Deep Learning for Bifurcation Detection: Extending Early Warning Signals to Dynamical Systems with Coloured Noise

Deep Learning for Bifurcation Detection: Extending Early Warning Signals to Dynamical Systems with Coloured Noise | 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 Deep Learning for Bifurcation Detection: Extending Early Warning Signals to Dynamical Systems with Coloured Noise Yazdan Babazadeh Maghsoodlo, Daniel Dylewsky, Madhur Anand, Chris T. Bauch This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5975924/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 Deep learning models have demonstrated remarkable success in recognising tipping points and providing early warning signals. However, there has been limited exploration of their application to dynamical systems governed by coloured noise, which characterizes many real-world systems. In this study, we demonstrated that, using the normal form theorem, it is possible to leverage the normal forms of three primary types of bifurcations (Fold, Transcritical, and Hopf) to construct a training set that enables deep learning architectures to perform effectively. Furthermore, we showed that this approach could accommodate coloured noise by replacing white noise with red noise during the training process. To evaluate the classifier trained on red noise compared to one trained on white noise, we tested their performance on mathematical models using ROC curves and AUC scores. Our findings reveal that the DL architecture can be effectively trained on colored noise inputs, as evidenced by high validation accuracy and minimal sensitivity to redness (ranging from 0.83 to 0.85). However, classifiers trained on white noise also demonstrate impressive performance in identifying tipping points in colored time series. This is further supported by high AUC scores (ranging from 0.9 to 1) for both classifiers across different colored stochastic time series. Applied Mathematics Deep learning Tipping points Early warning signals Dynamical systems Coloured noise Normal form theorem Bifurcations Fold bifurcation Transcritical bifurcation Hopf bifurcation Red noise White noise Training set construction ROC curves AUC scores 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-5975924","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":412247221,"identity":"4871714c-cd85-4528-89cc-5008c98cfc40","order_by":0,"name":"Yazdan Babazadeh Maghsoodlo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA70lEQVRIiWNgGAWjYDACCQY2EMXYD+XLEK9lZgOQPMDAwEO8lg0HiNVicLv52YMff+xkN9/IMfz8oeIeDz97A+OHH/i03DlmbtjDk2y87UaOscSBM8U8kj0HmCV78Gm5kWAmwSPBnLjtRu4GiYNtCTxAETa8zjO4kf5N8o9BfeLmGbmbfxz8B9HC+AevlhwzaZ6Ew4kbJHK3SRxsgGhhxmeL5J0z5cYyB44bzzjz/pvFmWMJQL8cbJbGFzt8t9u3PXzzp1q2vz0t+UZFTYIcP3vzwY9v8GhROIApxtiARwMDgzx+6VEwCkbBKBgFQAAAM1hUeNuKC6kAAAAASUVORK5CYII=","orcid":"","institution":"University of Waterloo","correspondingAuthor":true,"prefix":"","firstName":"Yazdan","middleName":"Babazadeh","lastName":"Maghsoodlo","suffix":""},{"id":412247222,"identity":"45bc4b5b-9f1d-457f-8a6f-6f592512511b","order_by":1,"name":"Daniel Dylewsky","email":"","orcid":"","institution":"Carleton University","correspondingAuthor":false,"prefix":"","firstName":"Daniel","middleName":"","lastName":"Dylewsky","suffix":""},{"id":412247223,"identity":"a561aa81-00c5-4e4e-910f-174a0bca80d9","order_by":2,"name":"Madhur Anand","email":"","orcid":"","institution":"University of Guelph","correspondingAuthor":false,"prefix":"","firstName":"Madhur","middleName":"","lastName":"Anand","suffix":""},{"id":412247224,"identity":"c0272f85-6980-47b5-b9fe-b892374e9eef","order_by":3,"name":"Chris T. 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