Examining the Limitations of Dynamic Mode Decomposition through Koopman Theory Analysis

The paper’s Theorem 3.8 proves that if one component of P(x(t)) keeps a constant nonzero sign on I, then a time state-space map ξ exists and ϕ(x)=e^{α ξ(x)+β} is a Koopman eigenfunction; it relies on Def. 3.3, Lemma 3.4, and Lemma 3.7 to formalize this construction. The candidate solution gives the same construction with more explicit details: it proves strict monotonicity of a coordinate via absolute continuity (Assumption 3.1 → Lemma 3.2), deduces injectivity of the trajectory, defines ξ as the inverse on X=x(I), and sets ϕ(x)=e^{α ξ(x)+β}, verifying d/dt ϕ(x(t))=αϕ(x(t)) (equivalently, K^τ_P ϕ(x(s))=e^{α τ}ϕ(x(s))). This matches the paper’s argument, just avoiding the differential-geometric phrasing (“simple and open curve”). Minor wording imprecision in the paper (“this entry is monotone”) is clarified by the model (x_i is strictly monotone because P_i(x(t)) keeps a strict sign). Overall, both are correct and essentially the same proof idea. See Def./Lemma statements and Theorem 3.8 in the paper .