Novel methods to distinguish between quasiperiodicity and a high periodicity

Novel methods to distinguish between quasiperiodicity and a high periodicity | 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 Novel methods to distinguish between quasiperiodicity and a high periodicity Ritam Bairagi, Debajyoti Guha, Soumitro Banerjee This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6503322/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Jan, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted 10 You are reading this latest preprint version Abstract Distinguishing between a quasiperiodic orbit and a high-periodic mode-locked orbit in a nonlinear system remains a challenge. In this paper, we present two approaches to address this issue. The first method analyzes how the distance between two nearby initial conditions located on the invariant closed curve evolves with time. For a high-periodic orbit, the distance asymptotically dies down, while for a quasiperiodic orbit, it remains of the same order of magnitude as the initial distance. We present an algorithm to utilize this property fruitfully to distinguish between the two types of orbits. The second method uses the spectral bifurcation diagram that plots the evolution of the spectral components as a parameter is varied. The gradual convergence of the spectral components allows one to identify the periodic windows. These methods provide convenient tools for distinguishing between high-periodicity and quasi-periodicity, subject to the limitations of machine precision. Quasiperiodicity Mode-locking Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 20 Jan, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 30 Sep, 2025 Reviews received at journal 22 Sep, 2025 Reviewers agreed at journal 03 Sep, 2025 Reviews received at journal 20 Jul, 2025 Reviewers agreed at journal 07 Jul, 2025 Reviewers agreed at journal 05 Jun, 2025 Reviewers invited by journal 05 May, 2025 Editor assigned by journal 30 Apr, 2025 Submission checks completed at journal 24 Apr, 2025 First submitted to journal 22 Apr, 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. 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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-6503322","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":452165212,"identity":"0dc6043a-a406-43d4-8f42-68aa7d94d718","order_by":0,"name":"Ritam Bairagi","email":"","orcid":"","institution":"Indian Institute of Science Education and Research Kolkata","correspondingAuthor":false,"prefix":"","firstName":"Ritam","middleName":"","lastName":"Bairagi","suffix":""},{"id":452165213,"identity":"4c6702b7-2a9a-4134-9e2d-5d82a5422642","order_by":1,"name":"Debajyoti Guha","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDUlEQVRIie3Rv0rEMBzA8V8oXJZ4tyZUuFdoOTg8KPgqueVGl1s6iPxiIFPBWZx8k5aCtyhdc4tY7gV0q3iIadFJUleHfLf8+ZCEAIRC/7GSqBcAzuZUYenGCdBhno+QSCcA2WlaVN8k+pNQ41Y3GVg5jH+Iv2lzb+L8WDNye1BVlz8vZhGQ1w7OLnxE2FavnkzNaLzGmj1ul0JDJArgWx9JbKX2ym0md44QI7OkBojdW9boJyjQXQz2FVbvnwOJPkZJo64FTjYMLMHyBOXSkcnoKcISnSqTsbTo3/IgF0ITsyoSP5k2u7bFIz+f093hrbuU6Q3Vte3yKy8BLn9NEey/x9+sHFkMhUKhUN8XU4Rd04YVV4gAAAAASUVORK5CYII=","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Debajyoti","middleName":"","lastName":"Guha","suffix":""},{"id":452165214,"identity":"aeaeff9b-48f0-464c-890b-bb602e35696c","order_by":2,"name":"Soumitro Banerjee","email":"","orcid":"","institution":"Indian Institute of Science Education and Research Kolkata","correspondingAuthor":false,"prefix":"","firstName":"Soumitro","middleName":"","lastName":"Banerjee","suffix":""}],"badges":[],"createdAt":"2025-04-22 10:38:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6503322/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6503322/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-025-11985-5","type":"published","date":"2026-01-20T15:57:13+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":101151850,"identity":"17cc7c04-cf17-4a4b-a14e-388cfd39a045","added_by":"auto","created_at":"2026-01-26 16:06:53","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4898878,"visible":true,"origin":"","legend":"","description":"","filename":"NLDFormatDistinguishingbetweenquasiperiodicityandhighperiodicity.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6503322/v1_covered_92cc2bdf-6184-4b59-8e1c-035480a00d42.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Novel methods to distinguish between quasiperiodicity and a high periodicity","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"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":"Quasiperiodicity, Mode-locking","lastPublishedDoi":"10.21203/rs.3.rs-6503322/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6503322/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Distinguishing between a quasiperiodic orbit and a high-periodic mode-locked orbit in a nonlinear system remains a challenge. 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