Identification and Computation of Slow Manifolds Using the Isostable Coordinate System

The paper defines the slow manifold W^s = {x | ψ_k(x)=0 for k>β} via principal Koopman eigenfunctions and notes invariance under forward flow because ψ̇_k = λ_k ψ_k implies ψ̇_k = 0 for k>β on W^s . It derives the backward-time dynamics on W^s by stacking the gradients I_k = ∂ψ_k/∂x as rows of a matrix and inverting to obtain dx/dt̃ = [I_1^T; …; I_N^T]^{-1} [−λ_1 ψ_1, …, −λ_β ψ_β, 0, …, 0]^T (its Eq. (19)) , and observes that this velocity is orthogonal to the span of the “fast” gradients {I_{β+1}, …, I_N} . The paper also gives the gradient dynamics İ_k = −(J^T − λ_k Id) I_k along forward time (its Eq. (6)), which in backward time becomes dI_k/dt̃ = (J^T − λ_k Id) I_k (its Eq. (20)) , and it introduces dual vectors g_k via I_j^T g_k = δ_jk, obtaining the forward-time decomposition dx/dt = Σ_j λ_j ψ_j g_j (its Eq. (32)) . The candidate solution reproduces exactly these steps with the same dual-frame construction and matrix notation, including the invariance (a), the backward-time formula (b), the orthogonality (c), the backward-time gradient dynamics (d), and the forward-time decomposition (e). Both also assume invertibility of the stacked-gradient matrix on the domain of interest . Minor presentational differences exist (the model derives (d) via differentiating ∇ψ·F = λψ, while the paper uses a variational argument), but the proofs are substantively the same.