Liouville Fractional Time-Delayed Neural Network with Two Inertial Terms. Circuits Syst Signal Process 40, 5280–5308 (2021). https://doi.org/10.1007/s00034-021-01717-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-021-01717-6