Temporal Integrity Limits in Discrete-Time Delayed Dissipative Systems

Temporal Integrity Limits in Discrete-Time Delayed Dissipative Systems | 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 Article Temporal Integrity Limits in Discrete-Time Delayed Dissipative Systems Nobuchika Yamaki, Tenna Churiki This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9000737/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 Delayed dissipative systems are ubiquitous in physical, biological, and engineered contexts, yet they are almost invariably analyzed, simulated, and controlled using discrete-time implementations. Temporal discretization is typically regarded as a numerical approximation whose effects vanish as the time step decreases. Here, we show that this assumption fails for systems with intrinsic delays. We introduce the Self-mapping Consistency Index (SMCI), a quantitative measure of dynamical identity across temporal resolutions, and apply it to a class of delayed dissipative feedback systems. Systematic parameter sweeps reveal a sharply defined temporal integrity boundary: as the temporal grain ΔT exceeds a critical value ΔT_c, trajectories generated under discrete-time evolution abruptly lose dynamical equivalence with their continuous-time counterparts. This identity collapse occurs discontinuously rather than through gradual degradation and is consistently accompanied by structural reorganization of the attractor and breakdown of predictive consistency. Across a wide range of parameters and numerical solvers, the critical temporal grain scales predominantly with the intrinsic delay time τ, following a quasi-universal relation ΔT_c / τ ≈ O(10⁻¹), with weak sensitivity to dissipation and nonlinearity. The persistence of this boundary under higher-order integration schemes demonstrates that identity collapse is not a numerical artifact but a fundamental constraint imposed by temporal discretization itself. These findings establish temporal resolution as a causal resource and identify an intrinsic limit on the faithful discrete-time representation of delayed dynamical systems. Physical sciences/Mathematics and computing Physical sciences/Physics Temporal Discretization Delayed Dissipative Systems Dynamical Identity Causal Structure Numerical Stability 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-9000737","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":614044622,"identity":"c1b807f7-6b8c-4b1c-96a0-6d5b20d89cb0","order_by":0,"name":"Nobuchika Yamaki","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIiWNgGAWjYFACHgYGxgYJAygvgYcfxiJei2QbkDpAWAsDXAuDwTECWszZe49J/NxhYczfwGP46UZNmozx/d6Dtz9U2OUxsDcffIBFi2XPuTTJ3jMSZhIHeIylc47l8Jgd40u2OHAmuZiB51iyARYtBjdyzKQZ2yRsGA7wGEjnsFUAtfCYSRxsO5DYIJFjJoFPizzQlt85/yp4jNtAWv4R1mJmcIDHTDq3LYfHgA2kpQGPljPnki172ySMDQ+zlVnn9qXxSBzLMbY4cyw5sQ2XX473Hrzxs63OcN7x5s23c74l2/M3nzG8UVFjl9iPI8QQgJkDYSTYPWx4lYMBO8JIbF4YBaNgFIyCkQsAmIthmvIjDLMAAAAASUVORK5CYII=","orcid":"","institution":"TNQ Tech","correspondingAuthor":true,"prefix":"","firstName":"Nobuchika","middleName":"","lastName":"Yamaki","suffix":""},{"id":614044623,"identity":"a88fedc8-7e79-4c3a-8432-eda88cb42e5a","order_by":1,"name":"Tenna Churiki","email":"","orcid":"","institution":"TNQ Tech","correspondingAuthor":false,"prefix":"","firstName":"Tenna","middleName":"","lastName":"Churiki","suffix":""}],"badges":[],"createdAt":"2026-03-01 10:23:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9000737/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9000737/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106959663,"identity":"05787d09-3440-42f5-8995-e94f93b832eb","added_by":"auto","created_at":"2026-04-15 09:13:06","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":514231,"visible":true,"origin":"","legend":"","description":"","filename":"TemporalIntegrityLimits.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9000737/v1_covered_4933c5a2-8c30-4e76-b8ea-0c9fd2e10ede.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Temporal Integrity Limits in Discrete-Time Delayed Dissipative Systems","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":"Temporal Discretization, Delayed Dissipative Systems, Dynamical Identity, Causal Structure, Numerical Stability","lastPublishedDoi":"10.21203/rs.3.rs-9000737/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9000737/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDelayed dissipative systems are ubiquitous in physical, biological, and engineered contexts, yet they are almost invariably analyzed, simulated, and controlled using discrete-time implementations. Temporal discretization is typically regarded as a numerical approximation whose effects vanish as the time step decreases. Here, we show that this assumption fails for systems with intrinsic delays.\u003c/p\u003e \u003cp\u003eWe introduce the Self-mapping Consistency Index (SMCI), a quantitative measure of dynamical identity across temporal resolutions, and apply it to a class of delayed dissipative feedback systems. Systematic parameter sweeps reveal a sharply defined temporal integrity boundary: as the temporal grain ΔT exceeds a critical value ΔT_c, trajectories generated under discrete-time evolution abruptly lose dynamical equivalence with their continuous-time counterparts. This identity collapse occurs discontinuously rather than through gradual degradation and is consistently accompanied by structural reorganization of the attractor and breakdown of predictive consistency.\u003c/p\u003e \u003cp\u003eAcross a wide range of parameters and numerical solvers, the critical temporal grain scales predominantly with the intrinsic delay time τ, following a quasi-universal relation ΔT_c / τ\u0026thinsp;\u0026asymp;\u0026thinsp;O(10⁻\u0026sup1;), with weak sensitivity to dissipation and nonlinearity. 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