About: Stable manifold theorem

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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. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1.

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rdfs:label	Stable manifold theorem (en) 穩定流形定理 (zh)
rdfs:comment	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. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1. (en) 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，是有關趨近給定的集合之結構。令為光滑函数，存在雙曲不動點。令為的穩定流形，則為不穩定流形。定理提到 * 為光滑流形，且切空间也和在點線性化的穩定空間（stable space）有相同維度。 * 為光滑流形，且切空间也和在點線性化的不穩定空間（unstable space）有相同維度。因此是穩定流形，而是不穩定流形。 (zh)
dct:subject	Dynamical systems Theorems in dynamical systems
Wikipage page ID	5494713 (xsd:integer)
Wikipage revision ID	995598423 (xsd:integer)
Link from a Wikipage to another Wikipage	Dynamical system Mathematics Orbit (dynamics) Eigenvalues and eigenvectors Tangent space Linearization Local diffeomorphism Center manifold Differential equation Jacobian matrix and determinant Dynamical systems Theorems in dynamical systems Lyapunov exponent Smooth manifold Stable manifold Unstable manifold Unstable set Unstable space Center manifold theorem Hyperbolic fixed point Smooth map Stable space
sameAs	Stable manifold theorem Stable manifold theorem Stable manifold theorem Stable manifold theorem Stable manifold theorem
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:PlanetMath
title	StableManifoldTheorem (en)
urlname	StableManifoldTheorem (en)
has abstract	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. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1. (en) 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，是有關趨近給定的集合之結構。令為光滑函数，存在雙曲不動點。令為的穩定流形，則為不穩定流形。定理提到 * 為光滑流形，且切空间也和在點線性化的穩定空間（stable space）有相同維度。 * 為光滑流形，且切空间也和在點線性化的不穩定空間（unstable space）有相同維度。因此是穩定流形，而是不穩定流形。 (zh)
gold:hypernym	Result
prov:wasDerivedFrom	wikipedia-en:Stable_manifold_theorem?oldid=995598423&ns=0
page length (characters) of wiki page	3395 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Stable_manifold_theorem
is Link from a Wikipage to another Wikipage of	Nancy Kopell Community matrix Hartman–Grobman theorem Paradox of enrichment Ramsey–Cass–Koopmans model Stable manifold Stable and unstable manifold theorem
is Wikipage redirect of	Stable and unstable manifold theorem
is foaf:primaryTopic of	wikipedia-en:Stable_manifold_theorem

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