Noname manuscript No. (will be inserted by the editor) Nonlinear Normal Modes based Study of Synchronization in Delay Coupled Limit Cycle Oscillators.

Noname manuscript No. (will be inserted by the editor) Nonlinear Normal Modes based Study of Synchronization in Delay Coupled Limit Cycle 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 Noname manuscript No. (will be inserted by the editor) Nonlinear Normal Modes based Study of Synchronization in Delay Coupled Limit Cycle Oscillators. Manoj Pandey, Govind M This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1725180/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 18 Jul, 2023 Read the published version in Nonlinear Dynamics → Version 1 posted 5 You are reading this latest preprint version Abstract Nonlinear normal modes (NNMs), a powerful analytical tool for studying nonlinear systems is used here to study synchronization dynamics in delay coupled oscillators, which exhibit limit cycle oscillations (LCO) and hysteresis when uncoupled. A method described by Shaw and Pierre is used to arrive at the NNMs which are found to be two dimensional invariant manifolds containing closed curves corresponding to either the in-phase or out-of-phase synchronized oscillations for both delayed and non-delayed, reactively or dissipatively coupled systems. Motion initiated on these NNMs, is found to evolve strictly on them, settling onto a stable orbit if existent or on to a quasi-periodic orbit lying between the two manifolds in the absence of one. The NNMs are used to decouple the governing equations of the system, giving the in-phase and out-of-phase synchronization frequencies directly. Parametric study in terms of strength of coupling, frequency detuning and delay time is performed using the averaged equations of the system, aided by continuation software AUTO, which also reveals the bifurcations in the system. These are verified using NNM, direct numerical integration and shooting method based Floquent analysis of the original system and are found to match well. The results obtained here agree with prior work in literature, at the same time extending them to hitherto unexplored parameter range and provide a new perspective on the same from the point of view of NNMs. Hence this study demonstrates the utility of NNMs for the study of synchronization dynamics in both delayed and non delay mutually coupled LCOs and can be useful for studying larger arrays of the same. Mathematics Subject Classification (2020) 70K75 · 34D06 · 34K24 Synchronization nonlinear normal modes delayed differential equations Full Text Cite Share Download PDF Status: Published Journal Publication published 18 Jul, 2023 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Major revisions 25 Sep, 2022 Reviewers agreed at journal 13 Jun, 2022 Reviewers invited by journal 12 Jun, 2022 Editor assigned by journal 07 Jun, 2022 First submitted to journal 04 Jun, 2022 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-1725180","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":112944018,"identity":"c26aaca1-646f-455d-83e5-44a7777109ca","order_by":0,"name":"Manoj Pandey","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAx0lEQVRIiWNgGAWjYLACHgYGOTZmKIeNWC3GbMzMJGpJbGBgJqwQDHRnNz978KbicHofO//RDQw1dgx80g34tZjdOWZuOOfM4dw2Zma2GwzHkhnYZA4Q0HIjwUyatw2mhe0AA5tEAiEt6d+kef8dTmcDa/lHlJYcoC0NhxPAWhjbiNFy50yZ5Jxj6YZAh5ndSOxL5iGs5Xb7Nok3Ndby8v0Hn9348M1OTn4GAS0MEmCyGcJJAMcRIQDRUkdY4SgYBaNgFIxcAADWzjtGrXzAvwAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-1032-6832","institution":"Indian Institute of Technology Madras","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Manoj","middleName":"","lastName":"Pandey","suffix":""},{"id":112944019,"identity":"24718314-b57b-466d-9166-75a00c4e71a1","order_by":1,"name":"Govind M","email":"","orcid":"","institution":"Indian Institute of Technology Madras","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Govind","middleName":"","lastName":"M","suffix":""}],"badges":[],"createdAt":"2022-06-04 10:47:01","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1725180/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1725180/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-023-08631-3","type":"published","date":"2023-07-18T21:40:30+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":22648419,"identity":"9aaba35c-9fa0-4f05-9d77-944763cad5b7","added_by":"auto","created_at":"2022-06-14 16:35:57","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":8362959,"visible":true,"origin":"","legend":"","description":"","filename":"NonlineardynamicsformatNNMND.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1725180/v1_covered.pdf"}],"financialInterests":"","formattedTitle":"Noname manuscript No. (will be inserted by the editor)\nNonlinear Normal Modes based Study of Synchronization in Delay Coupled Limit Cycle Oscillators.","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1725180/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"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},"keywords":"Synchronization, nonlinear normal modes, delayed differential equations","lastPublishedDoi":"10.21203/rs.3.rs-1725180/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1725180/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eNonlinear normal modes (NNMs), a powerful analytical tool for studying nonlinear systems is used here to study synchronization dynamics in delay coupled oscillators, which exhibit limit cycle oscillations (LCO) and hysteresis when uncoupled. A method described by Shaw and Pierre is used to arrive at the NNMs which are found to be two dimensional invariant manifolds containing closed curves corresponding to either the in-phase or out-of-phase synchronized oscillations for both delayed and non-delayed, reactively or dissipatively coupled systems. Motion initiated on these NNMs, is found to evolve strictly on them, settling onto a stable orbit if existent or on to a quasi-periodic orbit lying between the two manifolds in the absence of one. The NNMs are used to decouple the governing equations of the system, giving the in-phase and out-of-phase synchronization frequencies directly. Parametric study in terms of strength of coupling, frequency detuning and delay time is performed using the averaged equations of the system, aided by continuation software AUTO, which also reveals the bifurcations in the system. These are verified using NNM, direct numerical integration and shooting method based Floquent analysis of the original system and are found to match well. 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