Participation Factors for Nonlinear Autonomous Dynamical Systems in the Koopman Operator Framework

The candidate’s derivation mirrors the paper’s Appendix B/C proofs: start from the KMD expansion of the state observable f(x)=x (equilibrium: (17); limit cycle: (54)), take first-order variations with respect to the initial condition using smooth Koopman eigenfunctions, and identify coefficients as participation factors and generalized participations from Definitions 7–8 to obtain the displayed formulas for dx_k(t) and dz_j(t) ((26)–(27) for a stable equilibrium, and their LC analogues (55)–(56)). The paper executes this via the derivative of the solution map and the t=0 KMD identity for dx_k(0) ((49)–(53)), while the candidate subtracts two KMD series and Taylor-expands the eigenfunctions; both routes drop O(∥δx∥^2) terms and rely on the same convergence/smoothness assumptions. Hence, both are correct and essentially the same proof strategy, with slightly different bookkeeping of derivatives versus differences. See Theorem 1 and its proof (26)–(27) with (49)–(53) and Lemma 3 (17) for the EP case, and Theorem 2 with Lemma 4 (index set I) for the LC case (55)–(56) .