Bifurcation of limit cycles from a cubic reversible isochrone

Bifurcation of limit cycles from a cubic reversible isochrone | 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 Bifurcation of limit cycles from a cubic reversible isochrone Jihua Yang, Qipeng Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7389808/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Apr, 2026 Read the published version in Qualitative Theory of Dynamical Systems → Version 1 posted 16 You are reading this latest preprint version Abstract For the polynomial differential system ˙ x = −y + Chavarriga and García proved that when certain parameters of the system satisfy specific conditions, the origin is an isochronous center if and only if the system can be transformed into one of the five types: CR 1 , CR 2 , CR 3 , CR 4 , or CR 5. However, the bifurcation of limit cycles in these five isochronous systems has remained unexplored. In this paper, we focus on studying the limit cycle bifurcation of the CR 5 system under polynomial Liénard-type perturbations. Using Abelian integrals, we derive an upper bound for the number of limit cycles that can emerge from such perturbations and verify the existence of limit cycles for n = 1, 2, 3 through numerical simulations. The method we employed to obtain the algebraic structure of the Abelian integral differs in many aspects from other approaches. MSC 34C07; 34C05 cubic reversible isochronous system tangential Hilbert’s 16th problem Abelian integral limit cycle elliptic integral Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 Apr, 2026 Read the published version in Qualitative Theory of Dynamical Systems → Version 1 posted Editorial decision: Revision requested 08 Oct, 2025 Reviews received at journal 08 Oct, 2025 Reviews received at journal 25 Sep, 2025 Reviews received at journal 20 Sep, 2025 Reviews received at journal 19 Sep, 2025 Reviews received at journal 11 Sep, 2025 Reviewers agreed at journal 28 Aug, 2025 Reviewers agreed at journal 22 Aug, 2025 Reviewers agreed at journal 21 Aug, 2025 Reviewers agreed at journal 21 Aug, 2025 Reviewers agreed at journal 20 Aug, 2025 Reviewers agreed at journal 19 Aug, 2025 Reviewers invited by journal 19 Aug, 2025 Editor assigned by journal 19 Aug, 2025 Submission checks completed at journal 19 Aug, 2025 First submitted to journal 16 Aug, 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. 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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-7389808","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":506137174,"identity":"b010f4db-d4a4-463b-aaa2-4ee7ea040ef3","order_by":0,"name":"Jihua Yang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwElEQVRIiWNgGAWjYBACxmYeIGkgwSPP3tj48APxWgosZAx7DjcbSxBnD0jLhwobhhvpbQI8xGhgbuc9JvED6DDGmQ/bGCQY7OR0Gwg6jC9NsgeohV06se1BAUOysdkBglp4zCR4QLbMTmw3kGA4kLiNGC2Sf4BaGG4ebAOSRGqRBtnCcIOReC3G1jJALYY9icBANiDCL4b9ZwxvvvlTZy/Pfvzhww8VdnKEtTSgcA0IKAcBeSLUjIJRMApGwUgHAHPcOYqxVS2xAAAAAElFTkSuQmCC","orcid":"","institution":"Tianjin Normal University","correspondingAuthor":true,"prefix":"","firstName":"Jihua","middleName":"","lastName":"Yang","suffix":""},{"id":506137175,"identity":"db42e64a-a01b-49ac-9900-96add17df3dd","order_by":1,"name":"Qipeng Zhang","email":"","orcid":"","institution":"Tianjin Normal University","correspondingAuthor":false,"prefix":"","firstName":"Qipeng","middleName":"","lastName":"Zhang","suffix":""}],"badges":[],"createdAt":"2025-08-17 00:53:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7389808/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7389808/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s12346-026-01498-w","type":"published","date":"2026-04-06T15:58:23+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":106809158,"identity":"505d232e-63ae-4834-bba2-0bcd9d56475e","added_by":"auto","created_at":"2026-04-13 16:07:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":487141,"visible":true,"origin":"","legend":"","description":"","filename":"CR5.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7389808/v1_covered_6cd49996-fa91-435f-8a1f-ce6692e41efe.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Bifurcation of limit cycles from a cubic reversible isochrone","fulltext":[],"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":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"qualitative-theory-of-dynamical-systems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"qtds","sideBox":"Learn more about [Qualitative Theory of Dynamical Systems](http://link.springer.com/journal/12306)","snPcode":"12346","submissionUrl":"https://submission.nature.com/new-submission/12346/3","title":"Qualitative Theory of Dynamical Systems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"cubic reversible isochronous system, tangential Hilbert’s 16th problem, Abelian integral, limit cycle, elliptic integral","lastPublishedDoi":"10.21203/rs.3.rs-7389808/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7389808/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFor the polynomial differential system ˙ x = −y + Chavarriga and García proved that when certain parameters of the system satisfy specific conditions, the origin is an isochronous center if and only if the system can be transformed into one of the five types: CR 1 , CR 2 , CR 3 , CR 4 , or CR 5. 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