{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T08:19:57Z","timestamp":1783066797246,"version":"3.54.6"},"reference-count":46,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61304036"],"award-info":[{"award-number":["61304036"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009377","name":"Education Department of Hunan Province","doi-asserted-by":"publisher","award":["HNJG-20231214"],"award-info":[{"award-number":["HNJG-20231214"]}],"id":[{"id":"10.13039\/100009377","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017610","name":"Shenzhen Science and Technology Innovation Program","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100017610","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010877","name":"Shenzhen Science and Technology Innovation Commission","doi-asserted-by":"publisher","award":["JCYJ20250604175504005"],"award-info":[{"award-number":["JCYJ20250604175504005"]}],"id":[{"id":"10.13039\/501100010877","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.cnsns.2026.110129","type":"journal-article","created":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T22:19:57Z","timestamp":1778969997000},"page":"110129","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P1","title":["Stability and Hopf bifurcation of vertical magnetic bearing system with asymmetric displacement and velocity delays"],"prefix":"10.1016","volume":"162","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-5937-8133","authenticated-orcid":false,"given":"Zhizhou","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yueliang","family":"Pan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zheng","family":"Zi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuan","family":"Xie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaokun","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xitong","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lingling","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"3","key":"10.1016\/j.cnsns.2026.110129_bib0001","doi-asserted-by":"crossref","DOI":"10.1088\/2631-8695\/ad5e38","article-title":"Analysis of a passive vibration damper for high-speed superconducting magnetic bearings","volume":"6","author":"Baloochi","year":"2024","journal-title":"Eng Res Express"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110129_bib0002","doi-asserted-by":"crossref","first-page":"10147","DOI":"10.1109\/TTE.2024.3366551","article-title":"Unbalance control for high-speed active magnetic bearing systems without speed sensors","volume":"10","author":"Xu","year":"2024","journal-title":"IEEE Trans Transp Electrif"},{"issue":"12","key":"10.1016\/j.cnsns.2026.110129_bib0003","doi-asserted-by":"crossref","first-page":"3222","DOI":"10.3390\/en18123222","article-title":"Review on research and development of magnetic bearings","volume":"18","author":"Du","year":"2025","journal-title":"Energies"},{"key":"10.1016\/j.cnsns.2026.110129_bib0004","doi-asserted-by":"crossref","first-page":"12100","DOI":"10.1109\/ACCESS.2023.3241854","article-title":"Metaheuristic optimization techniques used in controlling of an active magnetic bearing system for high-speed machining application","volume":"11","author":"Gupta","year":"2023","journal-title":"IEEE Access"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110129_bib0005","doi-asserted-by":"crossref","first-page":"3287","DOI":"10.1007\/s00170-023-12389-8","article-title":"Magnetic bearing: structure, model, and control strategy","volume":"131","author":"Huang","year":"2024","journal-title":"Int J Adv Manuf Technol"},{"key":"10.1016\/j.cnsns.2026.110129_bib0006","doi-asserted-by":"crossref","first-page":"961","DOI":"10.1007\/s11071-011-0317-7","article-title":"Double hopf bifurcation of time-delayed feedback control for maglev system","volume":"69","author":"Zhang","year":"2012","journal-title":"Nonlinear Dyn"},{"key":"10.1016\/j.cnsns.2026.110129_bib0007","series-title":"2008 International conference on information and automation","first-page":"244","article-title":"Bifurcation control for maglev system with two state feedback delays","author":"Li","year":"2008"},{"key":"10.1016\/j.cnsns.2026.110129_bib0008","doi-asserted-by":"crossref","DOI":"10.1016\/j.ymssp.2022.109954","article-title":"Dynamic analysis and vibration control for a maglev vehicle-guideway coupling system with experimental verification","volume":"188","author":"Sun","year":"2023","journal-title":"Mech Syst Signal Process"},{"issue":"8","key":"10.1016\/j.cnsns.2026.110129_bib0009","first-page":"829","article-title":"Stability and hopf bifurcation of the maglev system with delayed speed feedback control","volume":"33","author":"Hong-Po","year":"2007","journal-title":"Acta Autom Sin"},{"issue":"21","key":"10.1016\/j.cnsns.2026.110129_bib0010","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.1016\/j.physd.2011.07.015","article-title":"Nonlinear analysis of a maglev system with time-delayed feedback control","volume":"240","author":"Zhang","year":"2011","journal-title":"Phys D Nonlinear Phenom"},{"issue":"8","key":"10.1016\/j.cnsns.2026.110129_bib0011","doi-asserted-by":"crossref","first-page":"6346","DOI":"10.21595\/jve.2017.18970","article-title":"Nonlinear dynamic analysis on maglev train system with flexible guideway and double time-delay feedback control","volume":"19","author":"Xu","year":"2017","journal-title":"J Vibroeng"},{"issue":"12","key":"10.1016\/j.cnsns.2026.110129_bib0012","doi-asserted-by":"crossref","first-page":"3935","DOI":"10.1142\/S0218127410028124","article-title":"Nonlinear dynamic behavior of a rotor active magnetic bearing","volume":"20","author":"Chen","year":"2010","journal-title":"Int J Bifurc Chaos"},{"issue":"12","key":"10.1016\/j.cnsns.2026.110129_bib0013","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1177\/1045389X08088666","article-title":"Nonlinear dynamics of magnetic bearing systems","volume":"19","author":"Ji","year":"2008","journal-title":"J Intell Mater Syst Struct"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110129_bib0014","doi-asserted-by":"crossref","first-page":"1539","DOI":"10.3390\/en18061539","article-title":"Speed estimation method of active magnetic bearings magnetic levitation motor based on adaptive sliding mode observer","volume":"18","author":"Gong","year":"2025","journal-title":"Energies"},{"issue":"5-6","key":"10.1016\/j.cnsns.2026.110129_bib0015","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1016\/j.mcm.2010.04.014","article-title":"Hopf bifurcation of the maglev time-delay feedback system via pseudo-oscillator analysis","volume":"52","author":"Zhang","year":"2010","journal-title":"Math Comput Model"},{"issue":"11","key":"10.1016\/j.cnsns.2026.110129_bib0016","doi-asserted-by":"crossref","first-page":"6106","DOI":"10.1016\/j.amc.2012.12.045","article-title":"Hopf bifurcation of time-delayed feedback control for maglev system with flexible guideway","volume":"219","author":"Zhang","year":"2013","journal-title":"Appl Math Comput"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110129_bib0017","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s11071-008-9432-5","article-title":"Stability and hopf bifurcation of the maglev system with delayed position and speed feedback control","volume":"57","author":"Zhang","year":"2009","journal-title":"Nonlinear Dyn"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110129_bib0018","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1007\/s00033-024-02376-8","article-title":"Bifurcation analysis of a delayed predator\u2013prey model with holling-III functional response","volume":"75","author":"Yang","year":"2024","journal-title":"Z F\u201dur Angew Math Phys"},{"issue":"05","key":"10.1016\/j.cnsns.2026.110129_bib0019","doi-asserted-by":"crossref","DOI":"10.1142\/S0218127415500741","article-title":"Stability and bifurcation analysis in a maglev system with multiple delays","volume":"25","author":"Zhang","year":"2015","journal-title":"Int J Bifurc Chaos"},{"issue":"24","key":"10.1016\/j.cnsns.2026.110129_bib0020","doi-asserted-by":"crossref","first-page":"22505","DOI":"10.1007\/s11071-024-10184-y","article-title":"Bifurcation analysis of a fractional-order eco-epidemiological system with two delays","volume":"112","author":"Zeng","year":"2024","journal-title":"Nonlinear Dyn"},{"key":"10.1016\/j.cnsns.2026.110129_bib0021","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.115673","article-title":"Complex dynamics in tick-borne disease transmission: a filippov-type control strategy model with multiple time delays","volume":"189","author":"Yu","year":"2024","journal-title":"Chaos Solit Fractals"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110129_bib0022","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1007\/s40808-025-02290-6","article-title":"Predator\u2013prey population dynamics with time delay and prey refuge effects","volume":"11","author":"Wayesa","year":"2025","journal-title":"Model Earth Syst Environ"},{"issue":"19","key":"10.1016\/j.cnsns.2026.110129_bib0023","doi-asserted-by":"crossref","first-page":"17585","DOI":"10.1007\/s11071-024-09923-y","article-title":"Multiple stability switches and hopf bifurcation in an age structured prey-predator system: effects of maturation and cooperative hunting delay","volume":"112","author":"Sahu","year":"2024","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110129_bib0024","first-page":"1","article-title":"Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic","volume":"2024","author":"Liu","year":"2024","journal-title":"J Inequalities Appl"},{"key":"10.1016\/j.cnsns.2026.110129_bib0025","first-page":"807","article-title":"Bifurcation and control of a predator-prey system with two time delays","volume":"16","author":"Lin","year":"2026","journal-title":"J Appl Anal Comput"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110129_bib0026","doi-asserted-by":"crossref","DOI":"10.3934\/era.2025045","article-title":"Hopf bifurcation analysis of a multiple delays stage-structure predator-prey model with refuge and cooperation","volume":"33","author":"Wu","year":"2025","journal-title":"Electron Res Arch"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110129_bib0027","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s12190-010-0383-x","article-title":"Hopf bifurcation in a three-species system with delays","volume":"35","author":"Meng","year":"2011","journal-title":"J Appl Math Comput"},{"issue":"7","key":"10.1016\/j.cnsns.2026.110129_bib0028","doi-asserted-by":"crossref","first-page":"8001","DOI":"10.1007\/s42417-024-01342-y","article-title":"Bistability, period n solutions, quasi-periodic, phase-locked solutions and chaos in an optoelectronic reservoir computing system with two delayed feedback loops","volume":"12","author":"Pei","year":"2024","journal-title":"J Vib Eng Technol"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110129_bib0029","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.jmaa.2005.02.034","article-title":"On stability crossing curves for general systems with two delays","volume":"311","author":"Gu","year":"2005","journal-title":"J Math Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110129_bib0030","article-title":"Hopf bifurcation analysis for a maglev system with two time delays","volume":"224","author":"Zhang","year":"2025","journal-title":"Mech Syst Signal Process"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110129_bib0031","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s12190-024-02240-y","article-title":"Stability and hopf bifurcation analysis of a networked SIR epidemic model with two delays","volume":"71","author":"Zhou","year":"2025","journal-title":"J Appl Math Comput"},{"key":"10.1016\/j.cnsns.2026.110129_bib0032","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.115956","article-title":"Impacts of microbial decomposition delays on the stability of paddy ecosystem during fallow season","volume":"192","author":"Zhou","year":"2025","journal-title":"Chaos Solit Fractals"},{"key":"10.1016\/j.cnsns.2026.110129_bib0033","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2023.113955","article-title":"Stability and hopf bifurcation of a multiple delayed predator\u2013prey system with fear effect, prey refuge and crowley\u2013martin function","volume":"175","author":"Liang","year":"2023","journal-title":"Chaos Solit Fractals"},{"issue":"9","key":"10.1016\/j.cnsns.2026.110129_bib0034","doi-asserted-by":"crossref","first-page":"3188","DOI":"10.1088\/1361-6544\/ab1f2f","article-title":"Diffusive spatial movement with memory and maturation delays","volume":"32","author":"Shi","year":"2019","journal-title":"Nonlinearity"},{"key":"10.1016\/j.cnsns.2026.110129_bib0035","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s10884-015-9432-3","article-title":"Stability switches in a logistic population model with mixed instantaneous and delayed density dependence","volume":"29","author":"Yan","year":"2017","journal-title":"J Dyn Differ Equ"},{"key":"10.1016\/j.cnsns.2026.110129_bib0036","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1007\/s11431-014-5536-y","article-title":"Stability switches and bogdanov-takens bifurcation in an inertial two-neuron coupling system with multiple delays","volume":"57","author":"Song","year":"2014","journal-title":"Sci China Technol Sci"},{"key":"10.1016\/j.cnsns.2026.110129_bib0037","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2022.112975","article-title":"Dynamics exploration for a fractional-order delayed zooplankton\u2013phytoplankton system","volume":"166","author":"Li","year":"2023","journal-title":"Chaos Solit Fractals"},{"key":"10.1016\/j.cnsns.2026.110129_bib0038","article-title":"Exploration of soliton solutions and dynamical analysis of the q-form zhanbota equation: an application to image encryption","author":"Li","year":"2025","journal-title":"Phys A Stat Mech Appl"},{"key":"10.1016\/j.cnsns.2026.110129_bib0039","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.compind.2019.03.001","article-title":"Hopf bifurcation analysis of maglev vehicle\u2013guideway interaction vibration system and stability control based on fuzzy adaptive theory","volume":"108","author":"Sun","year":"2019","journal-title":"Comput Ind"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110129_bib0040","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1006\/jsvi.2000.2916","article-title":"Bifurcation behavior of a rotor supported by active magnetic bearings","volume":"235","author":"Ji","year":"2000","journal-title":"J Sound Vib"},{"key":"10.1016\/j.cnsns.2026.110129_bib0041","series-title":"Theory and Applications of Hopf Bifurcation","volume":"vol. 41","author":"Hassard","year":"1981"},{"key":"10.1016\/j.cnsns.2026.110129_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.ymssp.2026.114158","article-title":"Stability and Hopf bifurcation analysis for a vertical active maglev bearing system with dual feedback delays","volume":"250","author":"Zhang","year":"2026","journal-title":"Mech Syst Signal Process"},{"key":"10.1016\/j.cnsns.2026.110129_bib0043","doi-asserted-by":"crossref","DOI":"10.1142\/S1793524525501013","article-title":"Dynamics analysis of a delayed diffusive predator\u2013prey model with memory-based diffusion and fear effect of prey","author":"Meng","year":"2025","journal-title":"Int J Biomath"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110129_bib0044","doi-asserted-by":"crossref","DOI":"10.1142\/S0218127426500719","article-title":"Bifurcations and Control of Plant\u2013Herbivore Systems with Spatiotemporal Nonlocal Effects in Arid Regions","volume":"36","author":"Wang","year":"2026","journal-title":"Int J Bifurc Chaos"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110129_bib0045","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1186\/s13662-025-03959-z","article-title":"Bifurcation and control of a delayed Leslie\u2013Gower fractional order predator\u2013prey model with fear effect and prey refuge","volume":"2025","author":"Wang","year":"2025","journal-title":"Adv Contin Discret Model"},{"issue":"07","key":"10.1016\/j.cnsns.2026.110129_bib0046","doi-asserted-by":"crossref","DOI":"10.1142\/S0218127424500901","article-title":"Hopf bifurcation and Turing instability of a delayed diffusive Zooplankton\u2013Phytoplankton model with hunting cooperation","volume":"34","author":"Meng","year":"2024","journal-title":"Int J Bifurc Chaos"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426004880?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426004880?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T07:53:44Z","timestamp":1783065224000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426004880"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":46,"alternative-id":["S1007570426004880"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110129","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Stability and Hopf bifurcation of vertical magnetic bearing system with asymmetric displacement and velocity delays","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110129","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110129"}}