The paper by Neil Fenichel focuses on the persistence and smoothness of invariant manifolds under perturbations of a flow. The main results provide sufficient conditions for the persistence of diffeomorphic invariant manifolds and a detailed study of their smoothness. The conditions are geometrically interpretable, such as requiring neighborhoods of points on the invariant manifold to be flattened in the direction of the manifold as they are carried forward by the flow. The theory applies to both compact and overflowing invariant manifolds, with the latter requiring more stringent hypotheses. Generalized Lyapunov type numbers are introduced to measure the asymptotic behavior of the flow, and the Uniformity Lemma ensures that hypotheses about this behavior lead to uniform conclusions. The paper also includes an example to illustrate the concepts and discusses the background, including earlier work by Hadamard, Levinson, Diliberto, McCarthy, and others. The proofs of the main theorems are detailed, showing how to construct and analyze the invariant manifolds under perturbations.The paper by Neil Fenichel focuses on the persistence and smoothness of invariant manifolds under perturbations of a flow. The main results provide sufficient conditions for the persistence of diffeomorphic invariant manifolds and a detailed study of their smoothness. The conditions are geometrically interpretable, such as requiring neighborhoods of points on the invariant manifold to be flattened in the direction of the manifold as they are carried forward by the flow. The theory applies to both compact and overflowing invariant manifolds, with the latter requiring more stringent hypotheses. Generalized Lyapunov type numbers are introduced to measure the asymptotic behavior of the flow, and the Uniformity Lemma ensures that hypotheses about this behavior lead to uniform conclusions. The paper also includes an example to illustrate the concepts and discusses the background, including earlier work by Hadamard, Levinson, Diliberto, McCarthy, and others. The proofs of the main theorems are detailed, showing how to construct and analyze the invariant manifolds under perturbations.