Koopman mode expansions between two invariant sets

Koopman mode expansions have been touted as a very useful way to represent nonlinear dynamics given the ability of dynamic mode decomposition to extract the modes from data. Here we explore how such an approach works for heteroclinic dynamics between two equilibria using a 1D nonlinear system (a pitchfork bifurcation) which allows explicit calculations. Well-defined Koopman mode expansions are found to exist around either equilibria but each fails at the same intermediary point between them indicating that there is no uniformly valid Koopman expansion for the dynamics. Results will be presented to indicate that this lack of uniformity carries over to the 3D Navier-Stokes equations as well.