Normally hyperbolic invariant manifold information
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]
^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
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