Principal component trajectories for modeling spectrally continuous dynamics as forced linear systems.

Delay embeddings of time series data have emerged as a promising coordinate basis for data-driven estimation of the Koopman operator, which seeks a linear representation for observed nonlinear dynamics. Recent work has demonstrated the efficacy of dynamic mode decomposition (DMD) for obtaining finite-dimensional Koopman approximations in delay coordinates. In this paper we demonstrate how nonlinear dynamics with sparse Fourier spectra can be (i) represented by a superposition of principal component trajectories and (ii) modeled by DMD in this coordinate space. For continuous or mixed (discrete and continuous) spectra, DMD can be augmented with an external forcing term. We present a method for learning linear control models in delay coordinates while simultaneously discovering the corresponding exogenous forcing signal in a fully unsupervised manner. This extends the existing DMD with control (DMDc) algorithm to cases where a control signal is not known a priori. We provide examples to validate the learned forcing against a known ground truth and illustrate their statistical similarity. Finally, we offer a demonstration of this method applied to real-world power grid load data to show its utility for diagnostics and interpretation on systems in which somewhat periodic behavior is strongly forced by unknown and unmeasurable environmental variables.

Discovering time-varying aerodynamics of a prototype bridge by sparse identification of nonlinear dynamical systems.

Vortex-induced vibrations (VIVs) have been observed on a long-span suspension bridge. The nonstationary wind in the field characterized by the time-varying mean wind speed is likely to lead to time-varying aerodynamics of the wind-bridge system during VIVs, which is different from VIVs induced by stationary or even steady wind in wind tunnels. In this paper, data-driven methods are proposed to reveal the time-varying aerodynamics of the wind-bridge system during VIV events based on field measurements on a long-span suspension bridge. First, a variant of the sparse identification of nonlinear dynamics algorithm is proposed to identify parsimonious, time-varying aerodynamical systems that capture VIV events of the bridge. Thus we are able to posit new, data-driven, and interpretable models highlighting the aeroelastic interactions between the wind and bridge. Second, a density-based clustering algorithm is applied to discovering the potential modes of dynamics during VIV events. As a result, the time-dependent model is obtained to reveal the evolution of the aerodynamics of the wind-bridge system over time during an entire VIV event. It is found that the level of self-excited effects of the wind-bridge system is significantly time varying with the real-time wind speed and bridge motion state. The simulations of VIVs by the obtained time-dependent models show high accuracies of the models with an averaged normalized mean-square error of 0.0023. The clustering of obtained models shows underlying distinct dynamical regimes of the wind-bridge system, which are distinguished by the level of self-excited effects.

Chaos as an intermittently forced linear system.

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.