  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 principle 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 singularl-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.
