Time-Delay Observables for Koopman: Theory and Applications

TL;DR

This paper explores the use of time-delay observables in the Koopman operator framework to represent nonlinear dynamical systems, providing universal representations and optimal finite-dimensional bases with practical implications.

Contribution

It introduces a universal, system-independent representation of Koopman operators using time-delay observables and identifies optimal finite-dimensional bases for certain systems.

Findings

Koopman operators admit universal representations in time-delay coordinates.

Certain systems have optimal finite-dimensional bases formed by these observables.

Analytic representations coincide with dynamic mode decomposition results.

Abstract

Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge. Koopman operator theory circumvents some of these issues by considering the dynamics in the space of observable functions on the state, in which the dynamics are intrinsically linear and thus amenable to standard techniques from numerical analysis and linear algebra. However, practical issues remain with this approach, as the space of observables is infinite-dimensional and selecting a subspace of functions in which to accurately represent the system is a nontrivial task. In this work we consider time-delay observables to represent nonlinear dynamics in the Koopman operator framework. We prove the surprising result that Koopman operators for different systems admit universal (system-independent) representations in these coordinates, and give analytic expressions for these representations. In addition, we show that for certain systems a restricted class of these observables form an optimal finite-dimensional basis for representing the Koopman operator, and that the analytic representation of the Koopman operator in these coordinates coincides with results computed by the dynamic mode decomposition. We provide numerical examples to complement our results. In addition to being theoretically interesting, these results have implications for a number of linearization algorithms for dynamical systems.