Fig. 2. Linear vs. nonlinear model reduction.

a Reduction of linear dynamics via Galerkin projection. The slowest spectral subspace, E¹ = E₁ (green), and the modal subspace, E₂ (black), span together the second slowest spectral subspace, E² = E₁ ⊕ E₂. The full dynamics (red curve) can be projected onto E¹ to yield a reduced slow model without transients. Projection of the full dynamics onto E² (blue curve) yields a reduced model that also captures the slowest decaying transient. Further, faster-decaying transients can be captured by projections onto slow spectral subspaces, E^k, with k > 1. b Reduction of nonlinearizable dynamics via restriction to spectral submanifolds (SSMs) in the ϵ = 0 limit of nonlinear, non-autonomous systems forced with ℓ frequencies. An SSM, W(E, Ωt; 0), is the unique, smoothest, nonlinear continuation of a nonresonant spectral subspace E. Specifically, the slowest SSM, W(E^k, Ωt; 0) (green), is the unique, smoothest, nonlinear continuation of the slowest spectral subspace, E^k. Nonlinearizability of the full dynamics follows if isolated stationary states coexist on at least one of the SSMs. The time-quasiperiodic SSMs for ϵ > 0, denoted W(E, Ωt; ϵ), are not shown here but they are $\mathcal{O}\left(ϵ\right)$C^r-close to the structures shown, as discussed by²⁹.