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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 Linear flow on the torus ： ウィキペディア英語版 Linear flow on the torus In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the ''n''-dimensional torus :$\backslash mathbb^n\; =\; \backslash underbrace\_n$ which is represented by the following differential equations with respect to the standard angular coordinates (θ1, θ2, ..., θ''n''): :$\backslash frac=\backslash omega\_1,\; \backslash quad\; \backslash frac=\backslash omega\_2,\backslash quad\; \backslash cdots,\; \backslash quad\; \backslash frac=\backslash omega\_n.$ The solution of these equations can explicitly be expressed as :$\backslash Phi\_\backslash omega^t(\backslash theta\_1,\; \backslash theta\_2,\; \backslash dots,\; \backslash theta\_n)=(\backslash theta\_1+\backslash omega\_1\; t,\; \backslash theta\_2+\backslash omega\_2\; t,\; \backslash dots,\; \backslash theta\_n+\backslash omega\_n\; t)\; \backslash mod\; 2\backslash pi.$ If we respesent the torus as R''n''/Z''n'' we see that a starting point is moved by the flow in the direction ω=(ω1, ω2, ..., ω''n'') at constant speed and when it reaches the border of the unitary ''n''-cube it jumps to the opposite face of the cube. For a linear flow on the torus either all orbits are periodic or all orbits are dense on a subset of the ''n''-torus which is a ''k''-torus. When the components of ω are rationally independent all the orbits are dense on the whole space. This can be easily seen in the two dimensional case: if the two components of ω are rationally independent then the Poincaré section of the flow on an edge of the unit square is an irrational rotation on a circle and therefore its orbits are dense on the circle, as a consequence the orbits of the flow must be dense on the torus. ==See also== *Completely integrable system *Ergodic theory *Quasiperiodic motion 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「Linear flow on the torus」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.