TY - JOUR

T1 - Koopman spectra in reproducing kernel Hilbert spaces

AU - Das, Suddhasattwa

AU - Giannakis, Dimitrios

N1 - Funding Information:
Dimitrios Giannakis received support from ONR YIP grant N00014-16-1-2649 , NSF grant DMS-1521775 , and DARPA grant HR0011-16-C-0116 . Suddhasattwa Das is supported as a postdoctoral research fellow from the first grant. The authors are also grateful to Corbinian Schlosser for his insightful feedback, and three anonymous referees for making a number of technical suggestions that have helped us improve the paper.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/9

Y1 - 2020/9

N2 - Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the L2 space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series is a non-trivial problem when the underlying system has a dense point spectrum, or a continuous spectrum behaving similarly to noise. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time series generated by such a system with unknown dynamics. Our main result gives necessary and sufficient conditions for a Fourier function, defined on N states sampled along an orbit of the dynamics, to be extensible to a Koopman eigenfunction on the whole state space, lying in a reproducing kernel Hilbert space (RKHS). In particular, we show that such an extension exists if and only if the RKHS norm of the Fourier function does not diverge as N→∞. In that case, the corresponding Fourier frequency is also a Koopman eigenfrequency, modulo a unique translate by a Nyquist frequency interval, and the RKHS extensions of the N-sample Fourier functions converge to a Koopman eigenfunction in RKHS norm. For Koopman eigenfunctions in L2 that do not have RKHS representatives, the RKHS extensions of Fourier functions at the corresponding eigenfrequencies are shown to converge in L2 norm. Numerical experiments on mixed-spectrum systems with weak periodic components demonstrate that this approach has significantly higher skill in identifying Koopman eigenfrequencies compared to conventional spectral estimation techniques based on the discrete Fourier transform.

AB - Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the L2 space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series is a non-trivial problem when the underlying system has a dense point spectrum, or a continuous spectrum behaving similarly to noise. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time series generated by such a system with unknown dynamics. Our main result gives necessary and sufficient conditions for a Fourier function, defined on N states sampled along an orbit of the dynamics, to be extensible to a Koopman eigenfunction on the whole state space, lying in a reproducing kernel Hilbert space (RKHS). In particular, we show that such an extension exists if and only if the RKHS norm of the Fourier function does not diverge as N→∞. In that case, the corresponding Fourier frequency is also a Koopman eigenfrequency, modulo a unique translate by a Nyquist frequency interval, and the RKHS extensions of the N-sample Fourier functions converge to a Koopman eigenfunction in RKHS norm. For Koopman eigenfunctions in L2 that do not have RKHS representatives, the RKHS extensions of Fourier functions at the corresponding eigenfrequencies are shown to converge in L2 norm. Numerical experiments on mixed-spectrum systems with weak periodic components demonstrate that this approach has significantly higher skill in identifying Koopman eigenfrequencies compared to conventional spectral estimation techniques based on the discrete Fourier transform.

KW - Ergodic dynamical systems

KW - Koopman operators

KW - Reproducing kernel Hilbert spaces

KW - Spectral estimation

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U2 - 10.1016/j.acha.2020.05.008

DO - 10.1016/j.acha.2020.05.008

M3 - Article

AN - SCOPUS:85086434533

VL - 49

SP - 573

EP - 607

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

IS - 2

ER -