The Adaptive Spectral Koopman Method for Dynamical Systems

The paper’s Theorem 3.2 shows that ASK solutions are real-valued when x, f, g (hence K) are real by pairing each complex eigen-contribution with its conjugate in the truncated reconstruction g_N(x(t)), explicitly using the middle entry of eigenvectors and a direct algebraic simplification to a real expression (Section 3.5; Lemma 3.1 and Theorem 3.2, and eqs. (14)–(16) for setup) . The candidate model solution proves the same claim via the conjugate-pair structure for real K, uniqueness of the Vc=g(Ξ) expansion with real right-hand side, and an explicit observation that the ASK amplitude a_j=c_j(v_j)_{mid} is invariant under eigenvector rescaling, so conjugate pairs contribute 2 Re(a_j e^{λ_j t}). Both arguments rely on the same core facts and reach the same conclusion; the model adds clarity on coefficient conjugacy and rescaling invariance, while the paper uses a concrete inner-product argument and explicit algebra to show the sum is real (see the paper’s computation of cm ν e^{λ t} + c̄m ν̄ e^{λ̄ t} yielding a real expression) .