Extraction of periodic structures from time series using Koopman analysis: An extension of normal-mode and Fourier analyses

Shervin Bagheri
Linné Flow Center
Department of Mechanics
KTH, Stockholm


In a classical linear normal modal analysis, the perturbation dynamics is described in terms of eigenmodes that span the local stable/unstable directions in a small neighborhood of an equilibrium point. This work is concerned with a modal analysis that is global and nonlinear. The flow dynamics is expanded into a finite set of coherent structures , where each mode represents all the points in the full spatial domain associated to one single frequency, damping coefficient and amplitude. The approach is based on the spectral characteristics of the linear infinite-dimensional Koopman operator. These modes, referred to as Koopman modes may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. We illustrate the method on an a few fluid flow examples and discuss the unresolved numerical and theoretical challenges.