Effects of Internal Resonance and Damping on Koopman Modes

The paper’s Proposition 1 claims that the products of Koopman eigenfunctions form a linearly independent set and uses this to assert a unique expansion of the identity observable in the eigenfunction basis. However, its proof relies on a time-domain contradiction that informally treats the x-dependent coefficients as if they were constants, concluding non-vanishing from the mere inequality of exponential rates; this misses a necessary step establishing that the coefficient functions cannot conspire to cancel for all x,t, and the ordering argument of real parts is not sufficient as written (see Eq. (33)–(39) and discussion around them) . By contrast, the candidate solution gives a standard, correct proof: with the canonical choice of linear eigenfunctions s_i(x)=w_i^T x (already foregrounded in the paper’s linear spectral setup), the monomials are just ordinary coordinate monomials, whose linear independence follows immediately by multi-index differentiation; the identity observable then expands uniquely using only degree-1 monomials for a linear system . The paper’s statement also imposes stronger-than-necessary sign assumptions and does not clearly specify the canonical choice of eigenfunctions needed to avoid algebraic dependence under resonances, whereas the model’s solution states these points explicitly and correctly. Consequently, the proposition’s intended conclusion is true in the canonical setting, but the paper’s proof is incomplete, while the model’s proof is sound and complete.