A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a hyperbolic set. The difference can be described heuristically as follows: For a manifold to be normally hyperbolic we are allowed to assume that the dynamics of itself is neutral compared with the dynamics nearby, which is not allowed for a hyperbolic set. NHIMs were introduced by Neil Fenichel in 1972.[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]
Definitionedit
Let M be a compactsmooth manifold, f: M → M a diffeomorphism, and Df: TM → TM the differential of f. An f-invariant submanifoldΛ 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 , the others being the stable bundle and the unstable bundle and denoted Es and Eu, respectively. 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 . Thus, there exist constants and c > 0 such that
^Fenichel, N (1972). "Persistence and Smoothness of Invariant Manifolds for Flows". Indiana Univ. Math. J. 21 (3): 193–226. doi:10.1512/iumj.1971.21.21017.
^Fenichel, N (1974). "Asymptotic Stability With Rate Conditions". Indiana Univ. Math. J. 23 (12): 1109–1137. doi:10.1512/iumj.1974.23.23090.
^Fenichel, N (1977). "Asymptotic Stability with Rate Conditions II". Indiana Univ. Math. J. 26 (1): 81–93. doi:10.1512/iumj.1977.26.26006.
^A. Katok and B. HasselblattIntroduction to the Modern Theory of Dynamical Systems, Cambridge University Press (1996), ISBN978-0521575577
M.W. Hirsch, C.C Pugh, and M. Shub Invariant Manifolds, Springer-Verlag (1977), ISBN978-3540081487doi:10.1007/BFb0092042
January 01, 1970
normally, hyperbolic, invariant, manifold, normally, hyperbolic, invariant, manifold, nhim, natural, generalization, hyperbolic, fixed, point, hyperbolic, difference, described, heuristically, follows, manifold, displaystyle, lambda, normally, hyperbolic, allo. A normally hyperbolic invariant manifold NHIM is a natural generalization of a hyperbolic fixed point and a hyperbolic set The difference can be described heuristically as follows For a manifold L displaystyle Lambda to be normally hyperbolic we are allowed to assume that the dynamics of L displaystyle Lambda itself is neutral compared with the dynamics nearby which is not allowed for a hyperbolic set NHIMs were introduced by Neil Fenichel in 1972 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 Definition editLet M be a compact smooth manifold f M M a diffeomorphism and Df TM TM the differential of f An f invariant submanifold L of M is said to be a normally hyperbolic invariant manifold if the restriction to L of the tangent bundle of M admits a splitting into a sum of three Df invariant subbundles one being the tangent bundle of L displaystyle Lambda nbsp the others being the stable bundle and the unstable bundle and denoted Es and Eu respectively 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 T L displaystyle T Lambda nbsp Thus there exist constants 0 lt l lt m 1 lt 1 displaystyle 0 lt lambda lt mu 1 lt 1 nbsp and c gt 0 such that T L M T L E s E u displaystyle T Lambda M T Lambda oplus E s oplus E u nbsp D f x E x s E f x s and D f x E x u E f x u for all x L displaystyle Df x E x s E f x s text and Df x E x u E f x u text for all x in Lambda nbsp D f n v c l n v for all v E s and n gt 0 displaystyle Df n v leq c lambda n v text for all v in E s text and n gt 0 nbsp D f n v c l n v for all v E u and n gt 0 displaystyle Df n v leq c lambda n v text for all v in E u text and n gt 0 nbsp and D f n v c m n v for all v T L and n Z displaystyle Df n v leq c mu n v text for all v in T Lambda text and n in mathbb Z nbsp See also editStable manifold Center manifold Hyperbolic fixed point Hyperbolic set Hyperbolic Lagrangian coherent structuresReferences edit Fenichel N 1972 Persistence and Smoothness of Invariant Manifolds for Flows Indiana Univ Math J 21 3 193 226 doi 10 1512 iumj 1971 21 21017 Fenichel N 1974 Asymptotic Stability With Rate Conditions Indiana Univ Math J 23 12 1109 1137 doi 10 1512 iumj 1974 23 23090 Fenichel N 1977 Asymptotic Stability with Rate Conditions II Indiana Univ Math J 26 1 81 93 doi 10 1512 iumj 1977 26 26006 A Katok and B HasselblattIntroduction to the Modern Theory of Dynamical Systems Cambridge University Press 1996 ISBN 978 0521575577 M W Hirsch C C Pugh and M Shub Invariant Manifolds Springer Verlag 1977 ISBN 978 3540081487 doi 10 1007 BFb0092042 Retrieved from https en wikipedia org w index php title Normally hyperbolic invariant manifold amp oldid 1169488674, wikipedia, wiki, book, books, library,