We review basic notions and properties of the eigenvalues and eigenfunctions of the nonautonomous Koopman operator in two formulations, process and skew product. Then we relate these formulations with the issues that arise when data-driven algorithms are applied to the evaluation of the nonautonomous Koopman eigenvalues and eigenvectors.
The first data-driven approach is the DMD application to nonautonomous Koopman eigenvalues and eigenvectors in the process formulation. In such approach the ambition to catch time dependence of these eigenvalues and eigenvectors fails as significant errors appear. We illustrate that on some examples and prove the structure of these errors.
The second data-driven approach is the DMD application to nonautonomous Koopman operator eigenvalues and eigenfunctions in the skew product formulation, i.e. when they are treated as eigenvalues and eigenfunctions of the underlying extended autonomous dynamical system. We illustrate this connection and excellent numerical results on several examples.