Global exponential periodicity of nonlinear neural networks with multiple time-varying delays

Huahai Qiu; Li Wan; Zhigang Zhou; Qunjiao Zhang; Qinghua Zhou; Huahai Qiu; Li Wan; Zhigang Zhou; Qunjiao Zhang; Qinghua Zhou

doi:10.3934/math.2023626

AIMS Mathematics

2023, Volume 8, Issue 5: 12472-12485. doi: 10.3934/math.2023626

Previous Article Next Article

Research article

Global exponential periodicity of nonlinear neural networks with multiple time-varying delays

1.
Research Center of Nonlinear Science, Research Center for Applied Mathematics and Interdisciplinary Sciences, School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan 430073, China
2.
School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China

Received: 26 December 2022 Revised: 16 March 2023 Accepted: 17 March 2023 Published: 27 March 2023
MSC : 32D40

Global exponential periodicity of nonlinear neural networks with multiple time-varying delays is investigated. Such neural networks cannot be written in the vector-matrix form because of the existence of the multiple delays. It is noted that although the neural network with multiple time-varying delays has been investigated by Lyapunov-Krasovskii functional method in the literature, the sufficient conditions in the linear matrix inequality form have not been obtained. Two sets of sufficient conditions in the linear matrix inequality form are established by Lyapunov-Krasovskii functional and linear matrix inequality to ensure that two arbitrary solutions of the neural network with multiple delays attract each other exponentially. This is a key prerequisite to prove the existence, uniqueness, and global exponential stability of periodic solutions. Some examples are provided to demonstrate the effectiveness of the established results. We compare the established theoretical results with the previous results and show that the previous results are not applicable to the systems in these examples.
- nonlinear neural network,
- multiple time-varying delays,
- periodic solution,
- exponential stability
Citation: Huahai Qiu, Li Wan, Zhigang Zhou, Qunjiao Zhang, Qinghua Zhou. Global exponential periodicity of nonlinear neural networks with multiple time-varying delays[J]. AIMS Mathematics, 2023, 8(5): 12472-12485. doi: 10.3934/math.2023626

Related Papers:

Abstract

Global exponential periodicity of nonlinear neural networks with multiple time-varying delays is investigated. Such neural networks cannot be written in the vector-matrix form because of the existence of the multiple delays. It is noted that although the neural network with multiple time-varying delays has been investigated by Lyapunov-Krasovskii functional method in the literature, the sufficient conditions in the linear matrix inequality form have not been obtained. Two sets of sufficient conditions in the linear matrix inequality form are established by Lyapunov-Krasovskii functional and linear matrix inequality to ensure that two arbitrary solutions of the neural network with multiple delays attract each other exponentially. This is a key prerequisite to prove the existence, uniqueness, and global exponential stability of periodic solutions. Some examples are provided to demonstrate the effectiveness of the established results. We compare the established theoretical results with the previous results and show that the previous results are not applicable to the systems in these examples.

References

[1]	P. N. Suganthan, E. K. Teoh, D. P. Mital, Pattern recognition by homomorphic graph matching using Hopfield neural networks, Image Vis. Comput., 13 (1995), 45–60. https://doi.org/10.1016/0262-8856(95)91467-R doi: 10.1016/0262-8856(95)91467-R
[2]	T. Deb, A. K. Ghosh, A. Mukherjee, Singular value decomposition applied to associative memory of Hopfield neural network, Mater. Today, 5 (2018), 2222–2228. https://doi.org/10.1016/j.matpr.2017.09.222 doi: 10.1016/j.matpr.2017.09.222
[3]	V. Donskoy, BOMD: Building optimization models from data (neural networks based approach), Quant. Finance Econ., 3 (2019), 608–623. https://doi.org/10.3934/QFE.2019.4.608 doi: 10.3934/QFE.2019.4.608
[4]	L. H. Huang, C. X. Huang, B. W. Liu, Dynamics of a class of cellular neural networks with time-varying delays, Phys. Lett. A, 345 (2005), 330–344. https://doi.org/10.1016/j.physleta.2005.07.039 doi: 10.1016/j.physleta.2005.07.039
[5]	X. D. Li, D. O'Regan, H. Akca, Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays, IMA J. Appl. Math., 80 (2015), 85–99. https://doi.org/10.1093/imamat/hxt027 doi: 10.1093/imamat/hxt027
[6]	S. Zhang, Y. G. Yu, Q. Wang, Stability analysis of fractional-order Hopfield neural networks with discontinuous activation functions, Neurocomputing, 171 (2016), 1075–1084. https://doi.org/10.1016/j.neucom.2015.07.077 doi: 10.1016/j.neucom.2015.07.077
[7]	C. J. Xu, P. L. Li, Global exponential convergence of neutral-type Hopfield neural networks with multi-proportional delays and leakage delays, Chaos Soliton. fract., 96 (2017), 139–144. https://doi.org/10.1016/j.chaos.2017.01.012 doi: 10.1016/j.chaos.2017.01.012
[8]	Y. H. Zhou, C. D. Li, H. Wang, Stability analysis on state-dependent impulsive Hopfield neural networks via fixed-time impulsive comparison system method, Neurocomputing, 316 (2018), 20–29. https://doi.org/10.1016/j.neucom.2018.07.047 doi: 10.1016/j.neucom.2018.07.047
[9]	S. Arik, A modified Lyapunov functional with application to stability of neutral-type neural networks with time delays, J. Franklin Inst., 356 (2019), 276–291. https://doi.org/10.1016/j.jfranklin.2018.11.002 doi: 10.1016/j.jfranklin.2018.11.002
[10]	F. X. Wang, X. G. Liu, M. L. Tang, L. F. Chen, Further results on stability and synchronization of fractional-order Hopfield neural networks, Neurocomputing, 346 (2019), 12–19. https://doi.org/10.1016/j.neucom.2018.08.089 doi: 10.1016/j.neucom.2018.08.089
[11]	W. Q. Shen, X. Zhang, Y. T. Wang, Stability analysis of high order neural networks with proportional delays, Neurocomputing, 372 (2020), 33–39. https://doi.org/10.1016/j.neucom.2019.09.019 doi: 10.1016/j.neucom.2019.09.019
[12]	O. Faydasicok, A new Lyapunov functional for stability analysis of neutral-type Hopfield neural networks with multiple delays, Neural Networks, 129 (2020), 288–297. https://doi.org/10.1016/j.neunet.2020.06.013 doi: 10.1016/j.neunet.2020.06.013
[13]	H. Wang, G. L. Wei, S. Wen, T. Huang, Generalized norm for existence, uniqueness and stability of Hopfield neural networks with discrete and distributed delays, Neural Networks, 128 (2020), 288–293. https://doi.org/10.1016/j.neunet.2020.05.014 doi: 10.1016/j.neunet.2020.05.014
[14]	Q. K. Song, Y. X. Chen, Z. J. Zhao, Y. R. Liu, F. E. Alsaadi, Robust stability of fractional-order quaternion-valued neural networks with neutral delays and parameter uncertainties, Neurocomputing, 420 (2021), 70–81. https://doi.org/10.1016/j.neucom.2020.08.059 doi: 10.1016/j.neucom.2020.08.059
[15]	Y. K. Deng, C. X. Huang, J. D. Cao, New results on dynamics of neutral type HCNNs with proportional delays, Math. Comput. Simul., 187 (2021), 51–59. https://doi.org/10.1016/j.matcom.2021.02.001 doi: 10.1016/j.matcom.2021.02.001
[16]	Q. K. Song, R. T. Zeng, Z. J. Zhao, Y. R. Liu, F. E. Alsaadi, Mean-square stability of stochastic quaternion-valued neural networks with variable coefficients and neutral delays, Neurocomputing, 471 (2022), 130–138. https://doi.org/10.1016/j.neucom.2021.11.033 doi: 10.1016/j.neucom.2021.11.033
[17]	S. Townley, A. Ilchmann, M. G. Weiss, W. Mcclements, A. C. Ruiz, D. H. Owens, et al., Existence and learning of oscillations in recurrent neural networks, IEEE Trans. Neural Networks, 11 (2000), 205–214. https://doi.org/10.1109/72.822523 doi: 10.1109/72.822523
[18]	H. Y. Zhao, Global exponential stability and periodicity of cellular neural networks with variable delays, Phys. Lett. A, 336 (2005), 331–341. https://doi.org/10.1016/j.physleta.2004.12.001 doi: 10.1016/j.physleta.2004.12.001
[19]	L. Q. Zhou, G. D. Hu, Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput., 195 (2008), 402–411. https://doi.org/10.1016/j.amc.2007.04.114 doi: 10.1016/j.amc.2007.04.114
[20]	L. Yang, Y. K. Li, Existence and exponential stability of periodic solution for stochastic Hopfield neural networks on time scales, Neurocomputing, 167 (2015), 543–550. https://doi.org/10.1016/j.neucom.2015.04.038 doi: 10.1016/j.neucom.2015.04.038
[21]	C. Wang, Piecewise pseudo-almost periodic solution for impulsive non-autonomous high-order Hopfield neural networks with variable delays, Neurocomputing, 171 (2016), 1291–1301. https://doi.org/10.1016/j.neucom.2015.07.054 doi: 10.1016/j.neucom.2015.07.054
[22]	A. M. Alimi, C. Aouiti, F. Chérif, F. Dridi, M. S. M'hamdi, Dynamics and oscillations of generalized high-order Hopfield neural networks with mixed delays, Neurocomputing, 321 (2018), 274–295. https://doi.org/10.1016/j.neucom.2018.01.061 doi: 10.1016/j.neucom.2018.01.061
[23]	S. Gao, R. Shen, T. R. Chen, Periodic solutions for discrete-time Cohen-Grossberg neural networks with delays, Phys. Lett. A, 383 (2019), 414–420. https://doi.org/10.1016/j.physleta.2018.11.016 doi: 10.1016/j.physleta.2018.11.016
[24]	F. C. Kong, Q. X. Zhu, K. Wang, J. J. Nieto, Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator, J. Franklin Inst., 356 (2019), 11605–11637. https://doi.org/10.1016/j.jfranklin.2019.09.030 doi: 10.1016/j.jfranklin.2019.09.030
[25]	M. Abdelaziz, F. Cherif, Piecewise asymptotic almost periodic solutions for impulsive fuzzy Cohen-Grossberg neural networks, Chaos Soliton. Fract., 132 (2020), 109575. https://doi.org/10.1016/j.chaos.2019.109575 doi: 10.1016/j.chaos.2019.109575
[26]	F. C. Kong, Y. Ren, R. Sakthivel, Delay-dependent criteria for periodicity and exponential stability of inertial neural networks with time-varying delays, Neurocomputing, 419 (2021), 261–272. https://doi.org/10.1016/j.neucom.2020.08.046 doi: 10.1016/j.neucom.2020.08.046
[27]	Z. W. Cai, L. H. Huang, Z. Y. Wang X. M. Pan, S. K. Liu, Periodicity and multi-periodicity generated by impulses control in delayed Cohen-Grossberg-type neural networks with discontinuous activations, Neural Networks, 143 (2021), 230–245. https://doi.org/10.1016/j.neunet.2021.06.013 doi: 10.1016/j.neunet.2021.06.013
[28]	H. Li, Y. G. Kao, I. Stamov, C. T. Shao, Global asymptotic stability and S-asymptotic $\omega$-periodicity of impulsive non-autonomous fractional-order neural networks, Appl. Math. Comput., 410 (2021), 126459. https://doi.org/10.1016/j.amc.2021.126459 doi: 10.1016/j.amc.2021.126459

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)