Model reduction for systems with random parameters using Spectral Submanifolds

Model reduction for systems with random parameters using Spectral Submanifolds | 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 Model reduction for systems with random parameters using Spectral Submanifolds Ahmed Amr Morsy, Paolo Tiso This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5938423/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Jun, 2025 Read the published version in Nonlinear Dynamics → Version 1 posted 10 You are reading this latest preprint version Abstract We present a method for studying the nonlinear dynamics of systems with random parameters. We use the theory of Spectral Submanifolds (SSMs) to perform model reduction. This enables reducing the nonlinear dynamics of high-dimensional models to low-dimensional invariant manifolds. To tackle the randomness of system parameters, we compute Polynomial Chaos Expansions (PCEs) of SSMs in a purely equation-driven approach. The resulting Parametric SSMs (PSSMs) can be systematically computed for arbitrary expansion orders. This capability, besides the optimality of PCEs in probabilistic settings, enables us to study systems with moderately large parameter perturbations. For mechanical systems, we use PSSMs to obtain parametric backbone curves and frequency response curves. Further, to perform uncertainty quantification, we derive closed-form expressions for convergent statistical moments of backbone curves without the need for any simulations. We illustrate the method with examples that include a slender beam subject to random manufacturing imperfections. nonlinear dynamics model reduction spectral submanifolds uncertainty quantification polynomial chaos expansion Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 Jun, 2025 Read the published version in Nonlinear Dynamics → Version 1 posted Editorial decision: Revision requested 21 Mar, 2025 Reviews received at journal 21 Mar, 2025 Reviews received at journal 04 Mar, 2025 Reviewers agreed at journal 01 Mar, 2025 Reviewers agreed at journal 01 Mar, 2025 Reviewers agreed at journal 04 Feb, 2025 Reviewers invited by journal 04 Feb, 2025 Editor assigned by journal 03 Feb, 2025 Submission checks completed at journal 01 Feb, 2025 First submitted to journal 31 Jan, 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. 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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-5938423","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":424037291,"identity":"c5a091cc-2c04-4280-bc07-8bcbd0d7d73d","order_by":0,"name":"Ahmed Amr Morsy","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArElEQVRIiWNgGAWjYPACGwYGCSDF2EC8ljQGHlK1HCZBi2577+PPPDXn5eyl2x8w/NxBhBazM8fNpHmO3TbmkTmQwNh7hhgtN9LYmHnYbif2SCQcYGZsI04L82eef+fqeyQSG4jWwiDN23YggUcimYFILWeOsUnO7Us27AHqPdhLlJbjbcwf3nyzk2efkf7wwU9itIAAEw+UcYBIDcAI/EG00lEwCkbBKBiRAAD8mjQlRpqczQAAAABJRU5ErkJggg==","orcid":"","institution":"ETH Zurich","correspondingAuthor":true,"prefix":"","firstName":"Ahmed","middleName":"Amr","lastName":"Morsy","suffix":""},{"id":424037292,"identity":"ac694aa9-fc4a-4c46-b64a-21c785b78bff","order_by":1,"name":"Paolo Tiso","email":"","orcid":"","institution":"ETH Zurich","correspondingAuthor":false,"prefix":"","firstName":"Paolo","middleName":"","lastName":"Tiso","suffix":""}],"badges":[],"createdAt":"2025-01-31 19:08:02","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5938423/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5938423/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11071-025-11396-6","type":"published","date":"2025-06-06T15:57:38+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":84243179,"identity":"31a4ae3b-4826-4fda-bd5b-4595fc502c75","added_by":"auto","created_at":"2025-06-09 16:12:49","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1365484,"visible":true,"origin":"","legend":"","description":"","filename":"PSSMmanuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5938423/v1_covered_576d6132-f971-4e6d-9730-b3114eceabf5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Model reduction for systems with random parameters using Spectral Submanifolds","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":"nonlinear dynamics, model reduction, spectral submanifolds, uncertainty quantification, polynomial chaos expansion","lastPublishedDoi":"10.21203/rs.3.rs-5938423/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5938423/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We present a method for studying the nonlinear dynamics of systems with random parameters. 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