The paper by Neil Fenichel explores the persistence and smoothness of invariant manifolds under perturbations of flows in Euclidean space. It addresses the challenge that even if an invariant manifold is analytic and asymptotically stable, a $ C^r $-close analytic flow may not have a diffeomorphic invariant manifold. The study focuses on sufficient conditions for the persistence of such manifolds and the smoothness of perturbed manifolds. Fenichel introduces generalized Lyapunov type numbers to measure the asymptotic behavior of flows on invariant manifolds. These numbers are defined and studied to derive uniform conditions for the persistence of invariant manifolds. The paper also discusses the smoothness of perturbed manifolds, particularly for overflowing invariant manifolds, which are defined as manifolds where the flow points strictly outward on the boundary. The paper presents a perturbation theorem for overflowing invariant manifolds, showing that under certain conditions, a $ C^r $-close flow will have a $ C^r $-diffeomorphic invariant manifold. It also examines the smoothness of stable and unstable manifolds and their behavior under perturbations. The study includes an example of a vector field with a stationary point and a closed orbit, demonstrating how the perturbation of the flow affects the smoothness of the invariant manifold. The paper also discusses the use of generalized Lyapunov type numbers to describe the perturbation theorem, which are asymptotic rates and do not depend on the choice of a metric. These numbers are used to analyze the behavior of flows on invariant manifolds and to derive conditions for the persistence of these manifolds under perturbations. The study concludes with a detailed analysis of the smoothness of invariant manifolds under perturbations, showing that the conditions for persistence and smoothness can be formulated in terms of generalized Lyapunov type numbers. The paper also provides a construction of invariant manifolds and their smoothness properties under perturbations, demonstrating the robustness of these manifolds under small changes in the flow.The paper by Neil Fenichel explores the persistence and smoothness of invariant manifolds under perturbations of flows in Euclidean space. It addresses the challenge that even if an invariant manifold is analytic and asymptotically stable, a $ C^r $-close analytic flow may not have a diffeomorphic invariant manifold. The study focuses on sufficient conditions for the persistence of such manifolds and the smoothness of perturbed manifolds. Fenichel introduces generalized Lyapunov type numbers to measure the asymptotic behavior of flows on invariant manifolds. These numbers are defined and studied to derive uniform conditions for the persistence of invariant manifolds. The paper also discusses the smoothness of perturbed manifolds, particularly for overflowing invariant manifolds, which are defined as manifolds where the flow points strictly outward on the boundary. The paper presents a perturbation theorem for overflowing invariant manifolds, showing that under certain conditions, a $ C^r $-close flow will have a $ C^r $-diffeomorphic invariant manifold. It also examines the smoothness of stable and unstable manifolds and their behavior under perturbations. The study includes an example of a vector field with a stationary point and a closed orbit, demonstrating how the perturbation of the flow affects the smoothness of the invariant manifold. The paper also discusses the use of generalized Lyapunov type numbers to describe the perturbation theorem, which are asymptotic rates and do not depend on the choice of a metric. These numbers are used to analyze the behavior of flows on invariant manifolds and to derive conditions for the persistence of these manifolds under perturbations. The study concludes with a detailed analysis of the smoothness of invariant manifolds under perturbations, showing that the conditions for persistence and smoothness can be formulated in terms of generalized Lyapunov type numbers. The paper also provides a construction of invariant manifolds and their smoothness properties under perturbations, demonstrating the robustness of these manifolds under small changes in the flow.