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Chaos as an intermittently forced linear system

Author

Listed:
  • Steven L. Brunton

    (University of Washington)

  • Bingni W. Brunton

    (University of Washington)

  • Joshua L. Proctor

    (Institute for Disease Modeling)

  • Eurika Kaiser

    (University of Washington)

  • J. Nathan Kutz

    (University of Washington)

Abstract

Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear.

Suggested Citation

  • Steven L. Brunton & Bingni W. Brunton & Joshua L. Proctor & Eurika Kaiser & J. Nathan Kutz, 2017. "Chaos as an intermittently forced linear system," Nature Communications, Nature, vol. 8(1), pages 1-9, December.
    • Handle: RePEc:nat:natcom:v:8:y:2017:i:1:d:10.1038_s41467-017-00030-8
    DOI: 10.1038/s41467-017-00030-8
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    Cited by:

    1. Riccardo Colantuono & Riccardo Colantuono & Massimiliano Mazzanti & Michele Pinelli, 2023. "Aviation and the EU ETS: an overview and a data-driven approach for carbon price prediction," SEEDS Working Papers 0123, SEEDS, Sustainability Environmental Economics and Dynamics Studies, revised Feb 2023.
    2. Soledad Le Clainche & José M. Vega, 2018. "Analyzing Nonlinear Dynamics via Data-Driven Dynamic Mode Decomposition-Like Methods," Complexity, Hindawi, vol. 2018, pages 1-21, December.
    3. Gyurhan Nedzhibov, 2024. "Delay-Embedding Spatio-Temporal Dynamic Mode Decomposition," Mathematics, MDPI, vol. 12(5), pages 1-18, March.
    4. Lampartová, Alžběta & Lampart, Marek, 2024. "Exploring diverse trajectory patterns in nonlinear dynamic systems," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    5. García-Rojas, Blanca E. & Ramirez-Dámaso, Gabriel & Caballero, Francisco & Femat, Ricardo, 2022. "Crisis-induced intermittency in Mexican dam flows," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    6. Jing Lu & Jingjun Jiang & Yidan Bai, 2024. "Deep Embedding Koopman Neural Operator-Based Nonlinear Flight Training Trajectory Prediction Approach," Mathematics, MDPI, vol. 12(14), pages 1-20, July.
    7. Zhang, Wenbo & Gu, Wei, 2024. "Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    8. Ali, Naseem & Cal, Raúl Bayoán, 2019. "Scale evolution, intermittency and fluctuation relations in the near-wake of a wind turbine array," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 215-229.
    9. Chau, Thi Tuyet Trang & Ailliot, Pierre & Monbet, Valérie, 2021. "An algorithm for non-parametric estimation in state–space models," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    10. Kanbur, Baris Burak & Kumtepeli, Volkan & Duan, Fei, 2020. "Thermal performance prediction of the battery surface via dynamic mode decomposition," Energy, Elsevier, vol. 201(C).

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