Chaos as an intermittently forced linear system

By Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, Eurika Kaiser, J. Nathan Kutz. 2016

645 citations recorded by the source corpus.

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.The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al.develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, background): “, 2009), nonlinear Laplacian spectral analysis (Giannakis and Majda, 2012), modeling emergent behavior (Roberts, 2014), Koopman analysis (Mezić, 2005; Budǐsić et al., 2012; Mezić, 2013; Brunton et al., 2017), automated inference of dynamics (Schmidt et al.”
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control (2016, background): “trolled setting). The use of time-delayed measurements is classical in system identication theory (see, e.g., [12]) but has also appeared in the context of Koopman operator approximation (see, e.g., [1, 23]). Assume therefore that we are given a collection of data X~ = [ 1 ;:::; K ]; Y~ = [+ 1 ;:::;”
Data Driven Governing Equations Approximation Using Deep Neural Networks (2018, methodology): “…conducted to eﬀectively deal with noises in data ([7, 40]), corruptions in data ([48]), partial diﬀerential equations [38, 41], etc. Methods have also been developed in conjunction with model selection approach ([28]), Koopman theory ([6]), and Gaussian process regression ([35]), to name a few.”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “Koopman theory has also been used to analyze chaotic dynamical systems from time-series data [85], [86], relying on the Takens embedding [87], which is related to sensor selection.”
Data-driven discovery of Koopman eigenfunctions for control (2017, background): “For chaotic systems, delay coordinates provides a promising embedding [5, 19, 109].”
Model selection for dynamical systems via sparse regression and information criteria (2017, methodology): “Time-delay coordinates have been successfully combined with SINDy to recover low-order representations of the system [1,40].”
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator (2016, methodology): “[3] proposed a new framework for Koopman analysis using Hankel-matrix representation of data.”
Modal Analysis of Fluid Flows: An Overview (2017, methodology): “Recent techniques also construct Koopman eigenfunctions using a regularized advection–diffusion operator [140] or approximate the Koopman operator in delay coordinates [133,148].”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, methodology): “In this experiment, LSTM was applied because it had been utilized for various nonlinear time-series, and Hankel DMD was used because it had been successfully utilized for analysis of chaotic systems [51].”
Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks (2018, background): “Recently, data-driven approximations of Koopman operators, including dynamic mode decomposition [26], diﬀusion maps [26], delay co-ordinates [8, 4, 1], and neural networks [27, 23, 15, 14, 17], have gained signiﬁcant attention.”