Nonlinear Dynamics of Dyads with Self-Regulating Internal States and Nonreciprocal, Time-Decaying Interactions

Nonlinear Dynamics of Dyads with Self-Regulating Internal States and Nonreciprocal, Time-Decaying Interactions | 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 Nonlinear Dynamics of Dyads with Self-Regulating Internal States and Nonreciprocal, Time-Decaying Interactions Arpan Dey This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9390873/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 introduce a minimal model for emotional and relational interactions between two individuals. The framework consists of four coupled nonlinear differential equations: two describing the time evolution of the individuals’ emotional states and two describing the directed relational feelings between them. The model incorporates key mechanisms relevant to interpersonal dynamics, including internal emotional regulation, nonreciprocal influence, time-decaying interaction strength, coupling between (internal) emotional dynamics and (external) relational dynamics, and stochastic perturbations. To summarize the evolving dyadic state, we introduce a bounded scalar metric that quantifies overall emotional and relational alignment between the two individuals. We further analyze the qualitative behavior of the system using numerical simulations. The model is minimal and operates under explicit assumptions. Within these limits, however, it provides a compact mathematical framework for studying coupled emotional and relational dynamics at the level of a dyad, complementing mean-field approaches that focus on large populations. Thermodynamics and statistical mechanics Mathematical Physics nonlinear dynamics dyadic systems stochastic differential equations emotion dynamics relational dynamics Full Text Additional Declarations The authors declare no competing interests. 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. 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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-9390873","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":621550961,"identity":"46200dcd-0ecb-4f0d-9710-9b8853f65449","order_by":0,"name":"Arpan Dey","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6ElEQVRIiWNgGAWjYNACNhseNobDB4AsCRlitaTJ8TMeSwBp4SFWy2FjyeYzBiAmYS38s88e/FxQxpy44diZz69u1FjwMLAfProBnxaJc3nJ0jPOsSVuOHN2m3XOMaDDeNLSbuC15gyPgTRvG0/ihhtntxnnsAG1SPCY4dUif4bH+Ddvm0Tihvtvnhnn/CNCi8EZHjOgLQbGkg1nmB/nthGhxfAMX5o1z7kEOX6GY2bMuX0SPGyE/CJ3hvfwbZ6y/6CofPw551udHD/74WP4vY8UEWwSYBK/clQtzB8Iqx4Fo2AUjIKRCACkaEe3ytwGwgAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0009-0005-6974-0642","institution":"University of Montpellier","correspondingAuthor":true,"prefix":"","firstName":"Arpan","middleName":"","lastName":"Dey","suffix":""}],"badges":[],"createdAt":"2026-04-11 23:55:12","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9390873/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9390873/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106960814,"identity":"2333d6cd-b742-4ed1-8af7-c0d3039e5200","added_by":"auto","created_at":"2026-04-15 09:23:14","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1119817,"visible":true,"origin":"","legend":"","description":"","filename":"dyadicemotionalmodelingfinal.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9390873/v1_covered_edb0c6b5-fbf0-4699-9496-bb2ea8943d41.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eNonlinear Dynamics of Dyads with Self-Regulating Internal States and Nonreciprocal, Time-Decaying Interactions\u003c/strong\u003e\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":"nonlinear dynamics, dyadic systems, stochastic differential equations, emotion dynamics, relational dynamics","lastPublishedDoi":"10.21203/rs.3.rs-9390873/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9390873/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe introduce a minimal model for emotional and relational interactions between two individuals. 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