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Chaos as an Intermittently Forced Linear System
Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, Eurika Kaiser, J. Nathan Kutz
TL;DR
The paper addresses how to identify useful structure in chaotic dynamics despite their strong nonlinearity and difficult prediction. It combines Takens delay embedding, sparse regression, and Koopman operator theory in HAVOK, a data-driven decomposition into an intermittently forced linear system. Across canonical and real-world examples, forcing is non-Gaussian and predictive of intermittent switching and bursting while separating approximately linear from strongly nonlinear phase-space regions.
Problem
Chaotic dynamics can be difficult to predict from uncertain measurements, while nonlinear Koopman-invariant representations are challenging or impossible in systems with multiple attracting structures or continuous spectra.
Method
HAVOK combines Takens delay embedding, sparse regression, and Koopman operator theory to identify an intermittently forced linear representation of chaos.
Results
Across Lorenz and real-world examples, non-Gaussian forcing with long tails predicts intermittent switching and bursting and marks coherent approximately linear and strongly nonlinear phase-space regions.
Takeaways & Limitations
Forcing activity provides a predictive signal for intermittent transients while linear models describe coherent regions of strongly nonlinear dynamics in delay coordinates.
Takeaways & Limitations
A finite-dimensional linear Koopman-invariant subspace cannot contain linear measurements of systems with multiple attractors, periodic orbits, or fixed points, and discrete spectra are not guaranteed.
Abstract
from arXiv · showhide
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.
1 Introduction
Chaotic systems exhibit organized structure despite difficult prediction, while data-driven methods address settings where governing equations are unavailable or elusive. The paper introduces HAVOK, combining delay embeddings, regression, and Koopman theory to represent chaos as a forced linear system.
- Chaotic dynamics are difficult to predict from uncertain measurements but remain highly organized, with coherent structure and patterns.
- Data-driven analysis is increasingly important because abundant data often coexist with elusive physical laws or governing equations.
- Takens embedding reconstructs an attractor from a single measurement, but its use for nonlinear model identification has remained disconnected from delay-based chaos characterization.
- HAVOK decomposes chaos into a forced linear system using time-delay embedding, regression models, and Koopman operator theory.
- The method partitions phase space into approximately linear regions with small forcing and strongly nonlinear regions with large forcing.
2 Background
This section introduces Koopman-based linear descriptions, data-driven regression methods, and time-delay embeddings as tools for analyzing nonlinear dynamics from measurements.
- 2.1 Koopman operator theory: The Koopman operator advances measurement functions linearly but is generally infinite dimensional, motivating finite-dimensional approximations.Finite matrix representations require measurement subspaces that remain invariant under the operator.
- 2.1 Koopman operator theory: Finite-dimensional Koopman-invariant representations are difficult to obtain and cannot contain linear measurements of systems with multiple attracting structures.Mixing chaotic systems may also lack a discrete Koopman spectrum.
- 2.2.1 Dynamic mode decomposition (DMD): Dynamic mode decomposition fits a linear operator between time-shifted snapshot matrices and can approximate Koopman modes.The best-fit operator minimizes the Frobenius-norm prediction error, often after low-rank SVD projection.
- 2.2.1 Dynamic mode decomposition (DMD): DMD commonly uses linear state measurements, but selecting nonlinear measurements that yield approximately closed Koopman systems remains an open problem.Nonlinear measurement functions may be informed by system knowledge or searched over function bases.
- 2.2.2 Sparse identification of nonlinear dynamics (SINDy): SINDy identifies sparse nonlinear dynamics from measurement data by using regression to select the few active, nonzero terms.The method is equation-free and exploits sparsity in the space of possible functions.
- 2.3 Time-delay embedding: Takens embedding enriches a measurement with delayed copies, allowing an attractor to be reconstructed under certain conditions.The reconstructed attractor is diffeomorphic to the original attractor, including in some cases from a single point measurement.
3 Decomposing chaos: Hankel alternative view of Koopman (HAVOK)
HAVOK represents strongly nonlinear chaotic dynamics as a linear model in leading delay coordinates with the final coordinate treated as intermittent forcing. Delay-coordinate construction and sparse regression reveal approximately linear coherent regions separated from rare switching or bursting events.
- Delay-coordinate construction: The method builds intrinsic measurement coordinates from a system’s time history rather than advancing instantaneous state measurements.These coordinates are obtained from a single measured time series using delay embedding and a Hankel-matrix SVD.
- Delay-coordinate construction: A low-rank SVD approximation of the Hankel matrix provides delay coordinates that are approximately invariant under the Koopman operator.The leading columns of U and V supply the measurement system and its corresponding time series.
- Forced linear representation: The model is linear in the first r −1 delay coordinates, while the last coordinate v_r is imposed as a forcing term.Sparse regression identifies a parsimonious linear model with dominant off-diagonal structure, whereas v_r is not accurately represented by linear or polynomial models.
- Forced linear representation: Non-Gaussian forcing with long tails corresponds to rare-event forcing that drives lobe switching in the Lorenz system.The forcing coordinate is therefore retained as an input rather than absorbed into a closed linear model.
- Phase-space organization: Small forcing marks coherent phase-space regions with approximately linear dynamics, whereas large forcing marks essential nonlinearities associated with intermittent switching or bursting.The forcing signal strongly predicts these events across the examples discussed in the paper.
4 HAVOK analysis illustrated on the chaotic Lorenz system
HAVOK reconstructs Lorenz chaos from a single measurement by embedding time delays, fitting linear dynamics to leading coordinates, and treating a low-energy coordinate as intermittent forcing. The forcing is usually small, but bursts precede lobe switching and distinguish approximately linear from strongly nonlinear regions.
- Delay embedding: HAVOK stacks delayed measurements into a Hankel matrix and uses SVD to obtain variance-ordered eigen-time-delay coordinates forming an embedded attractor.The resulting attractor is diffeomorphic to the original Lorenz attractor under Takens embedding conditions.
- Predictive forcing: The v15 forcing is nearly zero for most times and bursts when the trajectory is about to switch between Lorenz attractor lobes.The forcing-driven model reconstructs both the embedded attractor and the dominant coordinate v1.
- Predictive forcing: A threshold of 4 × 10^-6 identifies active forcing, whose bursts precede lobe switching by nearly one period, although some switching events are missed.Active forcing occurs on the outer portions of the attractor lobes, while inactive periods occupy predominantly linear regions.
- Linear and nonlinear regions: Small forcing corresponds to largely linear dynamics, whereas the forcing signal distills the system’s essential nonlinearity near lobe switching.The forcing distribution is compared with a scaled normal distribution and has non-Gaussian long-tail behavior associated with rare-event forcing.
- Connection to almost-invariant sets: The boundaries of HAVOK’s approximately linear and strongly nonlinear regions closely match the almost-invariant-set boundaries of the Perron-Frobenius operator.These boundaries mark qualitatively different dynamics and align with the two attractor-lobe structures.
5 HAVOK analysis on additional chaotic systems
HAVOK is applied across analytical, stochastic, delayed, and real-world systems. Across these examples, it captures attractor dynamics, predicts intermittent events, and uses large forcing as a predictive signal for switching and bursting.
- Example systems: The examples span Lorenz and Rössler systems, Mackey-Glass, forced Duffing, stochastic magnetic-field reversal, electrocardiogram, electroencephalogram, and measles data.The analysis therefore covers numerical models and systems characterized from real-world measurements.
- Cross-example results: In each example, HAVOK captures qualitative attractor dynamics and predicts large transients and intermittent phenomena.Large-forcing regions correspond to coherent phase-space regions, while small-forcing regions are well modeled by delay-coordinate Koopman linear dynamics.
- Cross-example results: Large forcing often precedes Lorenz lobe switching, magnetic-field reversal, and bursting measles outbreaks, making the forcing signal predictive.The reported predictive relationship spans both simulated systems and real-world data.
- Validation: HAVOK models generalize beyond training data and can reproduce attractor dynamics, including lobe switching, on an unused validation trajectory.The forcing is extracted from a relatively short window compared with a switching event, supporting measurement-based prediction in realistic applications.
5.1 Parameters
The paper’s examples include ODE, DDE, and SDE models alongside systems characterized purely by measurement data, with simulation and HAVOK parameters tabulated separately.
- Parameters: Simulation parameters are listed in Table 2, while system descriptions and HAVOK analysis parameters are summarized in Table 3.The measles data are interpolated from 2-week spacing to 0.2-week resolution and stacked into q = 50 rows.
- System classes: The example systems range from ordinary differential equations to delay and stochastic differential equations, plus systems characterized purely by measurement data.The forced Duffing oscillator is included as a weakly nonlinear, nonchaotic example for the examined parameters.
5.2 Description of all examples
The paper introduces the Duffing oscillator as a weakly nonlinear two-well system and identifies the Lorenz system as a canonical chaotic model.
- Forced Duffing oscillator: The Duffing oscillator has two potential wells separated by a saddle point and is used as a weakly nonlinear example.For the parameters examined, it is not chaotic.
- Model formulation: A second-order dynamical equation can be rewritten as a coupled system of first-order equations by introducing suspended variables.
- Lorenz system: The Lorenz system is presented as a canonical model for chaotic dynamics.Its equations are introduced as part of the paper’s collection of example systems.
5.2.3 R¨ossler system
The Rössler system produces oscillations in the x−y plane with occasional transient growth and decay in z, a form of bursting distinct from Lorenz lobe switching. The section also describes numerical simulation and measurement choices for analyzing this chaotic system.
- R¨ossler system: The Rössler trajectory oscillates in the x−y plane while showing occasional transient growth and decay in z.This behavior is termed bursting because the trajectory makes a transient from an attractor back to itself.
- R¨ossler system: Bursting differs from Lorenz lobe switching because it is a transient from an attractor to itself.
- Numerical simulation: The double pendulum is simulated with a variational integrator based on the Euler-Lagrange equations.The integrator uses a trapezoidal approximation to the action integral.
- Numerical simulation: For the double pendulum, the Lagrangian is defined as kinetic energy minus potential energy.
- Measurement: The measurement x(t) = cos(2θ1(t)) is used because the mean of θ1 drifts after a revolution.
5.2.6 Earth’s magnetic field reversal
The Earth’s magnetic field reverses over geological timescales, motivating models of rare switching events through turbulent magnetohydrodynamic dynamics and stochastic forcing. In the analyzed data, the forcing term becomes active before the dominant signal shows an obvious variance increase, enabling early prediction of reversals.
- Motivation: Earth’s magnetic field reversals are rare events occurring over geological timescales.The passage identifies understanding and predicting these events as an important geophysical challenge.
- Model: A simplified model represents turbulent magnetohydrodynamic fluctuations as stochastic forcing on a dynamo.
- Model: The magnetic-field model is formulated as a differential equation for the complex magnetic field A with nonlinear terms and stochastic forcing f.
- Model parameters: The stochastic model uses Gaussian random variables ξ with standard deviation 5 and specified coefficients μ, ν, B1–B4, and b1–b4.
- Prediction: The forcing term vr becomes large during field reversals and activates before v1 exceeds its expected variance.This provides a predictive signal before the dominant time series has a clear statistical signature.
5.2.7 Electrocardiogram (ECG)
The ECG analysis uses a single recorded heart signal embedded in time delays. The dataset contains 380 seconds sampled at 250 Hz.
- Data: An electrocardiogram measures the heart’s electrical activity and produces characteristic spiking pulses for each heartbeat.
- Context: The ECG signal has previously been analyzed with delay embeddings to quantify properties such as fractal dimension.
- Data: The analysis uses the qtdb/sel102 signal adapted from the PhysioNet database.
- Data: 380 seconds of ECG data were recorded at a sampling rate of 250 Hz.
5.2.8 Electroencephalogram (EEG)
EEG is recorded as scalp voltage, which provides only a rough proxy for brain activity, and the cited data are available through Sleep-EDF. The supplied passages also describe New York City measles data and forcing-based outbreak prediction.
- Electroencephalogram (EEG): An electroencephalogram records voltage from electrodes placed on the scalp as a nonintrusive measurement of brain activity.
- Electroencephalogram (EEG): The EEG voltage signal is only a rough proxy for brain activity.
- Data: The cited Sleep-EDF data are available through PhysioNet.
- Measles outbreaks: New York City measles outbreak data span 1928–1964 and are binned every two weeks.
- Measles outbreaks: The forcing signal becomes large directly before major measles outbreaks, including the largest outbreak after 1940.
6 Discussion
The discussion presents HAVOK as a data-driven representation of chaos using intermittently forced linear systems. Its forcing signal predicts transient events and separates approximately linear regions from strongly nonlinear ones.
- HAVOK identifies an intermittently forced linear-system representation of chaotic dynamics.
- The procedure combines machine-learning regressions, Takens’ embedding, and Koopman operator theory.
- Forcing activity predicts intermittent transients such as lobe switching and bursting.
- The forcing signal partitions phase space into coherent regions with approximately linear dynamics and regions with nonlinear dynamics.
- The analysis motivates searching for intrinsic measurement coordinates and simple linear representations of complex systems.