Blow-up of Koopman eigenfunctions at basin boundaries

We show that under non-degeneracy conditions on a dynamical system, all of its smooth, non-constant Koopman eigenfunctions blow up along the boundary of the basin of attraction of a fixed point. We give a simple, geometric version of this general result for the case of a basin boundary composed of stable manifolds of edge states. We also give a more technical version of this blow-up result that is applicable to more general basin boundaries. These findings improve available results on the necessary blow-up of at least one Koopman eigenfunction and clarify that examples showing non-trivial bounded Koopman eigenfunctions across basin boundaries are degenerate. Our findings also imply that reduced-order models based on approximating Koopman-invariant subspaces of a typical physical system necessarily break down before reaching any truly nonlinear (i.e. nonlinearizable) behaviour away from a fixed point. Furthermore, Koopman operator-based reduced models, as well as their data-driven numerical approximations, over different domains of attraction cannot be continuously patched together as often envisioned.