Time-Regularized Bifurcation Framework for Constructing Tulip Orbits in the CRTBP

Time-Regularized Bifurcation Framework for Constructing Tulip Orbits in the CRTBP | 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 Time-Regularized Bifurcation Framework for Constructing Tulip Orbits in the CRTBP Jiye Zhang, Xinyu Jiang, Yuan Yuan, Honghua Dai This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7847547/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 30 Apr, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted 9 You are reading this latest preprint version Abstract Tulip orbits, a recently identified family of complex periodic solutions in the circular restricted three-body problem (CRTBP), exhibit near-neutral stability, diverse geometrical structures, and multi-directional coverage of the secondary body, offering promising prospects for lunar observation and space domain awareness. Although prior studies suggest that these orbits can emerge through period-multiplying bifurcations from the L2 near rectilinear halo orbit (NRHO) subset, the singular dynamics associated with extremely low perilune distance hinder conventional bifurcation analysis to a high order. This work develops a time-regularized bifurcation framework to systematically construct tulip orbits, particularly those bifurcated from low-perilune NRHOs, and assess their performance in the potential deep-space missions. A key innovation lies in embedding a time regularization scheme based on the Sundman transformation into the bifurcation analysis. Specifically, the CRTBP equations of motion under the Sundman transformation and their first-order variational forms are derived to accurately compute state transition and monodromy matrices. Therefore, the numerical continuation of the L2 NRHO subset can then be extended to an extremely low peripsis, followed by the detection of a series of high-order period-multiplying bifurcations. Using an improved orbit family switching algorithm, tulip orbits with up to twenty petals can be robustly constructed and continued. The framework is also extended to multiple three-body systems, demonstrating the generality of the dynamical connections between tulip orbits and the bifurcations of NRHOs. Finally, preliminary mission-level assessments under a high-fidelity ephemeris model indicate that, compared with the widely used 9:2 NRHO and 2:1 distant retrograde orbit, tulip orbits can achieve complete lunar communication coverage with shorter communication gaps, while requiring station-keeping costs of a comparable magnitude. Tulip orbits CRTBP Time-regularized bifurcation framework Sundman transformation Deep-space missions Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 30 Apr, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 04 Jan, 2026 Reviews received at journal 01 Jan, 2026 Reviews received at journal 14 Dec, 2025 Reviewers agreed at journal 23 Nov, 2025 Reviewers agreed at journal 21 Oct, 2025 Reviewers invited by journal 21 Oct, 2025 Editor assigned by journal 20 Oct, 2025 Submission checks completed at journal 13 Oct, 2025 First submitted to journal 13 Oct, 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. 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-7847547","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":533424523,"identity":"4b4b9e3d-20a3-470b-a8fe-1aa722498408","order_by":0,"name":"Jiye Zhang","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"prefix":"","firstName":"Jiye","middleName":"","lastName":"Zhang","suffix":""},{"id":533424524,"identity":"50fbcdd9-f82e-4c65-8ce9-6525ed79233c","order_by":1,"name":"Xinyu Jiang","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"prefix":"","firstName":"Xinyu","middleName":"","lastName":"Jiang","suffix":""},{"id":533424525,"identity":"d073fddb-83f4-4653-b992-39038233809f","order_by":2,"name":"Yuan Yuan","email":"","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":false,"prefix":"","firstName":"Yuan","middleName":"","lastName":"Yuan","suffix":""},{"id":533424526,"identity":"f8c43b51-247d-4251-8c93-0eb53a10a475","order_by":3,"name":"Honghua Dai","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAt0lEQVRIiWNgGAWjYBACAwkGA4aECiiPh3gtZ0jWwthGihZz6eaNHx7Ou5O4XSKB8cHbNgZ5c0JaLOccK5ZI3PYsceeMBGbDuW0MhjsbCDnsRo4BUMvh3A03EtikedsYEgwOENZi/CNxDlgL+29itZhJJDZAbGEmUktamUXCscP1O3seNkvOOSdhuIGwluTNN3/UHDY2Z08++OFNmY08QVsQehkYG4CUBLHqwVpGwSgYBaNgFOAAAJprRQraN4KcAAAAAElFTkSuQmCC","orcid":"","institution":"Northwestern Polytechnical University","correspondingAuthor":true,"prefix":"","firstName":"Honghua","middleName":"","lastName":"Dai","suffix":""}],"badges":[],"createdAt":"2025-10-13 10:08:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7847547/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7847547/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-026-12465-0","type":"published","date":"2026-04-30T15:57:27+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":94866114,"identity":"f9b1305a-5fdd-422a-9a21-e3a0ea2c12c9","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":6268,"visible":true,"origin":"","legend":"","description":"","filename":"ed05c2b913604737ac45c97672af091a.json","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/112406f4cbf46db908cb1059.json"},{"id":94986582,"identity":"4c5e15a9-6568-421b-96c3-89ad21a4e1b9","added_by":"auto","created_at":"2025-11-03 07:00:29","extension":"xml","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":176653,"visible":true,"origin":"","legend":"","description":"","filename":"ed05c2b913604737ac45c97672af091a1enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/394ff1b505b58358a1552f40.xml"},{"id":94866106,"identity":"b77de774-f5f8-481a-b227-6d16944e33f4","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"pdf","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":88996,"visible":true,"origin":"","legend":"","description":"","filename":"BrouckediagramST.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/69a1a09ca42549c706ee660c.pdf"},{"id":94866097,"identity":"119803fb-ce2b-4518-a39f-f1cc2884e79b","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"pdf","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":400914,"visible":true,"origin":"","legend":"","description":"","filename":"DRO21CDepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/1c7af1c50ece6d9ee7fdac7f.pdf"},{"id":94866096,"identity":"60b6e602-52ea-4e9b-b9d9-9479da693b45","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"eps","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":815875,"visible":true,"origin":"","legend":"","description":"","filename":"DRO21CD.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/7434c49aabd2979ca0e9ca15.eps"},{"id":94866101,"identity":"0477c260-dc1a-4814-9398-d6053085bc33","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"pdf","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":399087,"visible":true,"origin":"","legend":"","description":"","filename":"DRO21Maxgapepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/7bb4b2c35f233435504eecb8.pdf"},{"id":94866121,"identity":"94e97367-c84e-4fec-8f96-80d845381bb8","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"eps","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":848472,"visible":true,"origin":"","legend":"","description":"","filename":"DRO21Maxgap.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/b0459214a2729d7f538c6238.eps"},{"id":94866120,"identity":"efe20d1b-8d09-4ace-889d-019b35029177","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"pdf","order_by":11,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":367396,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP16CDepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/15e8bcf0a00ad9b605d3dedf.pdf"},{"id":94866118,"identity":"b59c5786-21be-4b16-8a6a-b87935436dfc","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"pdf","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":414201,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP16Maxgapepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/f087c219e9f40a299bec97db.pdf"},{"id":94866098,"identity":"50cc5e55-e06a-4499-997b-ab289e5be96a","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"pdf","order_by":15,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":359788,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP20CDepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/b19b5a0290be36fb4406fe9b.pdf"},{"id":94986531,"identity":"d264ebe3-24e6-4880-bd1a-b81a999969bb","added_by":"auto","created_at":"2025-11-03 07:00:24","extension":"eps","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":666488,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP20CD.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/7576ae5fa6afae70a104abdc.eps"},{"id":94866105,"identity":"17cdb51e-aef4-4e33-910e-f5cca66b0bcd","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"pdf","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":397353,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP20Maxgapepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/07731248c35e0914ed4d006e.pdf"},{"id":94866125,"identity":"0e1c0f9a-622e-4084-9529-04e77e98d15e","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"eps","order_by":18,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":737459,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP20Maxgap.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/7019778b1e51822d154054ee.eps"},{"id":94985654,"identity":"ebb61c38-cfdb-41e6-ac6a-bc35fec22603","added_by":"auto","created_at":"2025-11-03 06:58:35","extension":"pdf","order_by":19,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":390922,"visible":true,"origin":"","legend":"","description":"","filename":"NRHO92CDepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/3fc14a0fe47e7248a9698caf.pdf"},{"id":94866133,"identity":"0c7417f6-c139-4ace-8007-ef55ebb00966","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"eps","order_by":20,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":746096,"visible":true,"origin":"","legend":"","description":"","filename":"NRHO92CD.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/c39644e92ae0117c04b95bfa.eps"},{"id":94986066,"identity":"a344dd67-8035-4eeb-85c5-c3ad40143fa7","added_by":"auto","created_at":"2025-11-03 06:59:44","extension":"pdf","order_by":21,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":379810,"visible":true,"origin":"","legend":"","description":"","filename":"NRHO92Maxgapepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/9cdeec0a7424d5542ed73a6d.pdf"},{"id":94866109,"identity":"f2750eb4-e36a-432b-b6af-062df8bbe82d","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"eps","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":727030,"visible":true,"origin":"","legend":"","description":"","filename":"NRHO92Maxgap.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/651c54d1cb80e205bbba48ec.eps"},{"id":94866162,"identity":"e532e2ff-1785-4309-9986-736241f949cd","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"pdf","order_by":23,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":765221,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP20Tresonant11epsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/0bd384eaf99a40eb099fa6a6.pdf"},{"id":94986117,"identity":"1ce3b1a7-c734-4544-9673-65fa0d8e4396","added_by":"auto","created_at":"2025-11-03 06:59:47","extension":"eps","order_by":24,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":702690,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOP20Tresonant11.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/129e5bfd12cc1713770003c3.eps"},{"id":94866130,"identity":"57bc7e17-3bdb-4d2f-be11-4db30123776a","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"pdf","order_by":25,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":62817,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOcontinuationclosureerrorstatisticalepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/fc2bd3e59ffeced39939524d.pdf"},{"id":94985749,"identity":"9b2faa70-4e01-49b1-93d6-a48c2364ec25","added_by":"auto","created_at":"2025-11-03 06:58:50","extension":"eps","order_by":26,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":177079,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOcontinuationclosureerrorstatistical.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/12bd1b6ab5f5411a5d3242f7.eps"},{"id":94985491,"identity":"b2b42c79-a874-4b8a-bcc9-e5900a061e1b","added_by":"auto","created_at":"2025-11-03 06:58:17","extension":"pdf","order_by":27,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":26465,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOcontinuationconditionnumstatisticalepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/1fcbd0dc1d89fb7bba5c99a6.pdf"},{"id":94866110,"identity":"e85b36db-dd7e-4d58-85c3-10a29875d539","added_by":"auto","created_at":"2025-10-31 13:55:16","extension":"eps","order_by":28,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":115107,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOcontinuationconditionnumstatistical.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/d91024be37670de6be367a5d.eps"},{"id":94866126,"identity":"839eefc9-fcb2-40a5-8fff-34b27cf0a698","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"pdf","order_by":29,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":26889,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOcontinuationtcstatisticalepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/90d9759ff46566e41d843eb6.pdf"},{"id":94866170,"identity":"bb83c07d-9076-451c-b297-a133d5c65b15","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"eps","order_by":30,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":67137,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOcontinuationtcstatistical.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/13c15530aa199b5cd8682f59.eps"},{"id":94985884,"identity":"8d0acaf9-cb58-40a7-bf72-1873178e85e6","added_by":"auto","created_at":"2025-11-03 06:59:11","extension":"pdf","order_by":32,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1243138,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOsGeometryepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/df87b268d5fe0f80fae5a134.pdf"},{"id":94866123,"identity":"a34166ad-561a-45a5-a929-310ec9702cf8","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"eps","order_by":33,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1022075,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOsGeometry.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/58f8a5693b0a522072f5bfb1.eps"},{"id":94866129,"identity":"da661e64-5cdd-42ac-8dc1-1716deba5dd7","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"pdf","order_by":34,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":624190,"visible":true,"origin":"","legend":"","description":"","filename":"NRHOsGeometry1epsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/8f514198ac630569f93780cb.pdf"},{"id":94986430,"identity":"fa2626ea-19e6-40f0-bded-1230fa2e7f7b","added_by":"auto","created_at":"2025-11-03 07:00:19","extension":"aux","order_by":39,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":26849,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.aux","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/c8fe5eabd3bcf6b4712a174a.aux"},{"id":94866152,"identity":"afc9afa9-5531-4c54-974e-cb27690d804e","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"bbl","order_by":40,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":20962,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.bbl","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/d9b06d8480d36fa60b30f2a0.bbl"},{"id":94866111,"identity":"5c3fa988-0b9c-40eb-b950-a0eb82d13c5a","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"blg","order_by":41,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3122,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.blg","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/d0638f23635fdcb9326c177c.blg"},{"id":94866148,"identity":"f689a8ae-f5a0-4eb3-8cf8-e4db8ea6b1ed","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"log","order_by":42,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":61116,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.log","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/2ac5ed1c9cbe1ddf11835e0b.log"},{"id":94986403,"identity":"228d94d8-5a49-4d52-b6bf-1e028e6c7c5d","added_by":"auto","created_at":"2025-11-03 07:00:16","extension":"out","order_by":43,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":6371,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.out","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/b53bcae1436acf548b598cab.out"},{"id":94866171,"identity":"c2439121-e81c-4089-a1e3-621c9f023c40","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"pdf","order_by":44,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":8970771,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/6b57d2cff49ef3fbf9c15e12.pdf"},{"id":94866135,"identity":"ed1c36ee-fdae-40ab-858c-d597129411e9","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"gz","order_by":45,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":257446,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.synctex.gz","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/6c92f592624f654c60fb5e3f.gz"},{"id":94986144,"identity":"03c4106c-2c31-4412-bb8f-13d19848b132","added_by":"auto","created_at":"2025-11-03 06:59:58","extension":"eps","order_by":46,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1173179,"visible":true,"origin":"","legend":"","description":"","filename":"TulipP20resonantevolution.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/9f4f81ba6e6e0d03fe54d8f1.eps"},{"id":94986086,"identity":"2ba75403-504a-467b-9fda-51bf588824a5","added_by":"auto","created_at":"2025-11-03 06:59:46","extension":"pdf","order_by":48,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":129195,"visible":true,"origin":"","legend":"","description":"","filename":"Tulipcontinuationsensitivityepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/70059d89fbb29147db70c01f.pdf"},{"id":94866172,"identity":"a44cf74b-1379-4d30-aa16-4917646c9ce0","added_by":"auto","created_at":"2025-10-31 13:55:19","extension":"pdf","order_by":50,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":145736,"visible":true,"origin":"","legend":"","description":"","filename":"Tulipcontinuationtcepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/a78e37540a7624221027bd3a.pdf"},{"id":94866168,"identity":"8f82547a-35dd-4e67-945c-bf18e3970ea9","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"pdf","order_by":52,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":24880,"visible":true,"origin":"","legend":"","description":"","filename":"TulipfamilyCrpepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/18441aafd0c14f6c33290a03.pdf"},{"id":94866139,"identity":"c70f0ad1-4f93-4471-98a2-5a2697aaf57d","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"eps","order_by":53,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":65004,"visible":true,"origin":"","legend":"","description":"","filename":"TulipfamilyCrp.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/a5d288e36e473ea1c629c9bd.eps"},{"id":94986157,"identity":"558d3597-dcb1-4ff5-b717-23d1a50acec1","added_by":"auto","created_at":"2025-11-03 07:00:00","extension":"pdf","order_by":55,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":6477769,"visible":true,"origin":"","legend":"","description":"","filename":"TulipfamilyPlutoCharonP2P5.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/ea465a145e4b1736a69dc39b.pdf"},{"id":94866141,"identity":"e602d82f-3a5f-4633-82c9-a247e3325ce9","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"pdf","order_by":56,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":25319,"visible":true,"origin":"","legend":"","description":"","filename":"Tulipfamilyrarpepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/88b09bb8a7cd7f4f6397d8bb.pdf"},{"id":94866158,"identity":"1f1e8fc3-eff1-4148-b5fa-592ba7e8e29f","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"pdf","order_by":58,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":61539,"visible":true,"origin":"","legend":"","description":"","filename":"Tulipfamilyresonant.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/d5307c448874c0fa77b38b2b.pdf"},{"id":94866166,"identity":"f0588666-b5fd-47f9-a776-92b3f75a00a6","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"pdf","order_by":59,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":32268,"visible":true,"origin":"","legend":"","description":"","filename":"Tulipssensitivityepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/5da9d97fa6e3c597beb821ce.pdf"},{"id":94986275,"identity":"a9460ae7-827d-42bb-b67e-d3434f35d5de","added_by":"auto","created_at":"2025-11-03 07:00:08","extension":"png","order_by":61,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":49070,"visible":true,"origin":"","legend":"","description":"","filename":"droephine.png","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/545804819e42e48ace98a399.png"},{"id":94866145,"identity":"6d351c55-c14b-43a7-bb42-dc413f303cb1","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"png","order_by":62,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":28979,"visible":true,"origin":"","legend":"","description":"","filename":"droephrot.png","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/5fb05afdfc84dbc060124ff7.png"},{"id":94866146,"identity":"ac06665f-101e-42c0-aa6a-feb2e84f028e","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"pdf","order_by":63,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3101,"visible":true,"origin":"","legend":"","description":"","filename":"figepsconvertedto.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/fc79664014e7b6b4e0b04a86.pdf"},{"id":94866173,"identity":"3eed8d52-282f-46be-ba52-f736cfdd7617","added_by":"auto","created_at":"2025-10-31 13:55:19","extension":"eps","order_by":64,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":91593,"visible":true,"origin":"","legend":"","description":"","filename":"fig.eps","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/1d35140047fbd8abdffc3806.eps"},{"id":94866151,"identity":"ad5a9ce1-cf4a-4849-95af-8f74fcf8772c","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"bst","order_by":65,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":146013,"visible":true,"origin":"","legend":"","description":"","filename":"snapacite.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/7bd531b25eee8d7f427576ba.bst"},{"id":94866137,"identity":"4f562b90-5454-459a-9ed1-905346420f38","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"bst","order_by":66,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":29828,"visible":true,"origin":"","legend":"","description":"","filename":"snaps.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/36f0bd0b7e9517ad967a5762.bst"},{"id":94866155,"identity":"31a407d9-d029-4ae0-9e67-cc81f4a7e879","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"bst","order_by":67,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":35515,"visible":true,"origin":"","legend":"","description":"","filename":"snbasic.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/6dd83e70fcbb054200339b38.bst"},{"id":94986501,"identity":"ebb60638-6d87-4203-b640-f63738cc9244","added_by":"auto","created_at":"2025-11-03 07:00:22","extension":"bst","order_by":68,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":33968,"visible":true,"origin":"","legend":"","description":"","filename":"snchicago.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/796dcddfaf0d7ef6135f6c34.bst"},{"id":94986193,"identity":"5815c01f-5644-4613-aee3-921d95e61ae0","added_by":"auto","created_at":"2025-11-03 07:00:02","extension":"cls","order_by":69,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":55857,"visible":true,"origin":"","legend":"","description":"","filename":"snjnl.cls","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/728036545a01178f399d91d4.cls"},{"id":94866156,"identity":"bbd2afcf-4718-4469-b7ac-a3679cd907b8","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"bst","order_by":70,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":64023,"visible":true,"origin":"","legend":"","description":"","filename":"snmathphysay.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/2803491cd4a9154c34ce14ac.bst"},{"id":94866144,"identity":"bae29b83-9e04-4f96-a65b-c8dfc0f391ed","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"bst","order_by":71,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":64166,"visible":true,"origin":"","legend":"","description":"","filename":"snmathphysnum.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/e073bd346e246466cc630626.bst"},{"id":94986204,"identity":"ad733456-17ec-4bdd-9b26-a5e0ec4f7ba3","added_by":"auto","created_at":"2025-11-03 07:00:03","extension":"bst","order_by":72,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":37333,"visible":true,"origin":"","legend":"","description":"","filename":"snnature.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/103312756ee52dc9bb91cb1e.bst"},{"id":94866140,"identity":"c355fcff-ce88-4ce4-9129-e36b553672de","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"bst","order_by":73,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":39951,"visible":true,"origin":"","legend":"","description":"","filename":"snvancouveray.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/4f3dc37168085e0b3313ac8f.bst"},{"id":94986170,"identity":"b53177a6-c5b8-49d4-9af7-612d12141b00","added_by":"auto","created_at":"2025-11-03 07:00:00","extension":"bst","order_by":74,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":40758,"visible":true,"origin":"","legend":"","description":"","filename":"snvancouvernum.bst","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/09409a6586be88d229ecb57e.bst"},{"id":94866138,"identity":"47290c10-260e-4fcd-8af7-c5e025564af3","added_by":"auto","created_at":"2025-10-31 13:55:17","extension":"png","order_by":76,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":46377,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinedroephine.png","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/867ea19da7ab707cd811de71.png"},{"id":94985501,"identity":"cb9c6bc4-f882-4775-8701-78ee3e5436b1","added_by":"auto","created_at":"2025-11-03 06:58:18","extension":"png","order_by":77,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":27133,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinedroephrot.png","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/d70a8975a649b030664dabc6.png"},{"id":94866161,"identity":"bd0bf1eb-2be9-4ea8-9e74-d40e060cd11c","added_by":"auto","created_at":"2025-10-31 13:55:18","extension":"xml","order_by":78,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":196238,"visible":true,"origin":"","legend":"","description":"","filename":"ed05c2b913604737ac45c97672af091a1structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/d25465538b79a7f2dd32c253.xml"},{"id":94866174,"identity":"3c2e1b88-593e-4673-891f-72f9d0d68237","added_by":"auto","created_at":"2025-10-31 13:55:19","extension":"html","order_by":79,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":233227,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1/a2ff42f0f76d0da2549b6299.html"},{"id":108438070,"identity":"47d18555-9a4f-42d0-9f44-ef97802be2f0","added_by":"auto","created_at":"2026-05-04 16:06:45","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":8284191,"visible":true,"origin":"","legend":"","description":"","filename":"TimeRegularizedBifurcationFrameworkforConstructingTulipOrbitsintheCRTBP.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7847547/v1_covered_5453f5fd-50e1-442d-a7f6-aebb6a1e623e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Time-Regularized Bifurcation Framework for Constructing Tulip Orbits in the CRTBP","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":"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":"Tulip orbits, CRTBP, Time-regularized bifurcation framework, Sundman transformation, Deep-space missions","lastPublishedDoi":"10.21203/rs.3.rs-7847547/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7847547/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTulip orbits, a recently identified family of complex periodic solutions in the circular restricted three-body problem (CRTBP), exhibit near-neutral stability, diverse geometrical structures, and multi-directional coverage of the secondary body, offering promising prospects for lunar observation and space domain awareness. Although prior studies suggest that these orbits can emerge through period-multiplying bifurcations from the L2 near rectilinear halo orbit (NRHO) subset, the singular dynamics associated with extremely low perilune distance hinder conventional bifurcation analysis to a high order. This work develops a time-regularized bifurcation framework to systematically construct tulip orbits, particularly those bifurcated from low-perilune NRHOs, and assess their performance in the potential deep-space missions. A key innovation lies in embedding a time regularization scheme based on the Sundman transformation into the bifurcation analysis. Specifically, the CRTBP equations of motion under the Sundman transformation and their first-order variational forms are derived to accurately compute state transition and monodromy matrices. Therefore, the numerical continuation of the L2 NRHO subset can then be extended to an extremely low peripsis, followed by the detection of a series of high-order period-multiplying bifurcations. Using an improved orbit family switching algorithm, tulip orbits with up to twenty petals can be robustly constructed and continued. The framework is also extended to multiple three-body systems, demonstrating the generality of the dynamical connections between tulip orbits and the bifurcations of NRHOs. Finally, preliminary mission-level assessments under a high-fidelity ephemeris model indicate that, compared with the widely used 9:2 NRHO and 2:1 distant retrograde orbit, tulip orbits can achieve complete lunar communication coverage with shorter communication gaps, while requiring station-keeping costs of a comparable magnitude.\u003c/p\u003e","manuscriptTitle":"Time-Regularized Bifurcation Framework for Constructing Tulip Orbits in the CRTBP","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-31 13:55:10","doi":"10.21203/rs.3.rs-7847547/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-01-05T00:56:45+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-01-01T07:07:12+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-12-14T14:29:49+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"187609772004429447065205898920456322393","date":"2025-11-23T13:39:16+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"86975466337224318585454385425943940652","date":"2025-10-21T09:21:58+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-10-21T05:45:48+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-10-21T03:27:56+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-10-14T02:29:04+00:00","index":"","fulltext":""},{"type":"submitted","content":"Nonlinear Dynamics","date":"2025-10-13T09:56:06+00:00","index":"","fulltext":""}],"status":"published","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}}],"origin":"","ownerIdentity":"f0c78993-56bc-479d-b82d-5009b3553ffe","owner":[],"postedDate":"October 31st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-05-04T16:06:33+00:00","versionOfRecord":{"articleIdentity":"rs-7847547","link":"https://doi.org/10.1007/s11071-026-12465-0","journal":{"identity":"nonlinear-dynamics","isVorOnly":false,"title":"Nonlinear Dynamics"},"publishedOn":"2026-04-30 15:57:27","publishedOnDateReadable":"April 30th, 2026"},"versionCreatedAt":"2025-10-31 13:55:10","video":"","vorDoi":"10.1007/s11071-026-12465-0","vorDoiUrl":"https://doi.org/10.1007/s11071-026-12465-0","workflowStages":[]},"version":"v1","identity":"rs-7847547","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7847547","identity":"rs-7847547","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}