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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク stable manifold ： ウィキペディア英語版 stable manifold In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. ==Definition== The following provides a definition for the case of a system that is either an iterated function or has discrete-time dynamics. Similar notions apply for systems whose time evolution is given by a flow. Let $X$ be a topological space, and $f\backslash colon\; X\backslash to\; X$ a homeomorphism. If $p$ is a fixed point for $f$, the stable set of $p$ is defined by :$W^s(f,p)\; =\backslash$ and the unstable set of $p$ is defined by :$W^u(f,p)\; =\backslash \; n\backslash to\; \backslash infty\; \backslash \}.$ Here, $f^$ denotes the inverse of the function $f$, i.e. $f\backslash circ\; f^=f^\backslash circ\; f\; =id\_$, where $id\_$ is the identity map on $X$. If $p$ is a periodic point of least period $k$, then it is a fixed point of $f^k$, and the stable and unstable sets of $p$ are :$W^s(f,p)\; =\; W^s(f^k,p)$ and :$W^u(f,p)\; =\; W^u(f^k,p).$ Given a neighborhood $U$ of $p$, the local stable and unstable sets of $p$ are defined by :$W^s\_\; n\backslash geq\; 0\backslash \}$ and :$W^u\_\}(f^,p,U).$ If $X$ is metrizable, we can define the stable and unstable sets for any point by :$W^s(f,p)\; =\; \backslash$ and :$W^u(f,p)\; =\; W^s(f^,p),$ where $d$ is a metric for $X$. This definition clearly coincides with the previous one when $p$ is a periodic point. Suppose now that $X$ is a compact smooth manifold, and $f$ is a $\backslash mathcal^k$ diffeomorphism, $k\backslash geq\; 1$. If $p$ is a hyperbolic periodic point, the stable manifold theorem assures that for some neighborhood $U$ of $p$, the local stable and unstable sets are $\backslash mathcal^k$ embedded disks, whose tangent spaces at $p$ are $E^s$ and $E^u$ (the stable and unstable spaces of $Df(p)$), respectively; moreover, they vary continuously (in a certain sense) in a neighborhood of $f$ in the $\backslash mathcal^k$ topology of $\backslash mathrm^k(X)$ (the space of all $\backslash mathcal^k$ diffeomorphisms from $X$ to itself). Finally, the stable and unstable sets are $\backslash mathcal^k$ injectively immersed disks. This is why they are commonly called stable and unstable manifolds. This result is also valid for nonperiodic points, as long as they lie in some hyperbolic set (stable manifold theorem for hyperbolic sets). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「stable manifold」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.