[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/autonomous-convergence-theorem-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki7\/autonomous-convergence-theorem-wikipedia\/","headline":"Autonomous convergence theorem – Wikipedia","name":"Autonomous convergence theorem – Wikipedia","description":"From Wikipedia, the free encyclopedia In mathematics, an autonomous convergence theorem is one of a family of related theorems which","datePublished":"2020-05-28","dateModified":"2020-05-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki7\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki7\/autonomous-convergence-theorem-wikipedia\/","wordCount":2204,"articleBody":"From Wikipedia, the free encyclopediaIn mathematics, an autonomous convergence theorem is one of a family of related  theorems which specify conditions guaranteeing global asymptotic stability of a  continuous  autonomous dynamical system.History[edit]The Markus\u2013Yamabe conjecture was formulated as an attempt to give conditions for global stability of continuous dynamical systems in two dimensions. However, the Markus\u2013Yamabe conjecture does not hold for dimensions higher than two, a problem which autonomous convergence theorems attempt to address. The first autonomous convergence theorem was constructed by Russell Smith.[1] This theorem was later refined by Michael Li and James Muldowney.[2]An example autonomous convergence theorem[edit]A comparatively simple autonomous convergence theorem is as follows:Let x{displaystyle x} be a  vector in some space X\u2286Rn{displaystyle Xsubseteq mathbb {R} ^{n}}, evolving according to an  autonomous differential equation x\u02d9=f(x){displaystyle {dot {x}}=f(x)}. Suppose that X{displaystyle X} is  convex and forward  invariant under f{displaystyle f}, and that there exists a  fixed point x^\u2208X{displaystyle {hat {x}}in X} such that f(x^)=0{displaystyle f({hat {x}})=0}. If there exists a logarithmic norm \u03bc{displaystyle mu } such that the Jacobian J(x)=Dxf{displaystyle J(x)=D_{x}f} satisfies \u03bc(J(x))"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki7\/autonomous-convergence-theorem-wikipedia\/#breadcrumbitem","name":"Autonomous convergence theorem – Wikipedia"}}]}]