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

Understand

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 (PCT) 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.