Spatio-Temporal Koopman Decomposition

Abstract

This paper deals with a new purely data-driven method, called the spatio-temporal Koopman decomposition, to approximate spatio-temporal data as a linear combination of (possibly growing or decaying exponentially) standing or traveling waves. The method combines (i) either standard singular value decomposition (SVD) or higher-order SVD and (ii) either standard dynamic mode decomposition (DMD) or an extension of this method by the authors, called higher-order DMD. In particular, for periodic or quasiperiodic attractors, the method gives the spatio-temporal pattern as a superposition of standing and/or traveling waves, which are identified in an efficient and robust way. Such superposition may give the whole pattern as a modulated, periodic or quasiperiodic, standing or traveling wave. The method is illustrated in some simple toy-model dynamics, and its performance is tested in the identification of standing and traveling waves in the Ginzburg–Landau equation and of azimuthal waves in a rotating spherical shell with thermal convection.