Chaos as an intermittently forced linear system

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.

Helicity Dynamics, Inverse, and Bidirectional Cascades in Fluid and Magnetohydrodynamic Turbulence: A Brief Review

“…Automatized current structure identification was performed in Servidio et al () and Klimas and Uritsky () for 2‐D MHD flows by detecting nulls of the magnetic field with a local hyperbolic topology as the plausible locus of reconnection events. Similarly, one can observe geodynamo field reversals using a decomposition of the data through a chaotic forcing with strong intermittent bursts (Brunton et al, ). Finally, Lagrangian models (also called α models) allow for computations at high Reynolds numbers by introducing a filter length scale.…”

Data-driven spectral analysis for coordinative structures in periodic systems with unknown and redundant dynamics

“…Practically, the previous work [36] used the hard threshold of SVD (1e-10). Another study in a similar algorithm [72] used the optimal hard threshold of SVD [73] when the noise level is unknown. Note that, obviously, the threshold of SVD is directly related to the dimension p of truncated SVD.…”

Subspace dynamic mode decomposition for stochastic Koopman analysis

“…rapidly becomes almost linearly dependent, and thus can be used to obtain a subspace that is approximately invariant to K. Based on the delayed measurements, we obtain a data matrix as a Hankel matrix. The use of delay coordinates for DMD was first discussed by Tu et al [19], and Brunton et al [18] mentioned DMD based on Hankel matrices, referring to the well-known Taken's theorem [46]. Susuki and Mezić [47] defined an approximation of the Koopman analysis using Prony's method, which also uses Hankel matrices.…”

“…It has been applied to a wide range of scientific and engineering subjects including fluid mechanics [9], power system analysis [10], medical care [11], epidemiology [12], robotic control [13], neuroscience [14], image processing [15], nonlinear system identification [16], finance [17], and chaotic systems [18]. DMD computes a set of modes along with the corresponding frequencies and decay rates, given a sequence of measurements from the target dynamics.…”