Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator

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.

Figures (9)

• Figure 1: Relation among two singularly-perturbed systems and associated slow/fast subsystems.
• Figure 2:
• Figure 4: Five flows derived from the fast and slow subsystems, and their limits which we analyze in this paper.
• Figure 5: $\angle \phi_{{\rm i}\omega}^\varepsilon$, and log-scaled derivatives of $\phi_{{\rm i}\omega}^\varepsilon$, with $F(x,y)=0$ (dash curves). The derivatives are computed using finite differences.
• Figure 6: $\ln \phi_\nu^\varepsilon$, and log-scaled derivatives of $\phi_\nu^\varepsilon$ with $F(x,y)=0$ (dash curves). The derivatives are computed using finite differences.

Theorems & Definitions (6)

• Lemma 1
• Lemma 2
• Theorem 1
• proof
• proof
• proof