Chaos as an intermittently forced linear system

ARTICLE DOI: 10.1038/s41467-017-00030-8 OPEN Chaos as an intermittently forced linear system Steven L. Brunton1, Bingni W. Brunton2, Joshua L. Proctor3, Eurika Kaiser1 & J. Nathan Kutz4 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. 1 Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA. 2 Department of Biology, University of Washington, Seattle, WA 98195, USA. 3 Institute for Disease Modeling, Bellevue, WA 98004, USA. 4 Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA. Correspondence and requests for materials should be addressed to S.L.B. (email: ) NATURE COMMUNICATIONS | 8: 19 | DOI: 10.1038/s41467-017-00030-8 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00030-8 D ynamical systems describe the changing world around us, modeling the interactions between quantities that co-evolve in time1. These dynamics often give rise to rich, complex behaviors that may be difficult to predict from uncertain measurements, a phenomena commonly known as chaos. Chaotic dynamics are ubiquitous in the physical, biological, and engineering sciences, and they have captivated amateurs and experts for over a century. The motion of planets2, weather and climate3, population dynamics4-6, epidemiology7, financial markets, earthquakes, and turbulence8, 9, are all compelling examples of chaos. Despite the name, chaos is not random, but is instead highly organized, exhibiting coherent structure and patterns10, 11. The confluence of big data and machine learning is driving a paradigm shift in the analysis and understanding of dynamical systems in science and engineering. Data are abundant, while physical laws or governing equations remain elusive, as is true for problems in climate science, finance, and neuroscience. Even in classical fields such as turbulence, where governing equations do exist, researchers are increasingly turning toward data-driven analysis12-16. Many critical data-driven problems, such as predicting climate change, understanding cognition from neural recordings, or controlling turbulence for energy efficient power production and transportation, are primed to take advantage of progress in the data-driven discovery of dynamics17–27. An early success of data-driven dynamical systems is the celebrated Takens embedding theorem9, which allows for the reconstruction of an attractor that is diffeomorphic to the original chaotic attractor from a time series of a single measurement. This remarkable result states that, under certain conditions, the full dynamics of a system as complicated as a turbulent fluid may be uncovered from a time series of a single point measurement. Delay embeddings have been widely used to analyze and characterize chaotic systems5–7, 28–31. They have also been used for linear system identification with the eigensystem realization algorithm (ERA)32 and in climate science with the singular spectrum analysis (SSA)33 and nonlinear Laplacian spectrum analysis34. ERA and SSA yield eigen-time-delay coordinates by applying principal component analysis to a Hankel matrix. However, these methods are not generally useful for identifying meaningful models of chaotic nonlinear systems, such as those considered here. In this work, we develop a universal data-driven decomposition of chaos into a forced linear system. This decomposition relies on time-delay embedding, a cornerstone of dynamical systems, but takes a new perspective based on regression models19 and modern Koopman operator theory35–37. The resulting method partitions phase space into coherent regions where the forcing is small and dynamics are approximately linear, and regions where the forcing is large. The forcing may be measured from time series data and strongly correlates with attractor switching and bursting phenomena in real-world examples. Linear representations of strongly nonlinear dynamics, enabled by machine learning and Koopman theory, promise to transform our ability to estimate, predict, and control complex systems in many diverse fields. A video abstract is available for this work at: https://youtu.be/ 831Ell3QNck, and code is available at: http://faculty.washington. edu/sbrunton/HAVOK.zip. Results Linear representations of nonlinear dynamics. Consider a dynamical system1 of the form d xðtÞ ¼ fðxðtÞÞ; dt ð1Þ where xðtÞ 2 Rn is the state of the system at time t and f represents the dynamic constraints that define the equations of motion. When 2 working with data, we often sample (1) discretely in time: Z ðkþ1ÞΔt xkþ1 ¼ Fðxk Þ ¼ xk þ fðxðτÞÞdτ; ð2Þ kΔt where xk = x(kΔt). The traditional geometric perspective of dynamical systems describes the topological organization of trajectories of (1) or (2), which are mediated by fixed points, periodic orbits, and attractors of the dynamics f. However, analyzing the evolution of measurements, y = g(x), of the state provides an alternative view. This perspective was introduced by Koopman in 193138, although it has gained traction recently with the pioneering work of Mezic et al.35, 36 in response to the growing abundance of measurement data and the lack of known governing equations for many systems of interest. Koopman analysis relies on the existence of a linear operator K for the dynamical system in (2), given by Δ Kg ¼ g  F ) Kgðxk Þ ¼ gðxkþ1 Þ: ð3Þ The Koopman operator K induces a linear system on the space of all measurement functions g, trading finite-dimensional nonlinear dynamics in (2) for infinite-dimensional linear dynamics in (3). Expressing nonlinear dynamics in a linear framework is appealing because of the wealth of optimal control techniques for linear systems and the ability to analytically predict the future. However, obtaining a finite-dimensional approximation of the Koopman operator is challenging in practice39, relying on intrinsic measurements related to the eigenfunctions of the Koopman operator K, which may be more difficult to obtain than (...truncated)