Local stability and bifurcation of multi-delay fractional-order bidirectional ring neural networks with reaction-diffusion

Yi Min; Yi Min

doi:10.3934/era.2026057

Electronic Research Archive

2026, Volume 34, Issue 2: 1238-1268. doi: 10.3934/era.2026057

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Research article

Local stability and bifurcation of multi-delay fractional-order bidirectional ring neural networks with reaction-diffusion

Yi Min ^{1,2
,
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1.
College of Mathematics and Physics, China Three Gorges University, Yichang 443002, China
2.
Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China

Received: 09 November 2025 Revised: 19 December 2025 Accepted: 04 January 2026 Published: 06 February 2026

This paper investigated the Hopf bifurcation in fractional-order ring neural networks incorporating multiple time delays (leakage, transmission, and distributed delays) and reaction-diffusion effects. By introducing virtual neurons into the original system, a new model capable of equivalently describing the influence of distributed time delays was constructed. The critical conditions for the emergence of pure imaginary roots in the characteristic equation were analytically determined using the Coates flow graph and the holistic element method. Furthermore, by selecting the leakage and transmission delays as bifurcation parameters, explicit criteria for local stability and the existence of Hopf bifurcation were established. The theoretical findings were substantiated through numerical simulations.
- reaction-diffusion,
- distributed-time delay,
- fractional order,
- Hopf bifurcation
Citation: Yi Min. Local stability and bifurcation of multi-delay fractional-order bidirectional ring neural networks with reaction-diffusion[J]. Electronic Research Archive, 2026, 34(2): 1238-1268. doi: 10.3934/era.2026057

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Abstract

This paper investigated the Hopf bifurcation in fractional-order ring neural networks incorporating multiple time delays (leakage, transmission, and distributed delays) and reaction-diffusion effects. By introducing virtual neurons into the original system, a new model capable of equivalently describing the influence of distributed time delays was constructed. The critical conditions for the emergence of pure imaginary roots in the characteristic equation were analytically determined using the Coates flow graph and the holistic element method. Furthermore, by selecting the leakage and transmission delays as bifurcation parameters, explicit criteria for local stability and the existence of Hopf bifurcation were established. The theoretical findings were substantiated through numerical simulations.

References

[1]	L. Pantanowitz, T. Pearce, I. Abukhiran, M. Hanna, S. Wheeler, T. R. Soong, et al., Nongenerative artificial intelligence in medicine: advancements and applications in supervised and unsupervised machine learning, Mod. Pathol., 38 (2025), 100680. https://doi.org/10.1016/j.modpat.2024.100680 doi: 10.1016/j.modpat.2024.100680
[2]	Y. H. Hu, X. L. Xu, L. Duan, M. Bilal, Q. Y. Wang, W. C. Dou, End-edge collaborative inference of convolutional fuzzy neural networks for big data-driven internet of things, IEEE Trans. Fuzzy Syst., 33 (2025), 203–217. https://doi.org/10.1109/tfuzz.2024.3412971 doi: 10.1109/tfuzz.2024.3412971
[3]	Y. X. He, E. Zio, Z. M. Yang, Q. Xiang, L. Fan, Q. He, et al., A systematic resilience assessment framework for multi-state systems based on physics-informed neural network, Reliab. Eng. Syst. Saf., 257 (2025), 110866. https://doi.org/10.1016/j.ress.2025.110866 doi: 10.1016/j.ress.2025.110866
[4]	J. H. Ma, H. L. Tu, Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomic models, Nonlinear Dyn., 76 (2013), 497–508. https://doi.org/10.1007/s11071-013-1143-x doi: 10.1007/s11071-013-1143-x
[5]	C. J. Xu, W. Ou, Q. Y. Cui, Y. C. Pang, M. X. Liao, J. W. Shen, et al., Theoretical exploration and controller design of bifurcation in a plankton population dynamical system accompanying delay, Discrete Contin. Dyn. Syst. - Ser. S, 18 (2025), 1182–1211. https://doi.org/10.3934/dcdss.2024036 doi: 10.3934/dcdss.2024036
[6]	C. J. Xu, W. Ou, Y. C. Pang, Q. Y. Cui, M. Rahman, M. Farman, et al., Hopf bifurcation control of a fractional-order delayed turbidostat model via a novel extended hybrid controller, Match-Commun. Math. Comput. Chem., 91 (2023), 367–413. https://doi.org/10.30546/1683-6154.22.4.2023.495 doi: 10.30546/1683-6154.22.4.2023.495
[7]	P. Baldi, A. F. Atiya, How delays affect neural dynamics and learning, IEEE Trans. Neural Networks, 5 (1994), 612–621. https://doi.org/10.1109/72.298231 doi: 10.1109/72.298231
[8]	T. Y. Cai, H. G. Zhang, F. S. Yang, Simplified frequency method for stability and bifurcation of delayed neural networks in ring structure, Neurocomputing, 121 (2013), 416–422. https://doi.org/10.1016/j.neucom.2013.05.022 doi: 10.1016/j.neucom.2013.05.022
[9]	C. D. Huang, J. D. Cao, M. Xiao, A. Alsaedi, T. Hayat, Effects of time delays on stability and Hopf bifurcation in a fractional ring-structured network with arbitrary neurons, Commun. Nonlinear Sci. Numer. Simul., 57 (2018), 1–13. https://doi.org/10.1016/j.cnsns.2017.09.005 doi: 10.1016/j.cnsns.2017.09.005
[10]	X. Xu, D. Y. Yu, Z. H. Wang, Inter-layer synchronization of periodic solutions in two coupled rings with time delay, Physica D, 396 (2019), 1–11. https://doi.org/10.1016/j.physd.2019.02.010 doi: 10.1016/j.physd.2019.02.010
[11]	B. B. Tao, M. Xiao, W. X. Zheng, J. D. Cao, J. W. Tang, Dynamics analysis and design for a bidirectional super-ring-shaped neural network with n neurons and multiple delays, IEEE Trans. Neural Networks Learn. Syst., 32 (2021), 2978–2992. https://doi.org/10.1109/tnnls.2020.3009166 doi: 10.1109/tnnls.2020.3009166
[12]	R. T. Xing, M. Xiao, Y. Z. Zhang, J. L. Qiu, Stability and Hopf bifurcation analysis of an (n + m)-neuron double-ring neural network model with multiple time delays, J. Syst. Sci. Complexity, 35 (2021), 159–178. https://doi.org/10.1007/s11424-021-0108-2 doi: 10.1007/s11424-021-0108-2
[13]	C. A. Desoer, The optimum formula for the gain of a flow graph or a simple derivation of Coates' formula, Proc. IRE, 48 (1960), 883–889. https://doi.org/10.1109/jrproc.1960.287625 doi: 10.1109/jrproc.1960.287625
[14]	Q. Dai, Exploration of bifurcation and stability in a class of fractional-order super-double-ring neural network with two shared neurons and multiple delays, Chaos, Solitons Fractals, 168 (2023), 113185. https://doi.org/10.1016/j.chaos.2023.113185 doi: 10.1016/j.chaos.2023.113185
[15]	Y. Z. Zhang, M. Xiao, W. X. Zheng, J. D. Cao, Large-scale neural networks with asymmetrical three-ring structure: stability, nonlinear oscillations, and Hopf bifurcation, IEEE Trans. Cybern., 52 (2022), 9893–9904. https://doi.org/10.1109/tcyb.2021.3109566 doi: 10.1109/tcyb.2021.3109566
[16]	H. S. Hou, C. Luo, Z. W. Mo, On fractional ring neural networks with multiple time delays: stability and Hopf bifurcation analysis, Chin. J. Phys., 90 (2024), 303–318. https://doi.org/10.1016/j.cjph.2024.05.030 doi: 10.1016/j.cjph.2024.05.030
[17]	Y. Z. Zhang, M. Xiao, J. D. Cao, W. X. Zheng, Stability and Hopf bifurcation for a quaternion-valued three-neuron neural network with leakage delay and communication delay, J. Franklin Inst., 360 (2023), 12969–12989. https://doi.org/10.1016/j.jfranklin.2023.09.052 doi: 10.1016/j.jfranklin.2023.09.052
[18]	H. L. Li, L. Zhang, C. Hu, H. J. Jiang, J. D. Cao, Global Mittag-Leffler synchronization of fractional-order delayed quaternion-valued neural networks: Direct quaternion approach, Appl. Math. Comput., 373 (2020), 125020. https://doi.org/10.1016/j.amc.2019.125020 doi: 10.1016/j.amc.2019.125020
[19]	M. F. Zhu, B. X. Wang, Y. H. Wu, Dynamical bifurcation of large-scale-delayed fractional-order neural networks with hub structure and multiple rings, IEEE Trans. Syst. Man Cybern.: Syst., 52 (2022), 1731–1743. https://doi.org/10.1109/tsmc.2020.3037094 doi: 10.1109/tsmc.2020.3037094
[20]	M. Hymavathi, T. F. Ibrahim, M. S. Ali, G. Stamov, I. Stamova, B. A. Younis, et al., Synchronization of fractional-order neural networks with time delays and reaction-diffusion terms via pinning control, Mathematics, 10 (2022), 3916. https://doi.org/10.3390/math10203916 doi: 10.3390/math10203916
[21]	W. Y. Xu, J. D. Cao, M. Xiao, D. W. C. Ho, G. H. Wen, A new framework for analysis on stability and bifurcation in a class of neural networks with discrete and distributed delays, IEEE Trans. Cybern., 45 (2015), 2224–2236. https://doi.org/10.1109/tcyb.2014.2367591 doi: 10.1109/tcyb.2014.2367591
[22]	X. H. Tian, R. Xu, Hopf bifurcation analysis of a reaction-diffusion neural network with time delay in leakage terms and distributed delays, Neural Process. Lett., 43 (2015), 173–193. https://doi.org/10.1007/s11063-015-9410-0 doi: 10.1007/s11063-015-9410-0
[23]	L. Wang, H. Y. Zhao, J. D. Cao, Synchronized bifurcation and stability in a ring of diffusively coupled neurons with time delay, Neural Networks, 75 (2016), 32–46. https://doi.org/10.1016/j.neunet.2015.11.012 doi: 10.1016/j.neunet.2015.11.012
[24]	J. Chen, M. Xiao, X. Q. Wu, Z. X. Wang, J. D. Cao, Spatiotemporal dynamics on a class of (n+1)-dimensional reactionÂ¨Cdiffusion neural networks with discrete delays and a conical structure, Chaos Solitons Fractals, 164 (2022), 112675. https://doi.org/10.1016/j.chaos.2022.112675 doi: 10.1016/j.chaos.2022.112675
[25]	J. J. He, M. Xiao, J. Zhao, Z. X. Wang, Y. Yao, J. D. Cao, Tree-structured neural networks: Spatiotemporal dynamics and optimal control, Neural Networks, 164 (2023), 395–407. https://doi.org/10.1016/j.neunet.2023.04.039 doi: 10.1016/j.neunet.2023.04.039
[26]	Y. X. Lu, M. Xiao, J. L. Liang, J. Chen, J. X. Lin, Z. X. Wang, et al., Spatiotemporal evolution of large-scale bidirectional associative memory neural networks with diffusion and delays, IEEE Trans. Syst. Man Cybern.: Syst., 54 (2024), 1388–1400. https://doi.org/10.1109/tsmc.2023.3326186 doi: 10.1109/tsmc.2023.3326186
[27]	J. Chen, M. Xiao, Y. H. Wan, C. D. Huang, F. Y. Xu, Dynamical bifurcation for a class of large-scale fractional delayed neural networks with complex ring-hub structure and hybrid coupling, IEEE Trans. Neural Networks Learn. Syst., 34 (2023), 2659–2669. https://doi.org/10.1109/tnnls.2021.3107330 doi: 10.1109/tnnls.2021.3107330
[28]	Y. X. Lu, M. Xiao, J. J. He, Z. X. Wang, Stability and bifurcation exploration of delayed neural networks with radial-ring configuration and bidirectional coupling, IEEE Trans. Neural Networks Learn. Syst., 35 (2024), 10326–10337. https://doi.org/10.1109/tnnls.2023.3240403 doi: 10.1109/tnnls.2023.3240403
[29]	Y. X. Lu, M. Xiao, X. Q. Wu, H. R. Karimi, X. P. Xie, J. D. Cao, et al., Tipping prediction of a class of large-scale radial-ring neural networks, Neural Networks, 181 (2025), 106820. https://doi.org/10.1016/j.neunet.2024.106820 doi: 10.1016/j.neunet.2024.106820
[30]	W. H. Li, X. B. Gao, R. X. Li, Dissipativity and synchronization control of fractional-order memristive neural networks with reaction-diffusion terms, Math. Methods Appl. Sci., 42 (2019), 7494–7505. https://doi.org/10.1002/mma.5873 doi: 10.1002/mma.5873
[31]	W. J. Sun, G. J. Ren, Y. G. Yu, X. D. Hai, Global synchronization of reaction-diffusion fractional-order memristive neural networks with time delay and unknown parameters, Complexity, 2020 (2020), 1–14. https://doi.org/10.1155/2020/4145826 doi: 10.1155/2020/4145826
[32]	J. Z. Lin, J. P. Li, R. Xu, Turing instability and pattern formation of a fractional Hopfield reaction-diffusion neural network with transmission delay, Nonlinear Anal.-Model. Control, 27 (2022), 1–18. https://doi.org/10.15388/namc.2022.27.27473 doi: 10.15388/namc.2022.27.27473
[33]	B. Y. Datsko, V. V. Gafiychuk, Chaotic dynamics in Bonhoffer–van der Pol fractional reaction–diffusion system, Signal Process., 91 (2011), 452–460. https://doi.org/10.1016/j.sigpro.2010.04.004 doi: 10.1016/j.sigpro.2010.04.004
[34]	B. Datsko, V. Gafiychuk, Oscillatory wave bifurcation and spatiotemporal patterns in fractional subhyperbolic reaction-diffusion systems, Commun. Nonlinear Sci. Numer. Simul., 142 (2025), 108601. https://doi.org/10.1016/j.cnsns.2025.108601 doi: 10.1016/j.cnsns.2025.108601
[35]	B. Datsko, V. Gafiychuk, Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point, Fract. Calc. Appl. Anal., 21 (2018), 237–253. https://doi.org/10.1515/fca-2018-0015 doi: 10.1515/fca-2018-0015
[36]	Z. H. Wu, Z. M. Wang, Y. L. Cai, W. M. Wang, Turing pattern formation in a fractional-order reaction-diffusion Holling-Ⅳ predator-prey model, Adv. Contin. Discrete Models, 2025 (2025). https://doi.org/10.1186/s13662-025-03998-6 doi: 10.1186/s13662-025-03998-6
[37]	Y. H. Wu, M. K. Ni, Existence of traveling pulses for a diffusive prey¨Cpredator model with strong Allee effect and weak distributed delay, Commun. Nonlinear Sci. Numer. Simul., 143 (2025), 108596. https://doi.org/10.1016/j.cnsns.2025.108596 doi: 10.1016/j.cnsns.2025.108596
[38]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, California, 1999.
[39]	O. Brandibur, E. Kaslik, Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model, Math. Methods Appl. Sci., 41 (2018), 7182–7194. https://doi.org/10.1002/mma.4768 doi: 10.1002/mma.4768
[40]	S. H. Liu, C. D. Wang, H. N. Wang, Y. H. Jing, J. D. Cao, Dynamical detections of a fractional-order neural network with leakage, discrete and distributed delays, Eur. Phys. J. Plus, 138 (2023), 575. https://doi.org/10.1140/epjp/s13360-023-04060-8 doi: 10.1140/epjp/s13360-023-04060-8

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