A normally hyperbolic invariant manifold (NHIM) is a natural generalization of a
hyperbolic fixed point In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbo ...
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
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
Neil is a masculine name of Gaelic and Irish origin. The name is an anglicisation of the Irish ''Niall'' which is of disputed derivation. The Irish name may be derived from words meaning "cloud", "passionate", "victory", "honour" or "champion".. ...
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.
[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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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 ma ...
, ''f'': ''M'' → ''M'' a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
, 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 satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
''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
, the others being the
stable bundle and the unstable bundle and denoted ''E''
''s'' and ''E''
''u'', respectively. With respect to some
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
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
. Thus, there exist constants
and ''c'' > 0 such that
:
:
:
:
and
:
See also
*
Stable manifold
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repello ...
*
Center manifold In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modellin ...
*
Hyperbolic fixed point In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbo ...
*
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 ...
* Hyperbolic
Lagrangian coherent structures
Lagrangian coherent structures (LCSs) are distinguished surfaces of trajectories in a dynamical system that exert a major influence on nearby trajectories over a time interval of interest. The type of this influence may vary, but it invariably cre ...
References
* M.W. Hirsch, C.C Pugh, and M. Shub ''Invariant Manifolds'', Springer-Verlag (1977), {{doi, 10.1007/BFb0092042
Dynamical systems