A Hartman–Grobman Theorem for Utkin-Selected Sliding Dynamics in Filippov Systems and Threshold Analysis in Dry Friction Oscillators

A Hartman–Grobman Theorem for Utkin-Selected Sliding Dynamics in Filippov Systems and Threshold Analysis in Dry Friction Oscillators | 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 Hartman–Grobman Theorem for Utkin-Selected Sliding Dynamics in Filippov Systems and Threshold Analysis in Dry Friction Oscillators Joshua Owolabi Adeleke This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8896048/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 prove a Hartman--Grobman theorem for \emph{Utkin-selected sliding dynamics} in codimension-one Filippov systems. Under uniform transversality and $C^1$ regularity, near a \emph{hyperbolic equilibrium} of the reduced sliding ODE on the switching manifold, the Utkin-selected evolution is locally \emph{orbit equivalent} to a linear product model: a transverse contracting drift $\dot\sigma=-1$ together with the linearized sliding flow $\dot z = Dz, z$ on the manifold. The proof combines smooth Hartman--Grobman theory for the sliding ODE with a transverse hitting/impact map that yields a product chart near the manifold. As an independent application of the reduction framework, we analyze a dry friction oscillator, derive explicit sticking thresholds under constant bias and harmonic forcing, and provide reproducible numerical validation. 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. 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-8896048","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":593671801,"identity":"7b8c7483-e2ce-4f56-bc48-037ec3a5f277","order_by":0,"name":"Joshua Owolabi Adeleke","email":"data:image/png;base64,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","orcid":"","institution":"Illinois Institute of Technology","correspondingAuthor":true,"prefix":"","firstName":"Joshua","middleName":"Owolabi","lastName":"Adeleke","suffix":""}],"badges":[],"createdAt":"2026-02-16 20:24:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8896048/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8896048/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104743358,"identity":"37bd051d-a13b-4c6c-9b70-8c46874235d8","added_by":"auto","created_at":"2026-03-16 16:56:23","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1205186,"visible":true,"origin":"","legend":"","description":"","filename":"LocalTheoryandPiecewiseNormalFormsforFilippovSystemswithMechanicalDryFriction.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8896048/v1_covered_28a043b4-f8a3-4267-a1f0-d5a97b13b5f8.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eA Hartman–Grobman Theorem for Utkin-Selected Sliding Dynamics in Filippov Systems and Threshold Analysis in Dry Friction Oscillators\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"","lastPublishedDoi":"10.21203/rs.3.rs-8896048/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8896048/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We prove a Hartman--Grobman theorem for \\emph{Utkin-selected sliding dynamics} in codimension-one Filippov systems. Under uniform transversality and $C^1$ regularity, near a \\emph{hyperbolic equilibrium} of the reduced sliding ODE on the switching manifold, the Utkin-selected evolution is locally \\emph{orbit equivalent} to a linear product model: a transverse contracting drift $\\dot\\sigma=-1$ together with the linearized sliding flow $\\dot z = Dz\\, z$ on the manifold. The proof combines smooth Hartman--Grobman theory for the sliding ODE with a transverse hitting/impact map that yields a product chart near the manifold. As an independent application of the reduction framework, we analyze a dry friction oscillator, derive explicit sticking thresholds under constant bias and harmonic forcing, and provide reproducible numerical validation.","manuscriptTitle":"A Hartman–Grobman Theorem for Utkin-Selected Sliding Dynamics in Filippov Systems and Threshold Analysis in Dry Friction Oscillators","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-16 09:11:06","doi":"10.21203/rs.3.rs-8896048/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":"24335eab-1fab-451f-b048-278664190895","owner":[],"postedDate":"March 16th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-03-19T17:23:16+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-16 09:11:06","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8896048","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8896048","identity":"rs-8896048","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}