On the structure of complex spectra and eigenfunctions of transfer and Koopman operators

TL;DR

This paper characterizes the eigenspectra and eigenfunctions of transfer and Koopman operators in noisy dynamical systems, providing insights into cycle detection and the effects of noise on spectral properties.

Contribution

It offers precise descriptions of eigenspectra and eigenfunctions under small noise, including their zero-noise limits and response behaviors, with algorithms for detecting cyclic motions.

Findings

Eigenfunctions localize support near cyclic motions.

Eigenvalues and eigenfunctions are robust to noise in the linear response regime.

Algorithms can detect cycle periods and locations from eigendata.

Abstract

Complex eigenspectra of transfer and Koopman operators describe rotational motion in dynamical systems. A particularly relevant situation in applications is when the rotation speed depends on the state-space position of the dynamics. We consider a canonical model of such dynamics in the presence of small noise, and provide precise characterisations of the eigenspectrum and eigenfunctions of the corresponding transfer operators. Further, we study the limiting behaviour of the eigenspectrum and eigenfunctions in the zero-noise limit, including their quadratic and linear response. Our results clarify the structure of transfer and Koopman operator eigenspectra, and provide new interpretations relevant to applications. Our theorems on support localisation of the eigenfunctions yield simple algorithms to detect the existence and state-space location of approximately cyclic motion with distinct periods. Our numerical results verify that information on the cycle periods and their locations determined by the operator eigendata is insensitive to noise level in the linear response regime. We believe that the dynamic mechanisms underlying the eigendata and their properties apply rather broadly and enhance our understanding of approximate cycle detection in dynamical systems with operator methods.