{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/chaos-as-an-intermittently-forced-linear","title":"Chaos as an Intermittently Forced Linear System","arxiv_id":"1608.05306","date":"2016-08-18","proceeding":null,"authors":["Steven L. Brunton","Bingni W. Brunton","Joshua L. Proctor","Eurika Kaiser","J. Nathan Kutz"],"abstract":"Understanding the interplay of order and disorder in chaotic systems is a central challenge in modern quantitative science. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines Takens' delay embedding with modern Koopman operator theory and sparse regression to obtain linear representations of strongly nonlinear dynamics. The result is a decomposition of chaotic dynamics into a linear model in the leading delay coordinates with forcing by low energy delay coordinates; we call this the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the canonical Lorenz system, as well as to real-world examples such as the Earth's magnetic field reversal, and data from electrocardiogram, electroencephalogram, and measles outbreaks. In each case, the forcing statistics are non-Gaussian, with long tails corresponding to rare events that trigger intermittent switching and bursting phenomena; this forcing is highly predictive, providing a clear signature that precedes these events. Moreover, the activity of the forcing signal demarcates large coherent regions of phase space where the dynamics are approximately linear from those that are strongly nonlinear.","url_abs":"http://arxiv.org/abs/1608.05306v1","url_pdf":"http://arxiv.org/pdf/1608.05306v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"chaos-as-an-intermittently-forced-linear","repo_url":"https://github.com/baronefr/rhavok-analysis","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"CC0-1.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1608.05306","atlas_url":"https://app.syntology.ai/?focus=1608.05306","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1608.05306"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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