Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator

TL;DR

This paper applies Koopman operator analysis to the singularly-perturbed van der Pol oscillator, revealing how spectral properties depend on perturbation and how the singular limit relates to the combined fast-slow dynamics.

Contribution

It demonstrates the spectral signature of the Koopman operator in singularly-perturbed systems and derives the singular limit through fast and slow subsystem analysis.

Findings

Koopman eigenvalues depend on the singular perturbation parameter.

Eigenfunctions reflect geometric properties of the perturbed system.

The singular limit of the Koopman operator is derived from fast and slow subsystems.

Abstract

The Koopman operator framework holds promise for spectral analysis of nonlinear dynamical systems based on linear operators. Eigenvalues and eigenfunctions of the Koopman operator, so-called Koopman eigenvalues and Koopman eigenfunctions, respectively, mirror global properties of the system's flow. In this paper we perform the Koopman analysis of the singularly-perturbed van der Pol system. First, we show the spectral signature depending on singular perturbation: how two Koopman {principal} eigenvalues are ordered and what distinct shapes emerge in their associated Koopman eigenfunctions. Second, we discuss the singular limit of the Koopman operator, which is derived through the concatenation of Koopman operators for the fast and slow subsystems. From the spectral properties of the Koopman operator for the {singularly}-perturbed system and the singular limit, we suggest that the Koopman eigenfunctions inherit geometric properties of the singularly-perturbed system. These results are applicable to general planar singularly-perturbed systems with stable limit cycles.