On the algebra of Koopman eigenfunctions and on some of their infinities

TL;DR

This paper explores the algebraic structure of Koopman eigenfunctions for reversible systems, proposing a method to efficiently compute and extend eigenfunctions, aiding in the analysis of complex dynamical behaviors.

Contribution

It introduces a novel approach leveraging the multiplicative group structure of eigenfunctions to accelerate their computation and continuation across singularities.

Findings

Enables larger sets of eigenfunctions from a small initial set.

Supports eigenfunction continuation across singularities.

Improves global representation learning from local data.

Abstract

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.