Learning Koopman Operators for Systems with Isolated Critical Points

The Koopman operator provides a way to transform a (potentially) nonlinear finite-dimensional dynamical system into an infinite-dimensional linear system by lifting the nonlinear state dynamics into a functional space of observables, where the dynamics are linear. Previous literature has claimed that it is not possible to represent nonlinear dynamics with multiple isolated critical points if the set of observables is finite and contains the state; more precisely, such a set cannot be invariant under the Koopman operator. In this paper, we investigate this claim in more detail and provide an analytical counterexample to disprove it. We also consider the convergence of discrete-time Koopman approximation error to the continuous-time error: we show both how this convergence occurs in general and how it can fail for systems with multiple isolated critical points. In particular, discontinuities in Koopman observables at the boundaries of basins of attraction may cause the continuoustime error to diverge; the discrete-time error also suffers from this as the sampling time step goes to zero.