A Minimal Set of Koopman Eigenfunctions -- Analysis and Numerics

TL;DR

This paper develops a comprehensive theoretical and numerical framework for Koopman eigenfunctions, introducing a minimal generating set and new coordinate systems to linearize nonlinear dynamics.

Contribution

It introduces a novel mathematical structure for Koopman eigenfunctions, defining minimal and independent sets, and establishes their equivalence and applications in system linearization.

Findings

Defined conditions for independence of Koopman eigenfunctions

Introduced a minimal generating set and proved its equivalence to maximal independent sets

Supported theory with numerical experiments demonstrating practical applications

Abstract

Research on Koopman operator theory has focused on three key areas for several decades: the mathematical structure of the Koopman eigenfunction space, the basis of this space, and the ability to represent nonlinear dynamics as linear. This study provides a thorough and comprehensive framework for these topics, including theoretical, analytical, and numerical approaches. A novel mathematical structure is introduced, which outlines permissible actions on the infinite set of Koopman Eigenfunction, under which this set is closed. Notions of generating and independent sets of Koopman eigenfunctions are defined. In addition, notions of a minimal generating set, and a maximal independent set are defined and are shown to be equivalent. This structure defines conditions for independence within the set of Koopman eigenfunctions. This independent set can be interpreted as a new coordinate system in which the dynamical system is linear. The theory also highlights the equivalence of a minimal set, flowbox representation, and conservation laws. Finally, the presented theory is supported by numerical experiments.