Higher order phase averaging for highly oscillatory systems

The candidate reconstruction matches the paper’s polynomial higher-order averaging with Gaussian weight: same mass matrix and inverse for Q = P2 (cf. M and M−1 in equation (2.36) ), the same contour/shift evaluation of the oscillatory moments R^m_α (Proposition 2.2 and its consequences in (2.31)–(2.34) ), and the same small-window and large-window limits (equations (2.40) and (2.41) ). The candidate’s identity I_n(c) = i^{−n} ∂_c^n e^{−(T^2 c^2)/2} is equivalent to the paper’s explicit polynomials times e^{−c^2 T^2/2} appearing in R^m_α, and the derivations lead to the same projected ODE (equation (2.39) ). One minor quibble is an optional “normalization” remark suggesting a −(3/2)T^4 factor for F_{m,2,2} in the T → ∞, c_m = 0 term of V̇0; the paper’s explicit formula has −3T^4 without such a factor (equation (2.41) ). Aside from this extraneous note, the logic and limits agree with the paper, including the invariance of V_k ≡ 0 (k ≥ 1) under the resonant system when initialized at zero, which recovers the phase-averaged V0 dynamics (see initialization policy for V̄k(0) in the polynomial basis and the T → ∞ discussion ).