Distinguished Limits in Vibrodynamics: from Nonuniqueness to Universality

The paper establishes that for x_τ + ε^λ x_s = ε u(x,s,τ) with 2π-periodic τ and smooth u, only two distinguished limits (DLs) yield closed averaged hierarchies: DL-1 (λ=1) and DL-2 (λ=2). It argues this via power-compatibility of ε and ε^λ with an integer-power series in ε (restricting to λ=N or λ=1/N), then derives the leading averaged equations: for DL-1, x0,s = ⟨u(x0,s,τ)⟩ and x1,s = (x1·∇0)⟨u⟩ + V0 with x̃1 = ũ0^τ; for DL-2, solvability at O(ε) enforces ⟨u(x0,s,τ)⟩=0 and the O(ε^2) average yields x0,s = V0 with x̃1 = ũ0^τ, where V0 ≡ ⟨(ũ0^τ·∇0)ũ0⟩ = 1/2⟨[ũ0,ũ0^τ]⟩. Non-integer paths are excluded by the power-compatibility requirement; λ=1/N with N≥2 is excluded because it forces x0,s≡0; and λ=N≥3 fails to produce a closed equation for x0 and is excluded. These statements, including the definitions of averaging, tilde-integration, restrictions x̃0≡0 and V0≠0, and the DL-1/DL-2 outcomes, appear explicitly in the paper’s Sections 3–5 (e.g., (3.1)–(3.4), (3.5)–(3.9), (4.1)–(4.12)) and summary items (List of Results) . The candidate solution independently reproduces the same case split and averaged equations, adding an explicit solvability-contradiction for λ=N≥3 using V0≠0 (namely, ⟨u1⟩ = V0), which tightens the paper’s exclusion claim without contradicting it. Thus both are correct and essentially follow the same asymptotic structure and conclusions.