# Stable manifold theorem

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{{Short description|Result in dynamical systems theory}}
In [mathematics](/source/mathematics), especially in the study of [dynamical system](/source/dynamical_system)s and [differential equation](/source/differential_equation)s, the '''stable manifold theorem''' is an important result about the structure of the set of [orbits](/source/Orbit_(dynamics)) approaching a given [hyperbolic fixed point](/source/hyperbolic_fixed_point). It roughly states that the existence of a [local diffeomorphism](/source/local_diffeomorphism) near a fixed point implies the existence of a local stable [center manifold](/source/center_manifold) containing that fixed point. This manifold has dimension equal to the number of [eigenvalues](/source/Eigenvalues_and_eigenvectors) of the [Jacobian matrix](/source/Jacobian_matrix_and_determinant) of the fixed point that are less than 1.<ref>{{cite book |first=Michael |last=Shub |title=Global Stability of Dynamical Systems |publisher=Springer |year=1987 |pages=65–66 |url=https://books.google.com/books?id=d-XgBwAAQBAJ&pg=PA65 }}</ref>

== Stable manifold theorem ==
Let 
:<math>f: U \subset \mathbb{R}^n \to \mathbb{R}^n</math>
be a [smooth map](/source/smooth_map) with hyperbolic fixed point at <math>p</math>. We denote by <math>W^{s}(p)</math> the [stable set](/source/stable_manifold) and by <math>W^{u}(p)</math> the [unstable set](/source/unstable_set) of <math>p</math>.

The theorem<ref>{{cite journal|last = Pesin|first = Ya B|title = Characteristic Lyapunov Exponents and Smooth Ergodic Theory|journal = [Russian Mathematical Surveys](/source/Russian_Mathematical_Surveys)|year = 1977|volume = 32|issue = 4|pages = 55–114|doi = 10.1070/RM1977v032n04ABEH001639|url = http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=1639|access-date = 2007-03-10|bibcode=1977RuMaS..32...55P| s2cid=250877457 |url-access = subscription}}</ref><ref>{{cite journal|last = Ruelle|first = David|title = Ergodic theory of differentiable dynamical systems|journal = Publications Mathématiques de l'IHÉS|year = 1979|volume = 50|pages = 27–58|url = http://www.numdam.org/numdam-bin/item?h=nc&id=PMIHES_1979__50__27_0|access-date = 2007-03-10|doi=10.1007/bf02684768| s2cid=56389695 }}</ref><ref>{{cite book| last = Teschl| given = Gerald|author-link=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[American Mathematical Society](/source/American_Mathematical_Society)| place = [Providence](/source/Providence%2C_Rhode_Island)| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}</ref> states that
* <math>W^{s}(p)</math> is a [smooth manifold](/source/smooth_manifold) and its [tangent space](/source/tangent_space) has the same dimension as the [stable space](/source/stable_space) of the [linearization](/source/linearization) of <math>f</math> at <math>p</math>.
* <math>W^{u}(p)</math> is a smooth manifold and its tangent space has the same dimension as the [unstable space](/source/unstable_space) of the linearization of <math>f</math> at <math>p</math>.

Accordingly <math>W^{s}(p)</math> is a '''[stable manifold](/source/stable_manifold)''' and <math>W^{u}(p)</math> is an '''[unstable manifold](/source/unstable_manifold)'''.

== See also ==
* [Center manifold theorem](/source/Center_manifold_theorem)
* [Lyapunov exponent](/source/Lyapunov_exponent)

== Notes==
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== References ==
*{{cite book |first=Lawrence |last=Perko |title=Differential Equations and Dynamical Systems |location=New York |publisher=Springer |edition=Third |year=2001 |isbn=0-387-95116-4 |pages=105–117 }}
*{{cite book |first=S. S. |last=Sritharan |title=Invariant Manifold Theory for Hydrodynamic Transition |publisher=John Wiley & Sons |year=1990 |isbn=0-582-06781-2 }}

== External links ==
*{{PlanetMath|title=StableManifoldTheorem|urlname=StableManifoldTheorem}}

Category:Dynamical systems
Category:Theorems in dynamical systems

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Adapted from the Wikipedia article [Stable manifold theorem](https://en.wikipedia.org/wiki/Stable_manifold_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Stable_manifold_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.