On the Equivalence of Koopman Eigenfunctions and Commuting Symmetries

TL;DR

This paper establishes a theoretical link between Koopman eigenfunctions and commuting symmetries, providing a constructive method for their computation and insights into the geometric structure of nonlinear systems.

Contribution

It offers a new geometric characterization of Koopman eigenfunctions through commuting symmetries and derives an explicit, convergent formula for principal eigenfunctions.

Findings

Explicit formula for Koopman eigenfunctions derived

Proved uniform convergence on the region of attraction

Established equivalence between eigenfunctions and symmetries

Abstract

The Koopman operator framework offers a way to represent a nonlinear system as a linear one. The key to this simplification lies in the identification of eigenfunctions. While various data-driven algorithms have been developed for this problem, a theoretical characterization of Koopman eigenfunctions from geometric properties of the flow is still missing. This paper provides such a characterization by establishing an equivalence between a set of Koopman eigenfunctions and a set of commuting symmetries -- both assumed to span the tangent spaces at every point on a simply connected open set. Based on this equivalence, we derive an explicit formula for the principal Koopman eigenfunctions and prove its uniform convergence on the region of attraction of a locally asymptotically stable equilibrium point, thereby offering a constructive method for computing Koopman eigenfunctions.