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渐进稳定 Connected to: {{::readMoreArticle.title}} 此條目没有列出任何参考或来源。 (2012年3月28日)維基百科所有的內容都應該可供查證。请协助補充可靠来源以改善这篇条目。无法查证的內容可能會因為異議提出而被移除。如果微分方程的解既是稳定的又是吸引的,则称该解是渐近稳定的。 稳定和吸引[编辑]设微分方程 d x d t = f ( t , x ) , x ( t 0 ) = x 0 {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t))=f(t,x),x(t_{0})=x^{0)) 满足解的存在唯一性定理的条件,其解 x ( t ) = x ( t , t 0 , x 0 ) {\displaystyle x(t)=x(t,t_{0},x^{0})} 的存在区间是 ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} 。 f ( t , x ) {\displaystyle f(t,x)} 还满足 f ( t , 0 ) = 0 {\displaystyle f(t,0)=0} ,保证 x ( t ) = 0 {\displaystyle x(t)=0} 是方程的解。 若 ∀ ϵ > 0 , ∃ δ = δ ( ϵ , t 0 ) , ∀ ‖ x 0 ‖ < δ , ‖ x ( t , t 0 , x 0 ) ‖ < ϵ {\displaystyle \forall \epsilon >0,\exists \delta =\delta (\epsilon ,t_{0}),\forall \lVert x^{0}\rVert <\delta ,\lVert x(t,t_{0},x^{0})\rVert <\epsilon } 则称零解是稳定的。 若 ∃ δ , ∀ x 0 ∈ S ( 0 , δ ) {\displaystyle \exists \delta ,\forall x^{0}\in S(0,\delta )} 和 ∀ ϵ > 0 , ∃ T = T ( ϵ , t 0 , x 0 ) {\displaystyle \forall \epsilon >0,\exists T=T(\epsilon ,t_{0},x^{0})} 并且当 t > t 0 + T {\displaystyle t>t_{0}+T} 时, ‖ x ( t , t 0 , x 0 ) ‖ < ϵ {\displaystyle \lVert x(t,t_{0},x^{0})\rVert <\epsilon } 则称零解是吸引的。 另见[编辑]李雅普诺夫稳定性 分类 分类:微分方程 {{bottomLinkPreText}} {{bottomLinkText}} This page is based on a Wikipedia article written by contributors (read/edit). Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses. Cover photo is available under {{::mainImage.info.license.name || 'Unknown'}} license. Cover photo is available under {{::mainImage.info.license.name || 'Unknown'}} license. Credit: (see original file).