{"paper_id":"299c2a4a-ee7f-4c36-8edb-8966cabb2ae5","body_text":"Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise | 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 Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise Lele Jia, Shuangbao Li, Liying Kou, Kongran Wu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3833974/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 This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable vibro-impact SD oscillator. Firstly, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod so that the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistble vibration characteristic, two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable vibro-impact SD oscillator through studying the persistence of the unique unperturbed non-smooth homoclinic structure. Secondly, the general framework of random Melnikov process for a class of bistable vibro-impact systems under Gaussian white noise is simply derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable vibro-impact SD oscillator. Thirdly, the effectiveness of a semi-analytical prediction by the Melnikov function is verified through the lagest Lyapunov exponent, bifurcation series, and 0 − 1 test. In addition, the sensitivity to the initial values of chaos is verified by the fractal of attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show rich dynamical geometric structures. A bistable vibro-impact SD oscillator Gaussian white noise Random Melnikov method Homoclinic chaos 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-3833974\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":true,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":265586872,\"identity\":\"5814b870-06b7-4c92-b96a-fa6420a371c5\",\"order_by\":0,\"name\":\"Lele Jia\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyklEQVRIiWNgGAWjYBACxmYGxgMMBUAWe2Pjww9EamE4wGAAZPEcbjaWINYmiBaJ9DYBHmKUM7czPzjwwcDG3uDmwzYGCQY7Od0Ggg5jMzg4wyAtcebsxLYHBQzJxmYHCGphMDjMY3A4gV86sd1AguFA4jbCWtg/HP5j8N+eTfJgmwQPcVqAVjAYHGDsl2AkXkvBwR6D5MSZPYnAQDYgwi+G/cc3PvhRYWdvcPz4w4cfKuzkCGtpQOEaEFAOAvJEqBkFo2AUjIKRDgBh3EKJnuTM6gAAAABJRU5ErkJggg==\",\"orcid\":\"\",\"institution\":\"Civil Aviation University of China\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Lele\",\"middleName\":\"\",\"lastName\":\"Jia\",\"suffix\":\"\"},{\"id\":265586873,\"identity\":\"7a541926-a0bd-442a-9602-ed09a136ac1f\",\"order_by\":1,\"name\":\"Shuangbao Li\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Civil Aviation University of China\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Shuangbao\",\"middleName\":\"\",\"lastName\":\"Li\",\"suffix\":\"\"},{\"id\":265586874,\"identity\":\"8e4042c3-9668-43c2-a446-1c5abad19371\",\"order_by\":2,\"name\":\"Liying Kou\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Civil Aviation University of China\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Liying\",\"middleName\":\"\",\"lastName\":\"Kou\",\"suffix\":\"\"},{\"id\":265586875,\"identity\":\"ca305ade-2bdc-4e0b-9b26-b1cbb49308f0\",\"order_by\":3,\"name\":\"Kongran Wu\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Civil Aviation University of China\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Kongran\",\"middleName\":\"\",\"lastName\":\"Wu\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2024-01-04 07:59:12\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-3833974/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-3833974/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":49663844,\"identity\":\"64f7bdfd-2801-45d5-8d74-c217ba89a85a\",\"added_by\":\"auto\",\"created_at\":\"2024-01-16 06:33:11\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":3460735,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"manuscript.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-3833974/v1_covered_14000ec8-ed62-4cf1-8a0b-65cffe739517.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise\",\"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\":\"info@researchsquare.com\",\"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\":\"A bistable vibro-impact SD oscillator, Gaussian white noise, Random Melnikov method, Homoclinic chaos\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-3833974/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-3833974/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable vibro-impact SD oscillator. Firstly, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod so that the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistble vibration characteristic, two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable vibro-impact SD oscillator through studying the persistence of the unique unperturbed non-smooth homoclinic structure. Secondly, the general framework of random Melnikov process for a class of bistable vibro-impact systems under Gaussian white noise is simply derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable vibro-impact SD oscillator. Thirdly, the effectiveness of a semi-analytical prediction by the Melnikov function is verified through the lagest Lyapunov exponent, bifurcation series, and 0 − 1 test. In addition, the sensitivity to the initial values of chaos is verified by the fractal of attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show rich dynamical geometric structures.\",\"manuscriptTitle\":\"Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2024-01-16 06:25:01\",\"doi\":\"10.21203/rs.3.rs-3833974/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"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\":\"19a9830b-cc39-484b-bff5-6316126851f8\",\"owner\":[],\"postedDate\":\"January 16th, 2024\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"posted\",\"subjectAreas\":[],\"tags\":[],\"updatedAt\":\"2024-01-16T06:25:01+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2024-01-16 06:25:01\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-3833974\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-3833974\",\"identity\":\"rs-3833974\",\"version\":[\"v1\"]},\"buildId\":\"qtupq5eGEP_6zYnWcrvyt\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}