Koopman Operator Family Spectrum for Nonautonomous Systems.

For any nonautonomous dynamical system, the family of Koopman operators, as well as related Koopman eigenvalues and eigenfunctions, is parameterized by a time pair. Therefore, a logical approach in the data-driven algorithms for the nonautonomous Koopman mode decomposition is the application of a dynamic mode decomposition (DMD) method on the moving stencils of snapshots in order to capture the time dependency. In this paper, we investigate the issues that arise in such an approach. These issues do not appear if we use the moving stencil approach as the model fitting method; they appear as significant errors in the computed nonautonomous Koopman operator eigenvalues. The first issue manifests itself in the hybrid dynamical systems when the moving stencil passes over a nonautonomous switching point. We show that such stencils can be detected through the Krylov subspace projection error and propose an algorithm that computes correct eigenvalues by avoiding such stencils. The second issue appears in the continuous-in-time nonautonomous systems. Even if we apply techniques of finding good observables that solve all issues in the autonomous case, the nonautonomous Koopman eigenvalues will still be computed with a significant error. In the presented theorems, we reveal the nature of this error and propose a second algorithm that is based on the reduction of the stencil size. The application of the two new data-driven algorithms on various nonautonomous systems shows complete recovery from the errors otherwise present in computation of the nonautonomous Koopman operator eigenvalues.