{"paper_id":"281ece4e-6834-4fec-a971-046a7fecef03","body_text":"Main sub-harmonic joint resonance of fractional quintic van der Pol-Duffing oscillator | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Main sub-harmonic joint resonance of fractional quintic van der Pol-Duffing oscillator Zhongkai Ren, Jiazhao Chen, Tingyu Wang, Zehua Zhang, Penghao Zhao, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4393503/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 10 Jul, 2024 Read the published version in Nonlinear Dynamics → Version 1 posted 9 You are reading this latest preprint version Abstract The main sub-harmonic joint resonance of the van der Pol-Duffing system with a quintic oscillator under dual-frequency excitation is investigated in this paper. The study examines the conditions for chaos and vibration resonance under different parameters. An approximate analytical solution for the principal sub-harmonic joint resonance of the system under dual-frequency excitation is obtained using the multi-scale method, while the Melnikov method provides necessary conditions for chaos in the system. Furthermore, based on the fast and slow variable separation method, vibration resonance of the system under various conditions is determined. Numerical simulations explore amplitude-frequency characteristics of total response at different excitation frequencies through analytical and simulation methods, with consistency between numerical and analytical results verified by plotting amplitude-frequency characteristic curves. Additionally, an analysis is conducted to investigate how fractional order, fractional differential coefficient, and cubic stiffness affect co-amplitude-frequency curves of the van der Pol-Duffing oscillator. The analysis reveals that a jump phenomenon exists in co-amplitude-harmonic resonance of this oscillator; moreover, changes in different parameters can alter both jump points and cause disappearance of such phenomena. Sub-critical fork bifurcation behavior as well as supercritical fork bifurcation behavior are studied along with vibration resonance caused by parameter variations. Results indicate that sub-critical fork bifurcation arises from changes in excitation term coefficient while supercritical fork bifurcation occurs due to fractional order variations. Furthermore, when different fractional order values are considered, there will be changes in resonance location, response amplitude gain, and vibration resonance mode within the system. The implementation of this measure enhances our comprehension of the vibration characteristics of the system, thereby refining the accuracy of the model and bolstering the stability of the system. Additionally, it serves as a preventive measure against resonance issues, which are particularly critical for mitigating the hazards associated with system resonance triggered by supercritical fork bifurcations. These hazards encompass potential structural damage and equipment failure. Multi-Scale Method Melnikov method Fast and slow variable separation method Vibration resonance Main sub-harmonic joint resonance Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 10 Jul, 2024 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 06 Jun, 2024 Reviews received at journal 26 May, 2024 Reviews received at journal 25 May, 2024 Reviewers agreed at journal 18 May, 2024 Reviewers agreed at journal 18 May, 2024 Reviewers invited by journal 17 May, 2024 Editor assigned by journal 14 May, 2024 Submission checks completed at journal 14 May, 2024 First submitted to journal 09 May, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-4393503\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":303953430,\"identity\":\"d9507c7d-2eaa-4c6b-9999-83a4ad2b152e\",\"order_by\":0,\"name\":\"Zhongkai Ren\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Taiyuan University of Technology\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Zhongkai\",\"middleName\":\"\",\"lastName\":\"Ren\",\"suffix\":\"\"},{\"id\":303953433,\"identity\":\"bb280f85-f95e-4c79-9cbe-3db7f913b8bb\",\"order_by\":1,\"name\":\"Jiazhao 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The study examines the conditions for chaos and vibration resonance under different parameters. An approximate analytical solution for the principal sub-harmonic joint resonance of the system under dual-frequency excitation is obtained using the multi-scale method, while the Melnikov method provides necessary conditions for chaos in the system. Furthermore, based on the fast and slow variable separation method, vibration resonance of the system under various conditions is determined. Numerical simulations explore amplitude-frequency characteristics of total response at different excitation frequencies through analytical and simulation methods, with consistency between numerical and analytical results verified by plotting amplitude-frequency characteristic curves. Additionally, an analysis is conducted to investigate how fractional order, fractional differential coefficient, and cubic stiffness affect co-amplitude-frequency curves of the van der Pol-Duffing oscillator. The analysis reveals that a jump phenomenon exists in co-amplitude-harmonic resonance of this oscillator; moreover, changes in different parameters can alter both jump points and cause disappearance of such phenomena. Sub-critical fork bifurcation behavior as well as supercritical fork bifurcation behavior are studied along with vibration resonance caused by parameter variations. Results indicate that sub-critical fork bifurcation arises from changes in excitation term coefficient while supercritical fork bifurcation occurs due to fractional order variations. Furthermore, when different fractional order values are considered, there will be changes in resonance location, response amplitude gain, and vibration resonance mode within the system. The implementation of this measure enhances our comprehension of the vibration characteristics of the system, thereby refining the accuracy of the model and bolstering the stability of the system. Additionally, it serves as a preventive measure against resonance issues, which are particularly critical for mitigating the hazards associated with system resonance triggered by supercritical fork bifurcations. 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