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In [[mathematics]], especially in the study of [[dynamical system]]s and [[differential equation]]s, the '''stable manifold theorem''' is an important result about the structure of the set of [[Orbit (dynamics)|orbits]] approaching a given [[hyperbolic fixed point]]. | |||
== Stable manifold theorem == | |||
Let | |||
:<math>f: U \subset \mathbb{R}^n \to \mathbb{R}^n</math> | |||
be a [[smooth map]] with hyperbolic fixed point at ''p''. We denote by <math>W^{s}(p)</math> the [[stable set]] and by <math>W^{u}(p)</math> the [[unstable set]] of ''p''. | |||
The theorem<ref>{{cite journal|last = Pesin|first = Ya B|title = Characteristic Lyapunov Exponents and Smooth Ergodic Theory|journal = [[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|accessdate = 2007-03-10|bibcode=1977RuMaS..32...55P}}</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|accessdate = 2007-03-10}}</ref><ref>{{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}</ref> states that | |||
* <math>W^{s}(p)</math> is a [[smooth manifold]] and its [[tangent space]] has the same dimension as the [[stable space]] of the [[linearization]] of ''f'' at ''p''. | |||
* <math>W^{u}(p)</math> is a smooth manifold and its tangent space has the same dimension as the [[unstable space]] of the linearization of ''f'' at ''p''. | |||
Accordingly <math>W^{s}(p)</math> is a '''[[stable manifold]]''' and <math>W^{u}(p)</math> is an '''[[unstable manifold]]'''. | |||
== See also == | |||
* [[Center manifold theorem]] | |||
* [[Lyapunov exponent]] | |||
== Notes== | |||
<references/> | |||
== References == | |||
* S. S. Sritharan, "Invariant Manifold Theory for Hydrodynamic Transition", John Wiley & Sons, (1990), ISBN 0-582-06781-2 | |||
== External links == | |||
*{{PlanetMath|title=StableManifoldTheorem|urlname=StableManifoldTheorem}} | |||
[[Category:Dynamical systems]] | |||
[[Category:Theorems in dynamical systems]] | |||
{{mathanalysis-stub}} | |||
Latest revision as of 19:29, 20 October 2013
In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point.
Stable manifold theorem
Let
be a smooth map with hyperbolic fixed point at p. We denote by the stable set and by the unstable set of p.
The theorem[1][2][3] states that
- is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of f at p.
- is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of f at p.
Accordingly is a stable manifold and is an unstable manifold.
See also
Notes
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
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References
- S. S. Sritharan, "Invariant Manifold Theory for Hydrodynamic Transition", John Wiley & Sons, (1990), ISBN 0-582-06781-2