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ON DYNAMIC MODE DECOMPOSITION: THEORY AND …

Manuscript submitted to the Journal of Computational DynamicsON DYNAMIC MODE DECOMPOSITION: THEORY AND APPLICATIONSJ onathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg,Dept. of Mechanical and Aerospace EngineeringPrinceton UniversityPrinceton, NJ 08544, USAS teven L. Brunton, and J. Nathan KutzDept. of Applied MathematicsUniversity of WashingtonSeattle, WA 98195, introduced in the fluid mechanics community, dynamicmode decomposition (DMD) has emerged as a powerful tool for analyzing thedynamics of nonlinear systems. However, existing DMD THEORY deals primarilywith sequential time series for which the measurement dimension is much largerthan the number of measurements taken. We present a theoretical frameworkin which we define DMD as the eigendecomposition of an approximating lin-ear operator. This generalizes DMD to a larger class of datasets, includingnonsequential time series. We demonstrate the utility of this approach by pre-senting novel sampling strategies that increase computational efficiency andmitigate the effects of noise, respectively.

At rst glance, it may seem dubious that a nonlinear system could be described by superposition of modes whose dynamics are governed by eigenvalues. After all, one needs a linear operator in order to talk about eigenvalues. However, it was shown in [3] that DMD is closely related to a spectral analysis of the Koopman operator.

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  Dome, Dynamics, Glance, Theory, Theory and, Decomposition, Dynamic mode decomposition

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