A Stability–Symmetry Approach to Periodic-Orbit Classification in the Three-Body Problem

A Stability–Symmetry Approach to Periodic-Orbit Classification in the Three-Body Problem | 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 A Stability–Symmetry Approach to Periodic-Orbit Classification in the Three-Body Problem khushboo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8283973/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We introduce a stability–symmetry classification scheme for periodic orbits in the planar three-body problem. The method combines three diagnostic measures, Floquet stability, discrete symmetry, and a braid-based complexity estimate, into a single quantity designed to highlight orbits with notable dynamical features. For each periodic orbit o, we compute (1) a stability index S (o) from Floquet multipliers, (2) a symmetry coefficient τ(o) reflecting time-reversal and spatial symmetries, and (3) a complexity measure C(o) derived from braid-theoretic word length. The combined indicator is defined as Q(o) = S (o) τ(o) C(o) . Applying this diagnostic to a collection of known orbits, we find that those with larger Q values tend to form a loosely defined cluster, separated from the remainder of the sample. Numerical experiments further suggest that chaotic trajectories appear to spend a considerable fraction of their evolution in neighborhoods of these elevated-Q orbits. The distribution of residence times displays approximate power-law behavior, with exponent β ≈ 9, consistent with hierarchical structure in the surrounding chaotic dynamics. These observations indicate that stability and symmetry, when considered together with topological complexity, can serve as practical tools for identifying periodic orbits that play a notable role in the phase-space dynamics. The classification scheme may provide a useful filtering step for future studies of orbit families and transport processes in the three-body problem. Keywords: three-body problem, orbit classification, stability analysis, symmetry methods, chaotic dynamics, celestial mechanics Mathematical Physics three-body problem orbit classification stability analysis symmetry methods chaotic dynamics celestial mechanics Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-8283973","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":555575951,"identity":"042bed2a-457b-45c5-a073-c172c06438b2","order_by":0,"name":"khushboo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+0lEQVRIiWNgGAWjYBACCQkGBmYGAwYGfvaGBIMPQBE2dmK1SPYceFA4A6SFmSgtQGBwI/HBZx4Qi5AWydk9xp8LCu7JMxxITtxs82ubPB8zA+OHjzm4tUjLnDGTnmFQbNjYcCzZOLfvtmEbMwOz5MxtuLXISeSYMfMYJDA2M/akGef23GYEamFj5sWvxfgzUIt9GzP/99+WPbftCWqRlsgxkAZqSexhY0gwZvhxO5GgFskZaWUgLckzeBgSDHsbbie3MTM24/WLxI3kzZ95/iTY7r//IMHgx5/btvPbmw9++IhHCypgbAOTDcSqB4E/pCgeBaNgFIyCkQIAUThNVe2XDokAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0006-7000-026X","institution":"JUET","correspondingAuthor":true,"prefix":"","firstName":"","middleName":"","lastName":"khushboo","suffix":""}],"badges":[],"createdAt":"2025-12-05 03:53:44","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8283973/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8283973/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":97673696,"identity":"83513d90-c248-485e-acfc-b928be904f93","added_by":"auto","created_at":"2025-12-08 09:41:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":715961,"visible":true,"origin":"","legend":"","description":"","filename":"Threebodyproblemkhushboo.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8283973/v1_covered_97076b1a-7464-43ea-b46d-b5965394b361.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Stability–Symmetry Approach to Periodic-Orbit Classification in the Three-Body Problem\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"JUET guna","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"three-body problem, orbit classification, stability analysis, symmetry methods, chaotic dynamics, celestial mechanics","lastPublishedDoi":"10.21203/rs.3.rs-8283973/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8283973/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe introduce a stability–symmetry classification scheme for periodic orbits in the planar three-body problem. The method combines three diagnostic measures, Floquet stability, discrete symmetry, and a braid-based complexity estimate, into a single quantity designed to highlight orbits with notable dynamical features. For each periodic orbit o, we compute (1) a stability index S (o) from Floquet multipliers, (2) a symmetry coefficient τ(o) reflecting time-reversal and spatial symmetries, and (3) a complexity measure C(o) derived from braid-theoretic word length. The combined indicator is defined as Q(o) = S (o) τ(o) C(o) . Applying this diagnostic to a collection of known orbits, we find that those with larger Q values tend to form a loosely defined cluster, separated from the remainder of the sample. Numerical experiments further suggest that chaotic trajectories appear to spend a considerable fraction of their evolution in neighborhoods of these elevated-Q orbits. The distribution of residence times displays approximate power-law behavior, with exponent β ≈ 9, consistent with hierarchical structure in the surrounding chaotic dynamics. These observations indicate that stability and symmetry, when considered together with topological complexity, can serve as practical tools for identifying periodic orbits that play a notable role in the phase-space dynamics. The classification scheme may provide a useful filtering step for future studies of orbit families and transport processes in the three-body problem. Keywords: three-body problem, orbit classification, stability analysis, symmetry methods, chaotic dynamics, celestial mechanics\u003c/p\u003e","manuscriptTitle":"A Stability–Symmetry Approach to Periodic-Orbit Classification in the Three-Body Problem","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-08 05:15:17","doi":"10.21203/rs.3.rs-8283973/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"f5adf9ab-bdae-4d56-b9fc-dd2402890c99","owner":[],"postedDate":"December 8th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":59129893,"name":"Mathematical Physics"}],"tags":[],"updatedAt":"2025-12-08T05:15:17+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-08 05:15:17","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8283973","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8283973","identity":"rs-8283973","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}