On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions

TL;DR

This paper explores how Koopman operator eigenfunctions can be used to match and rectify nonlinear dynamical systems, extending their application from prediction to systematic discovery of transformations.

Contribution

It introduces a novel approach leveraging Koopman spectral theory for matching and rectifying dynamical systems, including data-driven algorithms and illustrative examples.

Findings

Koopman eigenfunctions can identify conjugacies between systems.

The method extends Koopman analysis beyond prediction to system transformation.

Illustrative examples demonstrate the approach's effectiveness.

Abstract

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In this paper we will argue that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments. We believe, and document through illustrative examples, that this can nontrivially extend the use and applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards what can be considered as a systematic discovery of "Cole-Hopf-type" transformations for dynamics.