Universal resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems

The paper formulates the star-network Duffing system with weak forcing/damping and coupling (λ = O(γ^2)), derives the effective hub forcing Γ = γ a0(m)[1 + λN/(2−ω^2)] after linearizing the leaves, and applies the first-order Melnikov method to obtain the leaf and hub chaotic thresholds U(ω,0,m) and U(ω,λN,m) (Eqs. (6)–(7)) . In the Supplementary Material, the homoclinic orbit, the two Melnikov integrals (C = 4δ/3 and the √2πγωa0(m)sech term), and the scaled hub expressions are given explicitly, leading to the same threshold with the coupling-induced factor in the denominator . The candidate solution reproduces these steps and results almost verbatim: identical homoclinic data and integrals, the same scaling for the hub (α = 1 − λN), elimination of the quenched constant term (which indeed does not contribute since ∫ v_h dt = 0), and the same threshold U(ω,λN,m) with the factor (1 + λN/(2 − ω^2)) in the denominator. Their additional arguments (first-harmonic dominance; uniqueness of the minimizers in N and m; closed-form a0(m)) are consistent with the paper’s claims and the SM’s Fourier formula for a0(m) . Minor presentation differences aside, both derivations coincide in substance.