Further study on Hopf bifurcation and hybrid control strategy in BAM neural networks concerning time delay

Qingyi Cui; Changjin Xu; Wei Ou; Yicheng Pang; Zixin Liu; Jianwei Shen; Muhammad Farman; Shabir Ahmad; Qingyi Cui; Changjin Xu; Wei Ou; Yicheng Pang; Zixin Liu; Jianwei Shen; Muhammad Farman; Shabir Ahmad

doi:10.3934/math.2024647

AIMS Mathematics

2024, Volume 9, Issue 5: 13265-13290. doi: 10.3934/math.2024647

Previous Article Next Article

Research article

Further study on Hopf bifurcation and hybrid control strategy in BAM neural networks concerning time delay

1.
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
2.
Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550025, China
3.
School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
4.
Faculty of Arts and Sciences, Department of Mathematics, Near East University, 99138, Nicosia, Cyprus
5.
Department of Computer Science and Mathematics, Lebanese American University, 1107–2020, Beirut, Lebanon
6.
Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khyber Pakhtunkhwa, Pakistan

Received: 24 January 2024 Revised: 26 February 2024 Accepted: 01 March 2024 Published: 10 April 2024
MSC : 34C23, 34K18, 37GK15, 39A11, 92B20

Delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In this article, we proposed a class of new BAM neural networks involving time delay. The traits of solution and bifurcation behavior of the established BAM neural networks involving time delay were probed into. First, the existence and uniqueness is discussed using a fixed point theorem. Second, the boundedness of solution of the formulated BAM neural networks involving time delay was analyzed by applying an appropriate function and inequality techniques. Third, the stability peculiarity and bifurcation behavior of the addressed delayed BAM neural networks were investigated. Fourth, Hopf bifurcation control theme of the formulated delayed BAM neural networks was explored by virtue of a hybrid controller. By adjusting the parameters of the controller, we could control the stability domain and Hopf bifurcation onset, which was in favor of balancing the states of different neurons in engineering. To verify the correctness of gained major outcomes, computer simulations were performed. The acquired outcomes of this article were new and own enormous theoretical meaning in designing and dominating neural networks.
- BAM neural networks,
- trait of solution,
- Hopf bifurcation,
- stability,
- time delay,
- hybrid controller
Citation: Qingyi Cui, Changjin Xu, Wei Ou, Yicheng Pang, Zixin Liu, Jianwei Shen, Muhammad Farman, Shabir Ahmad. Further study on Hopf bifurcation and hybrid control strategy in BAM neural networks concerning time delay[J]. AIMS Mathematics, 2024, 9(5): 13265-13290. doi: 10.3934/math.2024647

Related Papers:

Abstract

Delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In this article, we proposed a class of new BAM neural networks involving time delay. The traits of solution and bifurcation behavior of the established BAM neural networks involving time delay were probed into. First, the existence and uniqueness is discussed using a fixed point theorem. Second, the boundedness of solution of the formulated BAM neural networks involving time delay was analyzed by applying an appropriate function and inequality techniques. Third, the stability peculiarity and bifurcation behavior of the addressed delayed BAM neural networks were investigated. Fourth, Hopf bifurcation control theme of the formulated delayed BAM neural networks was explored by virtue of a hybrid controller. By adjusting the parameters of the controller, we could control the stability domain and Hopf bifurcation onset, which was in favor of balancing the states of different neurons in engineering. To verify the correctness of gained major outcomes, computer simulations were performed. The acquired outcomes of this article were new and own enormous theoretical meaning in designing and dominating neural networks.

References

[1]	R. Zhao, B. X. Wang, J. G. Jian, Global $\mu$-stabilization of quaternion-valued inertial BAM neural networks with time-varying delays via time-delayed impulsive control, Math. Comput. Simul., 202 (2022), 223–245. http://dx.doi.org/10.1016/j.matcom.2022.05.036 doi: 10.1016/j.matcom.2022.05.036
[2]	C. J. Xu, D. Mu, Y. L. Pan, C. Aouiti, Y. C. Pang, L. Y. Yao, Probing into bifurcation for fractional-order BAM neural networks concerning multiple time delays, J. Computat. Sci., 62 (2022), 101701. http://dx.doi.org/10.1016/j.jocs.2022.101701 doi: 10.1016/j.jocs.2022.101701
[3]	J. J. Oliveira, Global stability criteria for nonlinear differential systems with infinite delay and applications to BAM neural networks, Chaos Soliton. Fract., 164 (2022), 112676. http://dx.doi.org/10.1016/j.chaos.2022.112676 doi: 10.1016/j.chaos.2022.112676
[4]	D. Z. Chen, Z. Q. Zhang, Finite-time synchronization for delayed BAM neural networks by the approach of the same structural functions, Chaos Soliton. Fract., 164 (2022), 112655. http://dx.doi.org/10.1016/j.chaos.2022.112655 doi: 10.1016/j.chaos.2022.112655
[5]	X. Y. Mao, X. M. Wang, H. Y. Qin, Stability analysis of quaternion-valued BAM neural networks fractional-order model with impulses and proportional delays, Neurocomputing, 509 (2022), 206–220. http://dx.doi.org/10.1016/j.neucom.2022.08.059 doi: 10.1016/j.neucom.2022.08.059
[6]	C. J. Xu, M. Farman, Z. X. Liu, Y. C. Pang, Numerical approximation and analysis of epidemic model with constant proportional caputo(CPC) operator, Fractals, 32 (2024), 2440014. http://dx.doi.org/10.1142/S0218348X24400140 doi: 10.1142/S0218348X24400140
[7]	P. L. Li, C. J. Xu, M. Farman, A. Akgul, Y. C. Pang, Qualitative and stability analysis of fractional order emotion panic spreading model insight of fractional operator, Fractals, 32 (2024), 2440011. http://dx.doi.org/10.1142/S0218348X24400115 doi: 10.1142/S0218348X24400115
[8]	C. J. Xu, D. Mu, Z. X. Liu, Y. C. Pang, M. X. Liao, P. L. Li, et al., Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks, Nonlinear Anal. Model. Control, 27 (2022), 1030–1053. http://dx.doi.org/10.15388/namc.2022.27.28491 doi: 10.15388/namc.2022.27.28491
[9]	Y. Cao, S. Ramajayam, R. Sriraman, R. Samidurai, Leakage delay on stabilization of finite-time complex-valued BAM neural network: Decomposition approach, Neurocomputing, 463 (2021), 505–513. http://dx.doi.org/10.1016/j.neucom.2021.08.056 doi: 10.1016/j.neucom.2021.08.056
[10]	M. Syed Ali, G. Narayanan, V. Shekher, A. Alsaedi, B. Ahmad, Global Mittag-Leffler stability analysis of impulsive fractional-order complex-valued BAM neural networks with time varying delays, Commun. Nonlinear Sci. Numer. Simul., 83 (2020), 105088. http://dx.doi.org/10.1016/j.cnsns.2019.105088 doi: 10.1016/j.cnsns.2019.105088
[11]	B. Kosko, Bidirectional associative memories, IEEE Trans. Syst. Man Cyber., 18 (1988), 49–60. http://dx.doi.org/10.1109/21.87054 doi: 10.1109/21.87054
[12]	W. G. Yang, Periodic solution for fuzzy Cohen-Grossberg bam neural networks with both time-varying and distributed delays and variable coefficients, Neural Process. Lett., 40 (2014), 51–73. http://dx.doi.org/10.1007/s11063-013-9310-0 doi: 10.1007/s11063-013-9310-0
[13]	J. Sprott, Chaotic dynamics on large networks, Chaos, 18 (2008), 023135. http://dx.doi.org/10.1063/1.2945229 doi: 10.1063/1.2945229
[14]	A. Vaishwar, B. K. Yadav, Stability and Hopf-bifurcation analysis of four dimensional minimal neural network model with multiple time delays, Chinese J. Phys., 77 (2022), 300–318. http://dx.doi.org/10.1016/j.cjph.2022.02.011 doi: 10.1016/j.cjph.2022.02.011
[15]	B. P. Belousov, A periodic reaction and its mechanism, New York: John Wiley, 1985.
[16]	A. M. Zhabotinskii, Periodic process of the oxidation of malonic acid in solution (Study of the kinetics of Belousov$^{, }$s), Biofizika, 9 (1964), 306–311.
[17]	Q. Din, T. Donchev, D. Kolev, Stability, Bifurcation analysis and chaos control in chlorine dioxide-iodine-malonic acid reaction, MATCH Commun. Math. Comput. Chem., 79 (2018), 577–606.
[18]	I. Lengyel, G. Ribai, I. R. Epstein, Experimental and modeling study of oscillations in the chlorine dioxide-iodine-malonic acid reaction, J. Amer. Chem. Soc., 112 (1990), 9104–9110. http://dx.doi.org/10.1021/ja00181a011 doi: 10.1021/ja00181a011
[19]	E. Mosekilde, Topics in Nonlinear Dynamics: Applications to Physics, Biology and Economic Systems, New Jersey: World Science Publisher, 1996. http://dx.doi.org/10.1142/3194
[20]	J. N. Wang, H. B. Shi, L. Xu, L. Zang, Hopf bifurcation and chaos of tumor-Lymphatic model with two time delays, Chaos Soliton. Fract., 157 (2022), 111922. http://dx.doi.org/10.1016/j.chaos.2022.111922 doi: 10.1016/j.chaos.2022.111922
[21]	N. C. Pati, B. Ghosh, Delayed carrying capacity induced subcritical and supercritical Hopf bifurcations in a predator-prey system, Math. Comput. Simul., 195 (2022), 171–196. http://dx.doi.org/10.1016/j.matcom.2022.01.008 doi: 10.1016/j.matcom.2022.01.008
[22]	C. J. Xu, Z. X. Liu, M. X. Liao, L. Y. Yao, Theoretical analysis and computer simulations of a fractional order bank data model incorporating two unequal time delays, Expert Syst. Appl., 199 (2022), 116859. http://dx.doi.org/10.1016/j.eswa.2022.116859 doi: 10.1016/j.eswa.2022.116859
[23]	C. J. Xu, W. Zhang, C. Aouiti, Z. X. Liu, L. Y. Yao, Further analysis on dynamical properties of fractional-order bi-directional associative memory neural networks involving double delays, Math. Meth. Appl. Sci., 45 (2022), 11736–11754. http://dx.doi.org/10.1002/mma.8477 doi: 10.1002/mma.8477
[24]	C. J. Xu, M. X. Liao, P. L. Li, Y. Guo, Z. X. Liu, Bifurcation properties for fractional order delayed BAM neural networks, Cogn. Comput., 13 (2021), 322–356. http://dx.doi.org/10.1007/s12559-020-09782-w doi: 10.1007/s12559-020-09782-w
[25]	C. J. Xu, W. Zhang, Z. X. Liu, L. Y. Yao, Delay-induced periodic oscillation for fractional-order neural networks with mixed delays, Neurocomputing, 488 (2022), 681–693. http://dx.doi.org/10.1016/j.neucom.2021.11.079 doi: 10.1016/j.neucom.2021.11.079
[26]	C. J. Xu, D. Mu, Z. X. Liu, Y. C. Pang, M. X. Liao, P. Li, Bifurcation dynamics and control mechanism of a fractional-order delayed Brusselator chemical reaction model, MATCH Commun. Math. Comput. Chem., 89 (2023), 73–106. http://dx.doi.org/10.46793/match.89-1.073X doi: 10.46793/match.89-1.073X
[27]	C. J. Xu, C. Aouiti, Z. X. Liu, P. L. Li, L. Y. Yao, Bifurcation caused by delay in a fractional-order coupled Oregonator model in chemistry, MATCH Commun. Math. Comput. Chem., 88 (2022), 371–396. http://dx.doi.org/10.46793/match.88-2.371X doi: 10.46793/match.88-2.371X
[28]	C. J. Xu, W. Zhang, C. Aouiti, Z. X. Liu, P. L. Li, Bifurcation dynamics in a fractional-order Oregonator model including time delay, MATCH Commun. Math. Comput. Chem., 87 (2022), 397–414. http://dx.doi.org/10.46793/match.87-2.397X doi: 10.46793/match.87-2.397X
[29]	C. J. Xu, J. T. Lin, Y. Y. Zhao, Q. Y. Cui, W. Ou, Y. C. Pang, et al., New results on bifurcation for fractional-order octonion-valued neural networks involving delays, Netw. Comput. Neural Syst., 2024. https://doi.org/10.1080/0954898X.2024.2332662 doi: 10.1080/0954898X.2024.2332662
[30]	C. R. Tian, Y. Liu, Delay-driven Hopf bifurcation in a networked Malaria model, Appl. Mathe. Lett., 132 (2022), 108092. http://dx.doi.org/10.1016/j.aml.2022.108092 doi: 10.1016/j.aml.2022.108092
[31]	H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, Z. D. Teng, Dynamical analysis of a fractional-order prey-predator model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435–449. http://dx.doi.org/10.1007/s12190-016-1017-8 doi: 10.1007/s12190-016-1017-8
[32]	M. Das, A. Maiti, G. P. Samanta, Stability analysis of a prey-predator fractional order model incorporating prey refuge, Ecol. Genet. Genom., 7–8 (2018), 33–46. http://dx.doi.org/10.1016/j.egg.2018.05.001 doi: 10.1016/j.egg.2018.05.001
[33]	Z. Z. Zhang, H. Z. Yang, Hybrid control of Hopf bifurcation in a two prey one predator system with time delay, Proc. Chinese Contr. Conf., 2014, 6895–6900. http://dx.doi.org/10.1109/ChiCC.2014.6896136 doi: 10.1109/ChiCC.2014.6896136
[34]	L. P. Zhang, H. N. Wang, M. Xu, Hybrid control of bifurcation in a predator-prey system with three delays, Acta Phys. Sinica, 60 (2011), 010506. http://dx.doi.org/10.7498/aps.60.010506 doi: 10.7498/aps.60.010506
[35]	Z. Liu, K. W. Chuang, Hybrid control of bifurcation in continuous nonlinear dynamical systems, Int. J. Bifur. Chaos, 15 (2005), 1895–3903. http://dx.doi.org/10.1142/S0218127405014374 doi: 10.1142/S0218127405014374
[36]	Y. Y. Ni, Z. Wang, X. Huang, Q. Ma, H. Shen, Intermittent sampled-data control for local stabilization of neural networks subject to actuator saturation: A work-interval-dependent functional approach, IEEE Trans. Neural Netw. Learn. Syst., 35 (2024), 1087–1097. http://dx.doi.org/10.1109/TNNLS.2022.3180076 doi: 10.1109/TNNLS.2022.3180076
[37]	L. Yao, Z. Wang, X. Huang, Y. X. Li, Q. Ma, H. Shen, Stochastic sampled-data exponential synchronization of Markovian jump neural networks with time-varying delay, IEEE Trans. Neural Netw. Learn. Syst., 34 (2023), 909–920. http://dx.doi.org/10.1109/TNNLS.2021.3103958 doi: 10.1109/TNNLS.2021.3103958
[38]	X. D. Si, Z. Wang, Y. J. Fan, H. Shen, Sampled-data-based bipartite leader-following synchronization of cooperation-competition neural networks via scheduled-interval looped-function, IEEE Trans. Circuits Syst. I, 70 (2023), 3723–3734. http://dx.doi.org/10.1109/TCSI.2023.3284858 doi: 10.1109/TCSI.2023.3284858
[39]	Q. Ni, J. C. Ji, B. Halkon, K. Feng, A. K. Nandi, Physics-Informed Residual Network (PIResNet) for rolling element bearing fault diagnostics, Mech. Syst. Signal Process., 200 (2023), 110544. http://dx.doi.org/10.1016/j.ymssp.2023.110544 doi: 10.1016/j.ymssp.2023.110544
[40]	P. L. Li, R. Gao, C. J. Xu, J. W. Shen, S. Ahmad, Y. Li, Exploring the impact of delay on Hopf bifurcation of a type of BAM neural network models concerning three nonidentical delays, Neural Process. Lett., 55 (2023), 5905–5921. http://dx.doi.org/10.1007/s11063-023-11392-0 doi: 10.1007/s11063-023-11392-0
[41]	C. J. Xu, Y. Y. Zhao, J. T. Lin, Y. C. Pang, Z. X. Liu, J. W. Shen, et al., Mathematical exploration on control of bifurcation for a plankton-oxygen dynamical model owning delay, J. Math. Chem., 2023, http://dx.doi.org/10.1007/s10910-023-01543-y doi: 10.1007/s10910-023-01543-y
[42]	W. Ou, C. J. Xu, Q. Y. Cui, Y. C. Pang, Z. X. Liu, J. W. Shen, et al., Hopf bifurcation exploration and control technique in a predator-prey system incorporating delay, AIMS Math., 9 (2023), 1622–1651. http://dx.doi.org/10.3934/math.2024080 doi: 10.3934/math.2024080
[43]	Q. Y. Cui, C. J. Xu, W. Ou, Y. C. Pang, Z. X. Liu, P. L. Li, et al., Bifurcation behavior and hybrid controller design of a 2D Lotka-Volterra commensal symbiosis system accompanying delay, Mathematics, 11 (2023), 4808. http://dx.doi.org/10.3390/math11234808 doi: 10.3390/math11234808
[44]	C. Maharajan, C. Sowmiya, C. J. Xu, Fractional order uncertain BAM neural networks with mixed time delays: An existence and Quasi-uniform stability analysis, J. Intell. Fuzzy Syst., 46 (2024), 4291–4313. http://dx.doi.org/10.3233/JIFS-234744 doi: 10.3233/JIFS-234744
[45]	C. J. Xu, M. Farman, A. Shehzad, Analysis and chaotic behavior of a fish farming model with singular and non-singular kernel, Inter. J. Biomath., 2023. http://dx.doi.org/10.1142/S179352452350105X doi: 10.1142/S179352452350105X
[46]	C. J. Xu, Y. C. Pang, Z. X. Liu, J. W. Shen, M. X. Liao, P. L. Li, Insights into COVID-19 stochastic modelling with effects of various transmission rates: Simulations with real statistical data from UK, Australia, Spain, and India, Phys. Scripta, 99 (2024), 025218. http://dx.doi.org/10.1088/1402-4896/ad186c doi: 10.1088/1402-4896/ad186c
[47]	C. J. Xu, M. X. Liao, P. L. Li, L. Y. Yao, Q. W. Qin, Y. L. Shang, Chaos control for a fractional-order Jerk system via time delay feedback controller and mixed controller, Fract. Fractional, 5 (2021), 257. http://dx.doi.org/10.3390/fractalfract5040257 doi: 10.3390/fractalfract5040257

Reader Comments

Your name:*

Email:*
© 2024 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)