A Fourier–Collocation Framework for Constructing Quasi-Periodic Orbit Families in the CR3BP

A Fourier–Collocation Framework for Constructing Quasi-Periodic Orbit Families in the CR3BP | 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 Fourier–Collocation Framework for Constructing Quasi-Periodic Orbit Families in the CR3BP Evan Lantier, Roshan Thomas Eapen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7724809/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 Trajectory design in multi-body dynamical environments increasingly relies on understanding quasi-periodic structures that shape long-term motion. This paper develops a Fourier–collocation framework for constructing quasi-periodic orbit (QPO) families in the Circular Restricted Three-Body Problem (CR3BP). Starting from a reference periodic orbit, fundamental frequencies are identified through frequency analysis and Floquet theory, enabling the construction of an initial Fourier-based torus. The torus is refined to enforce dynamical consistency with the CR3BP dynamics, and continuation in a perturbation parameter generates full QPO families. To improve accuracy, missing higher-order frequencies are systematically identified and incorporated by harmonic addition. The framework produces semi-analytical approximations that remain dynamically consistent over extended timescales, as verified through phase-point drift analyses. Results for Lissajous, quasi-halo, and quasi-distant retrograde orbit families demonstrate the utility of the developed framework. Although computationally intensive, the method establishes a scalable foundation for analyzing quasi-periodic structures in multi-body systems and is amenable to potential transitions to higher-fidelity models for trajectory design in the cislunar environment. CR3BP invariant tori quasi-periodic orbits (QPO) Fourier–collocation Lissajous orbits quasi-halo orbits quasi-distant retrograde orbits (Q-DRO) Full Text Additional Declarations No competing interests reported. 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. 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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-7724809","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":538535237,"identity":"399096c0-e660-4952-91f7-ec8024109193","order_by":0,"name":"Evan Lantier","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAx0lEQVRIiWNgGAWjYFCCw+2ffxhIyEE4bERpOdjGzFBhY0yKFkagljNpiQ1Ea+FnPNj2uLDtcPp2/jMGDB/KDhPWItlwsN14Ztvh3J0zcgwYZ5wjQovBgYMNErxALRtu8BgwAxmEtdhDtaQbnD9jwPyXGC0GwBCT5jmTlmBwIMeAmZEYLRIHDjYbzqiwMdw5I63gYM+5dMJa+GccPvjgg4GEvDn/4Y0PfpRZE9YCtAbmQgaGA0SoB1nTgNAyCkbBKBgFowArAAAhMEJmLmUElgAAAABJRU5ErkJggg==","orcid":"","institution":"Pennsylvania State University","correspondingAuthor":true,"prefix":"","firstName":"Evan","middleName":"","lastName":"Lantier","suffix":""},{"id":538535238,"identity":"b1be879c-9ea2-4468-86e5-8c014fda18dc","order_by":1,"name":"Roshan Thomas Eapen","email":"","orcid":"","institution":"Pennsylvania State University","correspondingAuthor":false,"prefix":"","firstName":"Roshan","middleName":"Thomas","lastName":"Eapen","suffix":""}],"badges":[],"createdAt":"2025-09-26 21:38:59","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7724809/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7724809/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":95606554,"identity":"38d02984-e5f7-4a19-884e-3d755d684ea4","added_by":"auto","created_at":"2025-11-11 07:20:31","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4130,"visible":true,"origin":"","legend":"","description":"","filename":"dcae01c47619434391aa0151ccd94899.json","url":"https://assets-eu.researchsquare.com/files/rs-7724809/v1/2f0bcbfc5428c9eaa9f26f5e.json"},{"id":99794669,"identity":"62065160-a3ca-4d08-92b3-60777ca708a6","added_by":"auto","created_at":"2026-01-08 13:35:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":9040892,"visible":true,"origin":"","legend":"","description":"","filename":"JASSFourierCollocationLantierEapen.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7724809/v1_covered_d9803543-d72a-4c37-b834-2e0a75f22f4e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Fourier–Collocation Framework for Constructing Quasi-Periodic Orbit Families in the CR3BP","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","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":"CR3BP, invariant tori, quasi-periodic orbits (QPO), Fourier–collocation, Lissajous orbits, quasi-halo orbits, quasi-distant retrograde orbits (Q-DRO)","lastPublishedDoi":"10.21203/rs.3.rs-7724809/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7724809/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Trajectory design in multi-body dynamical environments increasingly relies on understanding quasi-periodic structures that shape long-term motion. 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