An Extended Method of Multiple Scales

An Extended Method of Multiple Scales | 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 An Extended Method of Multiple Scales Nidish Narayanaa Balaji, Max J. Miller III, D. Dane Quinn This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7776236/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 11 You are reading this latest preprint version Abstract Perturbation techniques have long proven useful for analytically describing the dynamics of weakly nonlinear systems. The Method of Multiple Scales (MMS) is one such technique, popular for its versatility and structured approach to handling systems with multiple underlying time scales. However, in assuming an asymptotic series solution, classical MMS often has a restricted domain of applicability. For example, the generated solution for a nonlinear oscillator ultimately ends up as a perturbation about the homogeneous solution of the corresponding linear oscillator. While useful for capturing nonlinear phenomena such as amplitude dependent resonance, the solution will in general diverge from the direct numerical solution as the strength of the nonlinearity and/or the amplitude of the response increases. For example, in the lowest-order approximation the maximum amplitude of the response in resonance is independent of the strength of the nonlinearity, which does not match the response developed from the exact solution obtained from numerical integration. Higher-order solutions can of course be sought but immediately confront the practitioner with an abundance of choices to resolve under-determined equations. This is further complicated by the fact that classical asymptotic expansions lack uniform convergence properties by definition (therefore the name ``singular perturbation''). This work first examines how higher-order MMS has been historically addressed, and then proceeds to offer a novel take of its own. Inspiration is drawn from the exact solution for a harmonically forced and damped linear oscillator which can be expressed in the form of a rational polynomial fraction that is impossible to recover with traditional MMS (which uses a power law expansion). This motivates expansion of both the solution and the amplitude of the excitation in an ordering parameter. Based on this insight, the eXtended Method of Multiple Scales (XMMS) is introduced and applied to a damped-driven nonlinear (Duffing) oscillator. The XMMS approach is shown to consistently outperform traditional MMS and alternative higher-order approaches. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 26 Nov, 2025 Reviews received at journal 26 Nov, 2025 Reviews received at journal 25 Nov, 2025 Reviews received at journal 17 Nov, 2025 Reviewers agreed at journal 05 Nov, 2025 Reviewers agreed at journal 04 Nov, 2025 Reviewers agreed at journal 04 Nov, 2025 Reviewers invited by journal 04 Nov, 2025 Editor assigned by journal 02 Nov, 2025 Submission checks completed at journal 28 Oct, 2025 First submitted to journal 03 Oct, 2025 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-7776236","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":540478059,"identity":"3ed23ad4-fd95-44aa-b76d-b85d7ba40478","order_by":0,"name":"Nidish Narayanaa Balaji","email":"","orcid":"","institution":"Indian Institute of Technology-Madras","correspondingAuthor":false,"prefix":"","firstName":"Nidish","middleName":"Narayanaa","lastName":"Balaji","suffix":""},{"id":540478062,"identity":"b983a859-4a0e-44fe-b664-5dfaceba6955","order_by":1,"name":"Max J. 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The Method of Multiple Scales (MMS) is one such technique, popular for its versatility and structured approach to handling systems with multiple underlying time scales. However, in assuming an asymptotic series solution, classical MMS often has a restricted domain of applicability. For example, the generated solution for a nonlinear oscillator ultimately ends up as a perturbation about the homogeneous solution of the corresponding linear oscillator. While useful for capturing nonlinear phenomena such as amplitude dependent resonance, the solution will in general diverge from the direct numerical solution as the strength of the nonlinearity and/or the amplitude of the response increases. For example, in the lowest-order approximation the maximum amplitude of the response in resonance is independent of the strength of the nonlinearity, which does not match the response developed from the exact solution obtained from numerical integration. Higher-order solutions can of course be sought but immediately confront the practitioner with an abundance of choices to resolve under-determined equations. This is further complicated by the fact that classical asymptotic expansions lack uniform convergence properties by definition (therefore the name ``singular perturbation''). This work first examines how higher-order MMS has been historically addressed, and then proceeds to offer a novel take of its own. Inspiration is drawn from the exact solution for a harmonically forced and damped linear oscillator which can be expressed in the form of a rational polynomial fraction that is impossible to recover with traditional MMS (which uses a power law expansion). This motivates expansion of both the solution and the amplitude of the excitation in an ordering parameter. Based on this insight, the eXtended Method of Multiple Scales (XMMS) is introduced and applied to a damped-driven nonlinear (Duffing) oscillator. The XMMS approach is shown to consistently outperform traditional MMS and alternative higher-order approaches.\u003c/p\u003e","manuscriptTitle":"An Extended Method of Multiple Scales","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-14 04:56:37","doi":"10.21203/rs.3.rs-7776236/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-11-26T20:36:37+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-26T18:12:50+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-25T11:25:15+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-17T18:23:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"99944561287963870565432469668461269998","date":"2025-11-05T14:43:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"72376485454012874759581505384468503960","date":"2025-11-04T23:57:51+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"193504873699401754396330479662307996995","date":"2025-11-04T21:25:26+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-11-04T08:14:43+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-11-02T14:49:34+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-10-29T02:54:20+00:00","index":"","fulltext":""},{"type":"submitted","content":"Nonlinear Dynamics","date":"2025-10-03T20:51:02+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"1b546f6b-2b8c-4488-9c4f-6f64e3d940be","owner":[],"postedDate":"November 14th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2025-11-26T20:38:19+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-14 04:56:37","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7776236","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7776236","identity":"rs-7776236","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}