Koopman Analysis of the Singularly-Perturbed van der Pol Oscillator

The paper defines the singular-limit flow via the projection π to the attracting branches W∓ and the constrained slow flow S0_slow,τ, i.e., S̃0_slow,τ(x,y)=S0_slow,τ∘π(x,y) for τ>0, with domains D± determined by the fast-asymptotics (π is given explicitly) . In Appendix A it computes the half-slow-period as T0/2=3/2−ln 2, hence T0=3−2 ln 2 . The main spectral statement (Theorem 1) then asserts that the point spectrum of the singular-limit slow Koopman operator is exactly σ(Ũ0_slow,τ)={i n ω0 : n∈Z} and the associated eigenfunctions are smooth on D± . The candidate solution independently derives T0=3−2 ln 2 by explicit integration on W±, constructs a phase map φ measuring slow time-of-flight, establishes a conjugacy of the constrained flow to a rigid rotation modulo T0 so that nontrivial eigenpairs require e^{λT0}=1, and builds eigenfunctions f̃_n=e^{i n ω0 φ}∘π, also noting smoothness on D±—all in agreement with the paper’s framework (π, D±, and S̃0_slow,τ) . The only minor difference is a branch-wise sign convention in the paper’s explicit formula for ϕ̃0_iω on D−, which amounts to a choice of offset/orientation and does not affect the eigen-relation. Since the paper’s proof of Theorem 1 is omitted while the model supplies a clear proof sketch, I assess both as correct, with different proof presentations.