On the hysteresis and dynamics of a ring of three unidirectionally coupled bistable Duffing oscillators

On the hysteresis and dynamics of a ring of three unidirectionally coupled bistable Duffing oscillators | 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 On the hysteresis and dynamics of a ring of three unidirectionally coupled bistable Duffing oscillators José de Jesús Barba-Franco, Eduardo Gaspar-Sánchez, Ernesto Urenda-Cázares, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6597660/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 26 Feb, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted 9 You are reading this latest preprint version Abstract We study a ring of three Duffing oscillators coupled unidirectionally, focusing on the effects of the coupling strength, time-dependent damping and attractor following using time series, phase portrait, Fourier transformations, bifurcation diagrams, potential planes, Lyapunov exponents and hysteresis paths for two cases of damping: constant \((\alpha=0.4)\) and linearly increasing \((\alpha(t)=\frac{t}{\tau})\) . For constant damping term the dynamics shows quasiperiodic 2D torus, limit cycles, heteroclinic orbits, fixed points, chaos, and hyperchaos. These behavior shows the existence of hysteresis and an important change in the potential energy. If damping term is dependent of the time \((\alpha = \frac{t}{\tau} )\) , the dynamics of the system shows bottleneck behavior influenced by higher frequencies which persists for a small range of values of \((\sigma)\) . In resume, the potential energy in this set of three Duffing oscillators depends strongly of three factors: first the attractor tracking (past memories), second the coupling strength and third the damping term. Duffing Oscillator hysteresis coupling potential energy Pitchfork Bifurcation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 26 Feb, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 18 Jun, 2025 Reviews received at journal 10 Jun, 2025 Reviews received at journal 31 May, 2025 Reviewers agreed at journal 20 May, 2025 Reviewers agreed at journal 19 May, 2025 Reviewers invited by journal 19 May, 2025 Editor assigned by journal 10 May, 2025 Submission checks completed at journal 07 May, 2025 First submitted to journal 05 May, 2025 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-6597660","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":459195805,"identity":"49611bda-b955-44ad-96b8-4294d619efa6","order_by":0,"name":"José de Jesús Barba-Franco","email":"","orcid":"","institution":"Tecnológico Nacional de México","correspondingAuthor":false,"prefix":"","firstName":"José","middleName":"de Jesús","lastName":"Barba-Franco","suffix":""},{"id":459195806,"identity":"820b7f56-4b94-4f61-8673-460e615ba310","order_by":1,"name":"Eduardo Gaspar-Sánchez","email":"","orcid":"","institution":"University of 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Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Duffing Oscillator, hysteresis, coupling, potential energy, Pitchfork Bifurcation","lastPublishedDoi":"10.21203/rs.3.rs-6597660/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6597660/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe study a ring of three Duffing oscillators coupled unidirectionally, focusing on the effects of the coupling strength, time-dependent damping and attractor following using time series, phase portrait, Fourier transformations, bifurcation diagrams, potential planes, Lyapunov exponents and hysteresis paths for two cases of damping: constant \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\alpha=0.4)\\)\u003c/span\u003e\u003c/span\u003e and linearly increasing \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\alpha(t)=\\frac{t}{\\tau})\\)\u003c/span\u003e\u003c/span\u003e. For constant damping term the dynamics shows quasiperiodic 2D torus, limit cycles, heteroclinic orbits, fixed points, chaos, and hyperchaos. These behavior shows the existence of hysteresis and an important change in the potential energy. If damping term is dependent of the time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\alpha = \\frac{t}{\\tau} )\\)\u003c/span\u003e\u003c/span\u003e, the dynamics of the system shows bottleneck behavior influenced by higher frequencies which persists for a small range of values of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\sigma)\\)\u003c/span\u003e\u003c/span\u003e. In resume, the potential energy in this set of three Duffing oscillators depends strongly of three factors: first the attractor tracking (past memories), second the coupling strength and third the damping term. \u003c/p\u003e","manuscriptTitle":"On the hysteresis and dynamics of a ring of three unidirectionally coupled bistable Duffing oscillators","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-05-21 06:13:24","doi":"10.21203/rs.3.rs-6597660/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-06-18T11:55:28+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-06-10T14:12:03+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-05-31T10:44:38+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"41464412125147770837160868672367319028","date":"2025-05-20T08:39:11+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"139917983709892908715900016589998716264","date":"2025-05-19T13:38:36+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-05-19T13:19:30+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-05-10T13:54:14+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-05-07T15:11:50+00:00","index":"","fulltext":""},{"type":"submitted","content":"Nonlinear Dynamics","date":"2025-05-05T23:49:54+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"4b9eb4dd-4ef1-4137-b0bf-2b1677ffec67","owner":[],"postedDate":"May 21st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-03-02T16:01:37+00:00","versionOfRecord":{"articleIdentity":"rs-6597660","link":"https://doi.org/10.1007/s11071-025-12168-y","journal":{"identity":"nonlinear-dynamics","isVorOnly":false,"title":"Nonlinear Dynamics"},"publishedOn":"2026-02-26 15:58:28","publishedOnDateReadable":"February 26th, 2026"},"versionCreatedAt":"2025-05-21 06:13:24","video":"","vorDoi":"10.1007/s11071-025-12168-y","vorDoiUrl":"https://doi.org/10.1007/s11071-025-12168-y","workflowStages":[]},"version":"v1","identity":"rs-6597660","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6597660","identity":"rs-6597660","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}