On the Equivalence of Koopman Eigenfunctions and Commuting Symmetries

2025/08/04 by Jiang, Xinyuan, Li, Yan
#FOS: Electrical engineering #FOS: Physical sciences #Mathematical Physics (math-ph) #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2508.02437

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.

Citations

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Related

• A Minimal Set of Koopman Eigenfunctions -- Analysis and Numerics
• Smooth Koopman eigenfunctions
• Koopman Operator Methods for Global Phase Space Exploration of\n Equivariant Dynamical Systems
• On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions
• Solving nonlinear ordinary differential equations using the invariant manifolds and Koopman eigenfunctions