Koopman spectra in reproducing kernel Hilbert spaces

Abstract

Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the $L^2$ 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 $N\to \infty$, in which case the corresponding Fourier frequency is also a Koopman eigenfrequency. If such an RKHS extension does not exist, we can still construct an $L^2$ approximation of the eigenfunction. 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.

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• Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator
• Koopman Theory and Linear Approximation Spaces
• Limits of Learning Dynamical Systems
• Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces
• Linear quadratic control of nonlinear systems with Koopman operator learning and the Nyström method
• The spatiotemporal coupling in delay-coordinates dynamic mode decomposition
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• Uncovering Quasi-periodic Nature of Physical Systems: A Case Study of Signalized Intersections
• Smooth Koopman eigenfunctions
• A probabilistic approach to drift estimation from stochastic data