Not All Chaos is Equal: An In-Depth Evaluation of Ten Chaotic Maps in Dholes-Inspired Optimization for Constrained Engineering Problems

Not All Chaos is Equal: An In-Depth Evaluation of Ten Chaotic Maps in Dholes-Inspired Optimization for Constrained Engineering Problems | 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 Not All Chaos is Equal: An In-Depth Evaluation of Ten Chaotic Maps in Dholes-Inspired Optimization for Constrained Engineering Problems Dhara Buch, Hitarth Buch This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8671807/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 Chaos-enhanced search mechanisms are widely used to improve exploration and mitigate stagnation in metaheuristic optimization. However, their integration into the recently proposed Dholes-Inspired Optimization (DIO) algorithm has not been previously studied. This work presents the first systematic evaluation of chaotic DIO variants, achieved by embedding ten structurally diverse chaotic maps into DIO and benchmarking their performance across seventeen classical constrained engineering design problems. Each chaotic map is first analysed through dynamical diagnostics—including time-series behaviour, cobweb structure, return maps, and sampling-density patterns—to assess its ergodicity and distributional characteristics. All algorithmic variants are then evaluated under uniform experimental conditions (identical FE budgets, 51 runs, and fixed seed scheduling). Results demonstrate that Piecewise, Singer, Iterative, and Circle maps consistently enhance DIO’s convergence reliability, producing the lowest mean ranks and highest win counts across the testbed. Statistical tests (Friedman, Nemenyi, Wilcoxon) confirm the significance of these improvements. Representative convergence curves show that chaotic variants accelerate early exploration and achieve feasible designs more reliably on complex, multi-constraint landscapes. Across the seventeen constrained engineering problems, four chaotic maps—Piecewise, Singer, Iterative, and Circle—consistently form a statistically superior elite group, offering the most reliable performance improvements. These results provide clear, evidence-based guidance for practitioners seeking effective chaos mechanisms within DIO and similar swarm optimizers. The code is publicly available on https://in.mathworks.com/matlabcentral/fileexchange/182678-not-all-chaos-is-equal. Chaotic maps Optimization algorithms Constrained design Dholes-Inspired Optimization Metaheuristics Engineering benchmarks 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. 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-8671807","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":578879889,"identity":"8a9f7409-eddc-4c53-837d-300317650f25","order_by":0,"name":"Dhara Buch","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVRIie2RMWuDQBTHnwjnEnLrk36Jg0KwIPpVlIPrYvoRQqbrUuncb5LRIDgZux7YoS6dMjjGLWcSu0UdC73f8If3eD/egwdgMPxF1ALA2vrXArNbN5pWxK+Cc5X8Vg3KGMs6LbDbfQbvzqFpvdLfgJN/Q7O7r7hflXDTsuYfb8+PiEogLASDuLyvMJWsHixZc6YIILa5PiwBiOWkUvFQEft0UehxlpIFDAnRh2kFJ7a46oU/pZJHWAri6XAl/rBsTFmq9V51Mgjpa2HXWPiUUt403YgyEG912P1TCPTvmUHYh9XOGTUYDIZ/xxnlLFFr2cOB0gAAAABJRU5ErkJggg==","orcid":"","institution":"Government Engineering College, Rajkot","correspondingAuthor":true,"prefix":"","firstName":"Dhara","middleName":"","lastName":"Buch","suffix":""},{"id":578879890,"identity":"f64011b2-b033-4c4e-b592-0540e5181d7f","order_by":1,"name":"Hitarth Buch","email":"","orcid":"","institution":"Government Engineering College, Rajkot","correspondingAuthor":false,"prefix":"","firstName":"Hitarth","middleName":"","lastName":"Buch","suffix":""}],"badges":[],"createdAt":"2026-01-22 16:22:32","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-8671807/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8671807/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100979658,"identity":"1732fb18-9d6a-4d2c-a5e7-e7e9d07d9a95","added_by":"auto","created_at":"2026-01-23 11:55:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2705565,"visible":true,"origin":"","legend":"","description":"","filename":"R1EmeSpringChaoticDholePaper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8671807/v1_covered_8296a840-8f67-4ec0-b86b-c807a28dfd76.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eNot All Chaos is Equal: An In-Depth Evaluation of Ten Chaotic Maps in Dholes-Inspired Optimization for Constrained Engineering Problems\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":"Chaotic maps, Optimization algorithms, Constrained design, Dholes-Inspired Optimization, Metaheuristics, Engineering benchmarks","lastPublishedDoi":"10.21203/rs.3.rs-8671807/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8671807/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eChaos-enhanced search mechanisms are widely used to improve exploration and mitigate stagnation in metaheuristic optimization. However, their integration into the recently proposed Dholes-Inspired Optimization (DIO) algorithm has not been previously studied. This work presents the first systematic evaluation of chaotic DIO variants, achieved by embedding ten structurally diverse chaotic maps into DIO and benchmarking their performance across seventeen classical constrained engineering design problems. Each chaotic map is first analysed through dynamical diagnostics—including time-series behaviour, cobweb structure, return maps, and sampling-density patterns—to assess its ergodicity and distributional characteristics. All algorithmic variants are then evaluated under uniform experimental conditions (identical FE budgets, 51 runs, and fixed seed scheduling).\u003c/p\u003e\n\u003cp\u003eResults demonstrate that Piecewise, Singer, Iterative, and Circle maps consistently enhance DIO’s convergence reliability, producing the lowest mean ranks and highest win counts across the testbed. Statistical tests (Friedman, Nemenyi, Wilcoxon) confirm the significance of these improvements. Representative convergence curves show that chaotic variants accelerate early exploration and achieve feasible designs more reliably on complex, multi-constraint landscapes.\u003c/p\u003e\n\u003cp\u003eAcross the seventeen constrained engineering problems, four chaotic maps—Piecewise, Singer, Iterative, and Circle—consistently form a statistically superior elite group, offering the most reliable performance improvements. These results provide clear, evidence-based guidance for practitioners seeking effective chaos mechanisms within DIO and similar swarm optimizers. 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