Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1

Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1 | 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 Stability of Fixed Points in Generalized Fractional Maps of the Orders 0 < α < 1 Mark Edelman This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2039338/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 Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders 0 < α < 1 that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and logistic maps coincide with the results previously observed in numerical simulations. In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point - asymptotically period two bifurcation point. PACS 05.45.Pq · 45.10.Hj Mathematics Subject Classification (2000) 47H99 60G99 · 34A99 · 39A70 Fractional maps fixed points bifurcations Full Text Supplementary Files StabilityFixedPoint.pdf 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-2039338","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":137278274,"identity":"57cd2679-bda3-4705-90be-a3f8ee9553e8","order_by":0,"name":"Mark Edelman","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA40lEQVRIiWNgGAWjYFCCBMYDQFKOgRkhZEBICwNIizHpWhIbkITwa+FnTz5w6EbNnfT+du7kD4w7Dic2sDdvk8CnRbLnWcLhnGPPcmcc5t0mwXgGqIXnWBleLQY3cgwO57Adzt3AzLuNgbENqEUixwyvFvsb+R8O5/w7nG7AzLv5A1iL/Bv8WgwkchgO57YdTgBq2SABsYUHvxaJM88MDuf2HTYE+yWxLd24jSet2AKfFv725IePc74dlufvP7v5w8c2a9l+9sMbb+DTggoSGJoZ2IhXDgF1pGoYBaNgFIyCEQAAiopOLg5fQtQAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-5190-3651","institution":"Yeshiva University Stern College for Women","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Mark","middleName":"","lastName":"Edelman","suffix":""}],"badges":[],"createdAt":"2022-09-07 00:05:47","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2039338/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2039338/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":26880080,"identity":"f49f1f05-6c14-4b20-8e79-b16fc53ef07d","added_by":"auto","created_at":"2022-09-23 14:20:33","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":240975,"visible":true,"origin":"","legend":"","description":"","filename":"NODYD2202613reviewer.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2039338/v1_covered.pdf"},{"id":26880079,"identity":"f5da0c73-a320-4958-b88b-0cd088da3281","added_by":"auto","created_at":"2022-09-23 14:20:25","extension":"pdf","order_by":6,"title":"","display":"","copyAsset":false,"role":"supplement","size":189220,"visible":true,"origin":"","legend":"","description":"","filename":"StabilityFixedPoint.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2039338/v1/3ff8700e9bdd608a5a8e9420.pdf"}],"financialInterests":"","formattedTitle":"Stability of Fixed Points in Generalized Fractional Maps of the Orders\n\n0 \u0026lt; α \u0026lt; 1","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-2039338/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"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":false,"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":"Fractional maps, fixed points, bifurcations","lastPublishedDoi":"10.21203/rs.3.rs-2039338/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2039338/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eCaputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. 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In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point - asymptotically period two bifurcation point.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ePACS \u003c/strong\u003e05.45.Pq · 45.10.Hj\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMathematics Subject Classification (2000)\u003c/strong\u003e 47H99 60G99 · 34A99 · 39A70\u003c/p\u003e","manuscriptTitle":"Stability of Fixed Points in Generalized Fractional Maps of the Orders\n0 \u0026lt; α \u0026lt; 1","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2022-09-23 14:20:24","doi":"10.21203/rs.3.rs-2039338/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"53f0376a-897d-48e3-b157-2b86b46280d8","owner":[],"postedDate":"September 23rd, 2022","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2022-11-02T13:12:42+00:00","versionOfRecord":[],"versionCreatedAt":"2022-09-23 14:20:24","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-2039338","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2039338","identity":"rs-2039338","version":["v1"]},"buildId":"ehx78VzkSd0WSzXnipQa-","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}