Data-driven nonlinear model reduction to spectral submanifolds in mechanical systems

The paper states and uses the forced 2D-SSM normal form ρ̇ = −α(ρ)ρ + f sin ψ, ψ̇ = ω(ρ) − Ω + (f/ρ) cos ψ, and explicitly derives the frequency–response relation Ω = ω(ρ) ± sqrt(f^2/ρ^2 − α(ρ)^2) with sin ψ = α(ρ)ρ/f, as well as the ‘forced backbone’ condition at maximal amplitude f = α(ρmax)ρmax with cos ψ = 0 and Ω = ω(ρmax) . The candidate solution reproduces exactly these steps and conclusions. The only nuance is a minor sign/convention slip in the paper’s prose about the quadrature phase (“θ = Ωt − π/2”) which, when combined with Eq. (9)–(10), would imply sin(Ωt − θ) = 1 instead of the required −1 at the amplitude boundary; mathematically, the condition cos ψ = 0 and sin ψ = 1 (i.e., ψ = π/2) is the consistent quadrature inference from the model equations. Aside from this notational sign, both arguments coincide.