A General Approach to Nonautonomous Shadowing for Nonlinear Dynamics

The paper proves Theorem 3.1 via a Banach fixed point on a full z-variable that includes the center component, yielding a contraction constant q = c(2D+1) + 2cD/λ and the standard bounds; the construction x = y + z̄ then delivers P^3(x−y) = 0, sup|x−y| ≤ Cδ, residual in Im P^3, and the estimate (3.3) as stated. The model’s solution instead solves only the Es⊕Eu components using a Green operator G and leaves the center as an explicit residual r(t) ∈ Im P^3; with 2cD/λ < 1 (implied by q < 1) this also yields the same conclusions and bounds. There is a small slip in the model: it incorrectly asserts (I−P^3)Φ = Φ; the correct statement is that G solves w′ = A w + (I−P^3)Φ, and then the leftover center component equals r(t) as desired. With this minor correction, the model’s proof is sound and achieves the same result as the paper by a slightly different route. See Theorem 3.1 and its proof steps (the fixed-point operator T, bounds (3.4)–(3.12), the construction of z̄ and x, and the residual estimate (3.3)) in the paper for direct comparison .