itself is neutral compared with the dynamics nearby, which is not allowed for a hyperbolic set.
[1] In this and subsequent papers,[2][3] Fenichel proves that NHIMs possess stable and unstable manifolds and more importantly, NHIMs and their stable and unstable manifolds persist under small perturbations.
Thus, in problems involving perturbation theory, invariant manifolds exist with certain hyperbolicity properties, which can in turn be used to obtain qualitative information about a dynamical system.
[4] Let M be a compact smooth manifold, f: M → M a diffeomorphism, and Df: TM → TM the differential of f. An f-invariant submanifold Λ of M is said to be a normally hyperbolic invariant manifold if the restriction to Λ of the tangent bundle of M admits a splitting into a sum of three Df-invariant subbundles, one being the tangent bundle of
With respect to some Riemannian metric on M, the restriction of Df to Es must be a contraction and the restriction of Df to Eu must be an expansion, and must be relatively neutral on