{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:51:50Z","timestamp":1762509110084,"version":"build-2065373602"},"reference-count":60,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T00:00:00Z","timestamp":1762300800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12002120","12572034","12072222","11672349","11902287"],"award-info":[{"award-number":["12002120","12572034","12072222","11672349","11902287"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100006407","name":"Natural Science Foundation of Henan Province","doi-asserted-by":"publisher","award":["252300420315"],"award-info":[{"award-number":["252300420315"]}],"id":[{"id":"10.13039\/501100006407","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Academic and Technical Leader of Henan University of Urban Construction","award":["YCJXSJSDTR202308"],"award-info":[{"award-number":["YCJXSJSDTR202308"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study investigates stochastic bifurcation in a generalized tristable Rayleigh\u2013Duffing oscillator with fractional inertial force under both additive and multiplicative recycling noises. The system\u2019s dynamic behavior is influenced by its inherent spatial symmetry, represented by the potential function, as well as by temporal symmetry breaking caused by fractional memory effects and recycling noise. First, an approximate integer-order equivalent system is derived from the original fractional-order model using a harmonic balance method, with minimal mean square error (MSE). The steady-state probability density function (sPDF) of the amplitude is then obtained via stochastic averaging. Using singularity theory, the conditions for stochastic P bifurcation (SPB) are identified. For different fractional derivative\u2019s orders, transition set curves are constructed, and the sPDF is qualitatively analyzed within the regions bounded by these curves\u2014especially under tristable conditions. Theoretical results are validated through Monte Carlo simulations and the Radial Basis Function Neural Network (RBFNN) approach. The findings offer insights for designing fractional-order controllers to improve system response control.<\/jats:p>","DOI":"10.3390\/sym17111874","type":"journal-article","created":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T13:19:29Z","timestamp":1762348769000},"page":"1874","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Response Control and Bifurcation Phenomenon of a Tristable Rayleigh\u2013Duffing System with Fractional Inertial Force Under Recycling Noises"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3790-4201","authenticated-orcid":false,"given":"Yajie","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Henan University of Urban Construction, Pingdingshan 467036, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guoguo","family":"Tian","sequence":"additional","affiliation":[{"name":"School of Foreign Studies, Henan University of Urban Construction, Pingdingshan 467036, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiqiang","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Tianjin University, Tianjin 300072, China"},{"name":"Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongtao","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Tianjin University, Tianjin 300072, China"},{"name":"Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Hao","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao 066004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiangyun","family":"Zhang","sequence":"additional","affiliation":[{"name":"Basic Course Department, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shengli","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Transportation Engineering, East China Jiaotong University, Nanchang 330013, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1007\/BF02877067","article-title":"Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion","volume":"44","author":"Xu","year":"2001","journal-title":"Sci. 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