Graph-Based Analysis of Chaotic Dynamics in the Double Pendulum System

Graph-Based Analysis of Chaotic Dynamics in the Double Pendulum System | 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 Graph-Based Analysis of Chaotic Dynamics in the Double Pendulum System Sumeyye Bakim, Nurten Urlu Ozalan, Erdi Gulbahce This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5928823/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 Understanding the interplay between spatial and kinematic properties in chaotic systems is crucial for advancing nonlinear dynamics, yet remains a challenging problem. The double pendulum, as a classic example of deterministic chaos, provides a rich platform for exploring these dynamics, making its study highly relevant to researchers in nonlinear systems. Previous works have extensively analyzed the trajectories of the double pendulum using traditional numerical and analytical methods, offering insights into its chaotic behavior. However, these approaches often fail to quantitatively link spatial relationships with physical motion under varying initial conditions, leaving gaps in our understanding of how geometry and dynamics interact. This study addresses these challenges by introducing a novel graph-theoretic framework to analyze the motion of the double pendulum. By representing trajectories as undirected graphs where nodes correspond to discrete positions of the pendulum bob and edges are defined based on a tunable threshold distance parameter ϵ we systematically explore the sensitivity of graph connectivity to ϵ. This methodology emphasizes the importance of careful parameter tuning to preserve trajectory continuity and ensure valid graph-based metrics. Through this framework, we hypothesize that graph representations will reveal meaningful correlations between spatial relationships (encoded as edge weights) and kinematic properties, providing a quantitative link between geometric and physical features. Additionally, sensitivity analysis is expected to uncover power-law relationships between ϵ and trajectory resolution, demonstrating the robustness of graph-based methods in capturing both regular and chaotic dynamics. These findings underscore the potential of graph theory as a versatile tool for analyzing the complex behavior of nonlinear systems, offering fresh insights into the intricate dynamics of chaotic phenomena such as those in the double pendulum. graph theory double pendulum chaotic dynamics trajectory analysis 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-5928823","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":409495023,"identity":"a300b1da-92e0-4b66-9e8f-579745c5cf96","order_by":0,"name":"Sumeyye Bakim","email":"","orcid":"","institution":"KTO Karatay University","correspondingAuthor":false,"prefix":"","firstName":"Sumeyye","middleName":"","lastName":"Bakim","suffix":""},{"id":409495024,"identity":"0bfdbdf8-8606-41dd-a03f-4a4de114d2df","order_by":1,"name":"Nurten Urlu Ozalan","email":"","orcid":"","institution":"KTO Karatay University","correspondingAuthor":false,"prefix":"","firstName":"Nurten","middleName":"Urlu","lastName":"Ozalan","suffix":""},{"id":409495025,"identity":"52ea3c8b-cce9-46d3-8e76-cb4b79465ecf","order_by":2,"name":"Erdi Gulbahce","email":"data:image/png;base64,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","orcid":"","institution":"KTO Karatay University","correspondingAuthor":true,"prefix":"","firstName":"Erdi","middleName":"","lastName":"Gulbahce","suffix":""}],"badges":[],"createdAt":"2025-01-30 08:53:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5928823/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5928823/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":75326060,"identity":"956ee99c-505d-450e-a79a-4d6554cba318","added_by":"auto","created_at":"2025-02-03 11:17:25","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":866247,"visible":true,"origin":"","legend":"","description":"","filename":"QualitativeTheoryofDynamicalSystems.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5928823/v1_covered_31bc2d2e-c7f2-4750-bb89-9bdf5d3e2011.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Graph-Based Analysis of Chaotic Dynamics in the Double Pendulum System","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"graph theory, double pendulum, chaotic dynamics, trajectory analysis","lastPublishedDoi":"10.21203/rs.3.rs-5928823/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5928823/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Understanding the interplay between spatial and kinematic properties in chaotic systems is crucial for advancing nonlinear dynamics, yet remains a challenging problem. 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