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

Abstract: Complex spectra of transfer and Koopman operators describe rotational motion in dynamical systems. A particularly relevant situation in applications is where the rotation speed depends on the position in the phase space. 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 for applications. Our theorems on support localisation of the eigenfunctions yield simple algorithms to detect the existence and phase-space location of approximately cyclic motion with distinct periods. Our response results demonstrate that information on the cycle periods and their locations determined by the operator eigendata is largely insensitive to the noise level. We believe the mechanisms creating the eigendata apply more broadly and enhance our understanding of approximate cycle detection in dynamical systems with operator methods.