On the algebra of Koopman eigenfunctions and on some of their infinities
For researchers studying nonlinear dynamical systems, this work provides a method to learn consistent global Koopman representations from locally sampled data, particularly useful for multistable systems with sparse measurements.
The paper exploits the multiplicative group property of Koopman eigenfunctions to accelerate numerical computation of eigenspaces, enabling richer representations and handling of singularities for global consistency from local data.
For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic numerical computation of the eigenspaces of the operator. Given a small set of (so-called ``principal'') eigenfunctions that are approximated conventionally, we can obtain a much larger set by constructing polynomials of the principal eigenfunctions. This enriches the set, and thus allows us to more accurately represent application-specific observables. Often, eigenfunctions exhibit localized singularities (e.g. in simple, one-dimensional problems with multiple steady states) or extended ones (e.g. in simple, two-dimensional problems possessing a limit cycle, or a separatrix); we discuss eigenfunction matching/continuation across such singularities. By handling eigenfunction singularities and enabling their continuation, our approach supports learning consistent global representations from locally sampled data. This is particularly relevant for multistable systems and applications with sparse or fragmented measurements.