Chaos as an Intermittently Forced Linear System

TL;DR

This paper introduces HAVOK, a data-driven method that decomposes chaotic systems into linear models with intermittent forcing, enabling better understanding and prediction of complex nonlinear dynamics across various real-world systems.

Contribution

The paper presents HAVOK, a novel approach combining delay embedding, Koopman theory, and sparse regression to linearize and analyze chaotic systems with intermittent forcing.

Findings

HAVOK effectively models chaos as an intermittently forced linear system.

Forcing signals are non-Gaussian with long tails, predicting rare events.

The method distinguishes linear from nonlinear regions in phase space.

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.