A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a hyperbolic fixed point and a

hyperbolic set In dynamical systems theory, a subset Λ of a smooth manifold ''M'' is said to have a hyperbolic structure with respect to a smooth map ''f'' if its tangent bundle may be split into two invariant subbundles, one of which is contracting and th ...

. The difference can be described heuristically as follows: For a manifold

\Lambda

to be normally hyperbolic we are allowed to assume that the dynamics of

\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. In this and subsequent papers, 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 In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

.A. Katok and B. Hasselblatt''Introduction to the Modern Theory of Dynamical Systems'', Cambridge University Press (1996),

Definition

Let ''M'' be a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

, ''f'': ''M'' → ''M'' a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

, and ''Df'': ''TM'' → ''TM'' the differential of ''f''. An ''f''-invariant

submanifold In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...

''Λ'' of ''M'' is said to be a normally hyperbolic invariant manifold if the restriction to ''Λ'' of the

tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...

of ''M'' admits a splitting into a sum of three ''Df''-invariant subbundles, one being the tangent bundle of

\Lambda

, the others being the

stable bundle In mathematics, a stable vector bundle is a (holomorphic vector bundle, holomorphic or algebraic vector bundle, algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stabl ...

and the unstable bundle and denoted ''E''^''s'' and ''E''^''u'', respectively. With respect to some

Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

on ''M'', the restriction of ''Df'' to ''E''^''s'' must be a contraction and the restriction of ''Df'' to ''E''^''u'' must be an expansion, and must be relatively neutral on

T\Lambda

. Thus, there exist constants

0 < \lambda < \mu^ < 1

and ''c'' > 0 such that :

T_\Lambda M = T\Lambda\oplus E^s\oplus E^u

(Df)_x E^s_x = E^s_\text(Df)_x E^u_x = E^u_ \textx\in \Lambda,

\, Df^nv\,  \le c\lambda^n\, v\, \textv\in E^s\textn> 0,

\, Df^v\,  \le c\lambda^n \, v\, \textv\in E^u\textn>0,

and :

\, Df^n v\,  \le c\mu^ \, v\, \textv\in T\Lambda\textn \in \mathbb.

References

* M.W. Hirsch, C.C Pugh, and M. Shub ''Invariant Manifolds'', Springer-Verlag (1977), {{doi, 10.1007/BFb0092042 Dynamical systems

Definition

See also

References