查看“︁穩定流形定理”︁的源代码

'''穩定流形定理'''（stable manifold theorem）是[[數學]]定理，[[动力系统]]及[[微分方程]]有關，是有關趨近給定{{link-en|雙曲不動點|hyperbolic fixed point}}的{{link-en|軌道 (動力系統|Orbit (dynamics)|軌道}}集合之結構。

令
:<math>f: U \subset \mathbb{R}^n \to \mathbb{R}^n</math>
為[[光滑函数]]，存在雙曲不動點<math>p</math>。令<math>W^{s}(p)</math>為<math>p</math>的[[穩定流形]]，<math>W^{u}(p)</math>則為不穩定流形。

定理<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|archive-date = 2007-09-27|archive-url = https://web.archive.org/web/20070927104152/http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=1639}}</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|doi = 10.1007/bf02684768|archive-date = 2016-03-03|archive-url = https://web.archive.org/web/20160303173924/http://www.numdam.org/numdam-bin/item?h=nc&id=PMIHES_1979__50__27_0}}</ref><ref>{{cite book| last = Teschl| given = Gerald| title = Ordinary Differential Equations and Dynamical Systems| publisher = [[美國數學學會|American Mathematical Society]]| place = [[普罗维登斯|Providence]]| year = 2012| isbn = 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/| access-date = 2020-02-19| archive-date = 2012-06-26| archive-url = https://web.archive.org/web/20120626043727/http://www.mat.univie.ac.at/~gerald/ftp/book-ode/| dead-url = yes}}</ref>提到
* <math>W^{s}(p)</math>為[[微分流形|光滑流形]]，且[[切空间]]也和<math>f</math>在<math>p</math>點[[線性化]]的穩定空間（stable space）有相同維度。
* <math>W^{u}(p)</math>為光滑流形，且切空间也和<math>f</math>在<math>p</math>點線性化的不穩定空間（unstable space）有相同維度。

因此<math>W^{s}(p)</math>是[[穩定流形]]，而<math>W^{u}(p)</math>是不穩定流形。

== 相關條目 ==
* [[中心流形定理]]
* [[李亚普诺夫指数]]

== 註解==
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== 參考資料 ==
*{{cite book |first=Lawrence |last=Perko |title=Differential Equations and Dynamical Systems |url=https://archive.org/details/differentialequa00perk_020 |location=New York |publisher=Springer |edition=Third |year=2001 |isbn=0-387-95116-4 |pages=[https://archive.org/details/differentialequa00perk_020/page/n119 105]–117 }}
*{{cite book |first=S. S. |last=Sritharan |title=Invariant Manifold Theory for Hydrodynamic Transition |url=https://archive.org/details/invariantmanifol0000srit |location= |publisher=John Wiley & Sons |year=1990 |isbn=0-582-06781-2 }}

== 外部連結 ==
*{{PlanetMath|title=StableManifoldTheorem|urlname=StableManifoldTheorem}}

[[Category:动力系统]]
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