A Koopman-Hill framework for the bifurcation analysis of nonlinear dynamical systems in codimension-1 and -2

A Koopman-Hill framework for the bifurcation analysis of nonlinear dynamical systems in codimension-1 and -2 | 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 Koopman-Hill framework for the bifurcation analysis of nonlinear dynamical systems in codimension-1 and -2 Fabia BAYER, Roberto ALCORTA This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7048070/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Mar, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted 9 You are reading this latest preprint version Abstract This paper proposes a full numerical bifurcation analysis framework built on the harmonic balance method. For Floquet stability analysis, the framework leverages the Koopman-Hill projection method, enabling a "best of both worlds" computational strategy wherein results from typically time-based (monodromy matrix, bifurcation test functions) and frequency-based (exploiting Hill’s method for extended systems) are combined optimally. In particular, the detection and localization of bifurcations in codimension-2 and higher is achieved through frequency-based, direct rank-one updates (Wielandt deflation) on the mon-odromy matrix. Furthermore, this work details the application of the Koopman-Hill projection to different formulations of dynamical systems, in such a way that the proposed techniques are straightforwardly applicable to cases of wide practical interest, namely: second order ODEs and dynamical systems involving a state-dependent mass matrix. The robustness and performance of the novel framework are tested on three benchmark examples and compared to the traditional sorting-based Hill method, showing clear evidence in favor of using the Koopman-Hill projection in terms of reduced computation times and more precise results. Floquet harmonic balance numerical continuation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 Mar, 2026 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 14 Oct, 2025 Reviews received at journal 14 Oct, 2025 Reviewers agreed at journal 02 Oct, 2025 Reviews received at journal 04 Aug, 2025 Reviewers agreed at journal 14 Jul, 2025 Reviewers invited by journal 10 Jul, 2025 Editor assigned by journal 08 Jul, 2025 Submission checks completed at journal 06 Jul, 2025 First submitted to journal 04 Jul, 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. 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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-7048070","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":483690966,"identity":"804f385c-568e-4526-a3cc-77489da2185b","order_by":0,"name":"Fabia BAYER","email":"","orcid":"","institution":"University of Stuttgart","correspondingAuthor":false,"prefix":"","firstName":"Fabia","middleName":"","lastName":"BAYER","suffix":""},{"id":483690969,"identity":"05eec2b4-306c-44e1-bd3e-c6028cee4c32","order_by":1,"name":"Roberto ALCORTA","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIie3RwUrDMBjA8W8Esktg14yN9BVSAq1i8VkyCs1lA4+KoIHCdprnPYaP4Mihl7KzR0tfYN4meDBp50VadTeR/KHJR+kPEgrg8/3JyHEfthvDzXbllsFLP5H2QdAMoiXcLYj/isz0TyRerbf1zSGBUY4qO1yqh8mq2gM3LAaE9x1kWu7SsJQZUIOFHdLFcloKaok41whtOgil82ispQEwBNsBLZZ0DpN3bmaPTyPTdTBL4jdHgpbcK0xVfYCGINRDooEjvCVGYioj+i0hu3Sss4yE7i46K0J7sOgMuBLc9JDhevuqk4SxIq8qndwGwUbVz3B9wXiRd5LPyNcX+fFPndDdid/7fD7fP+4DGcVVK7Ajxh4AAAAASUVORK5CYII=","orcid":"","institution":"Univ Lyon, INSA Lyon, CNRS, LaMCoS","correspondingAuthor":true,"prefix":"","firstName":"Roberto","middleName":"","lastName":"ALCORTA","suffix":""}],"badges":[],"createdAt":"2025-07-04 15:23:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7048070/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7048070/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-025-12086-z","type":"published","date":"2026-03-03T15:58:48+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":104250666,"identity":"58f73ff8-47c1-4c59-aa13-5e57210f5817","added_by":"auto","created_at":"2026-03-09 16:05:11","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3196138,"visible":true,"origin":"","legend":"","description":"","filename":"KoopmanHillcodim2paperFBRA.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7048070/v1_covered_575e1a9f-b7a5-4326-9328-8216bc7c0934.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Koopman-Hill framework for the bifurcation analysis of nonlinear dynamical systems in codimension-1 and -2","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":"Floquet, harmonic balance, numerical continuation","lastPublishedDoi":"10.21203/rs.3.rs-7048070/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7048070/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper proposes a full numerical bifurcation analysis framework built on the harmonic balance method. 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