Warning Signs for Non-Markovian Bifurcations: Color Blindness and Scaling Laws

The paper proves that along the attracting branch of the slow manifold the adiabatic variance V∞(y) diverges with universal exponents that depend on the bifurcation normal form and the noise type: white noise yields exponents (−1, −1, −1/2), colored OU noise shows no divergence (0, 0, 0), and α-regular Volterra noises (including fBM and Rosenblatt with H = α + 1/2) yield exponents (−2H, −2H, −H). These statements, derived via linearization and an adiabatic limit for a nonlocal variance equation, are explicit in the paper’s formulas (normal-form derivatives a(y): y, y, −2√−y; white-noise scaling; OU “color blindness”; α-regular and fBM/Rosenblatt scalings and Table 1) . The candidate solution independently linearizes around the attracting manifold and computes the stationary variances of the resulting linear filters: OU for white noise; a 2×2 OU Lyapunov equation for colored noise; and a Volterra/Young-integral representation for α-regular drivers. It reproduces all exponents and constants specialized to fBM/Rosenblatt (V∞ = σ² H Γ(2H)/|a(y)|^{2H}), exactly matching the paper. The only minor discrepancy is an intermediate constant for generic α-regular noise (a factor α(2α+1) versus the paper’s cαΓ(2α)), which does not affect exponents and reduces to the same H Γ(2H) constant under the standard fBM/Rosenblatt normalization. Overall, proofs differ in technique but agree on results.