{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T08:36:44Z","timestamp":1769935004209,"version":"3.49.0"},"reference-count":36,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T00:00:00Z","timestamp":1747094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research and Innovation Team Program of the Sichuan University of Science and Engineering","award":["SUSE652B002"],"award-info":[{"award-number":["SUSE652B002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley\u2013Dickson construction. The existence and uniqueness of the solution for the considered fractional-order delayed quaternion neural networks are proven by using the compression mapping theorem. It is demonstrated that the solutions of the involved fractional delayed quaternion neural networks are bounded by constructing appropriate functions. Some sufficient conditions for the stability and Hopf bifurcation of the considered fractional-order delayed quaternion neural networks are established by utilizing the stability theory of fractional differential equations and basic bifurcation knowledge. To validate the rationality of the theoretical results, corresponding simulation results and bifurcation diagrams are provided. The relationship between the order of appearance of bifurcation phenomena and the order is also studied, revealing that bifurcation phenomena occur later as the order increases. The theoretical results established in this paper are of significant guidance for the design and improvement of neural networks.<\/jats:p>","DOI":"10.3390\/axioms14050366","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T06:40:23Z","timestamp":1747118423000},"page":"366","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Stability and Hopf Bifurcation of Fractional-Order Quaternary Numerical Three-Neuron Neural Networks with Different Types of Delays"],"prefix":"10.3390","volume":"14","author":[{"given":"Qiankun","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5706-2443","authenticated-orcid":false,"given":"Tianzeng","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Hainan Normal University, Haikou 571127, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaowen","family":"Tan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TQE.2022.3174017","article-title":"Neural-Network Decoders for Quantum Error Correction Using Surface Codes: A Space Exploration of the Hardware Cost-Performance Tradeoffs","volume":"3","author":"Overwater","year":"2022","journal-title":"IEEE Trans. Quantum Eng."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Khan, A.I., Shah, J.L., and Bhat, M.M. (2020). CoroNet: A deep neural network for detection and diagnosis of COVID-19 from chest x-ray images. Comput. Methods Programs Biomed., 196.","DOI":"10.1016\/j.cmpb.2020.105581"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1559","DOI":"10.1109\/TIP.2022.3144017","article-title":"Weighted feature fusion of convolutional neural network and graph attention network for hyperspectral image classification","volume":"31","author":"Dong","year":"2022","journal-title":"IEEE Trans. Image Process."},{"key":"ref_4","first-page":"2935","article-title":"Memristor-based neural network circuit of full-function pavlov associative memory with time delay and variable learning rate","volume":"50","author":"Sun","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Urbaniak, I., and Wolter, M. (2020). Quality assessment of compressed and resized medical images based on pattern recognition using a convolutional neural network. Commun. Nonlinear Sci. Numer. Simul., 95.","DOI":"10.1016\/j.cnsns.2020.105582"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1109\/TNSE.2022.3223930","article-title":"Privacy protection of medical data based on multi-scroll memristive Hopfield neural network","volume":"10","author":"Yu","year":"2022","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s11424-021-0108-2","article-title":"Stability and Hopf bifurcation analysis of an (n + m)-neuron double-ring neural network model with multiple time delays","volume":"35","author":"Xing","year":"2022","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1080\/00207390410001686571","article-title":"A brief historical introduction to fractional calculus","volume":"35","author":"Debnath","year":"2004","journal-title":"Int. J. Math. Educ. Sci. Technol."},{"key":"ref_9","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1007\/s40314-021-01595-3","article-title":"Qualitative analysis of Caputo fractional integro-differential equations with constant delays","volume":"40","author":"Bohner","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Yang, X.J., Gao, F., and Ju, Y. (2020). General Fractional Derivatives with Applications in Viscoelasticity, Academic Press.","DOI":"10.1016\/B978-0-12-817208-7.00011-X"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1108\/09615531311289088","article-title":"A random walk solution for fractional diffusion equations","volume":"23","author":"Zielinski","year":"2013","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1016\/j.ijleo.2018.11.087","article-title":"Fractional model of the dielectric dispersion","volume":"180","author":"Ortega","year":"2019","journal-title":"Optik"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Herrmann, R. (2013). Infrared spectroscopy of diatomic moleculesa fractional calculus approach. Int. J. Mod. Phys. B, 27.","DOI":"10.1142\/S0217979213500197"},{"key":"ref_15","first-page":"2357","article-title":"Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel","volume":"14","author":"Kachhia","year":"2021","journal-title":"Discret. Contin. Dyn. Syst.-Ser. S"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"13897","DOI":"10.1007\/s10462-023-10474-8","article-title":"A survey of fractional calculus applications in artificial neural networks","volume":"56","author":"Joshi","year":"2023","journal-title":"Artif. Intell. Rev."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wu, J., Campbell, S.A., and B\u00e9lair, J. (2022). Time-delayed neural networks: Stability and oscillations. Encyclopedia of Computational Neuroscience, Springer.","DOI":"10.1007\/978-1-0716-1006-0_513"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Braverman, E., Tun\u00e7, C., and Tun\u00e7, O. (2025). On global stability of nonlinear systems with unbounded and distributed delays and a dominating non-delay term. Commun. Nonlinear Sci. Numer. Simul., 143.","DOI":"10.1016\/j.cnsns.2025.108590"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3007","DOI":"10.1007\/s00034-020-01610-8","article-title":"Effect of sparse representation of time series data on learning rate of time-delay neural networks","volume":"40","author":"Mikhael","year":"2021","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_20","unstructured":"Ji, X.A., Molnr, T.G., Avedisov, S.S., and Orosz, G. (2021, January 13). Learning the dynamics of time delay systems with trainable delays. Proceedings of the 3rd Conference on Learning for Dynamics and Control, Virtual."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3317","DOI":"10.1109\/TAC.2019.2940865","article-title":"A dynamic feedback framework for control of time-delay nonlinear systems with unstable zero dynamics","volume":"65","author":"Lin","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2725","DOI":"10.1109\/TCSI.2021.3062970","article-title":"Damping power system electromechanical oscillations using time delays","volume":"68","author":"Tzounas","year":"2021","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2019.08.001","article-title":"Chaos in time delay systems, an educational review","volume":"824","author":"Wernecke","year":"2019","journal-title":"Phys. Rep."},{"key":"ref_24","unstructured":"Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981). Theory and Applications of Hopf Bifurcation, CUP Archive."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Voight, J. (2021). Quaternion Algebras, Springer Nature.","DOI":"10.1007\/978-3-030-56694-4"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2957","DOI":"10.1007\/s10462-019-09752-1","article-title":"A survey of quaternion neural networks","volume":"53","author":"Parcollet","year":"2020","journal-title":"Artif. Intell. Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"12969","DOI":"10.1016\/j.jfranklin.2023.09.052","article-title":"Stability and Hopf bifurcation for a quaternion-valued three-neuron neural network with leakage delay and communication delay","volume":"360","author":"Zhu","year":"2023","journal-title":"J. Frankl. Inst."},{"key":"ref_28","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1810","DOI":"10.1016\/j.camwa.2009.08.019","article-title":"Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized MittagCLeffler stability","volume":"59","author":"Li","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Bandyopadhyay, B., and Kamal, S. (2015). Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach, Springer.","DOI":"10.1007\/978-3-319-08621-7"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s12190-016-1017-8","article-title":"Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge","volume":"54","author":"Li","year":"2017","journal-title":"J. Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s11071-006-9094-0","article-title":"Stability analysis of linear fractional differential system with multiple time delays","volume":"48","author":"Deng","year":"2007","journal-title":"Nonlinear Dyn."},{"key":"ref_33","first-page":"963","article-title":"Stability results for fractional differential equations with applications to control processing","volume":"2","author":"Matignon","year":"1996","journal-title":"Comput. Eng. Syst. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.neunet.2019.05.002","article-title":"Novel bifurcation results for a delayed fractional-order quaternion-valued neural network","volume":"117","author":"Huang","year":"2019","journal-title":"Neural Netw."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1080\/0025570X.2022.2125254","article-title":"An Accessible Proof of Hurwitzs Sums of Squares Theorem","volume":"95","author":"Brown","year":"2022","journal-title":"Math. Mag."},{"key":"ref_36","first-page":"1","article-title":"A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order","volume":"1","author":"Bhalekar","year":"2011","journal-title":"Fract. Calc. Appl. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/366\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:31:50Z","timestamp":1760031110000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/366"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,13]]},"references-count":36,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050366"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050366","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,13]]}}}