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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 Bendixson–Dulac theorem ： ウィキペディア英語版 Bendixson–Dulac theorem In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a $C^1$ function $\backslash varphi(x,\; y)$ (called the Dulac function) such that the expression :$\backslash frac\; +\; \backslash frac$ has the same sign ($\backslash neq\; 0$) almost everywhere in a simply connected region of the plane, then the plane autonomous system : $\backslash frac\; =\; f(x,y),$ : $\backslash frac\; =\; g(x,y)$ has no periodic solutions lying entirely within the region. "Almost everywhere" means everywhere except possibly in a set of measure 0, such as a point or line. The theorem was first established by Swedish mathematician Ivar Bendixson in 1901 and further refined by French mathematician Henri Dulac in 1933 using Green's theorem. ==Proof== Without loss of generality, let there exist a function $\backslash varphi(x,\; y)$ such that :$\backslash frac\; +\backslash frac\; >0$ in simply connected region $R$. Let $C$ be a closed trajectory of the plane autonomous system in $R$. Let $D$ be the interior of $C$. Then by Green's Theorem, :$\backslash iint\; \_^\; +\backslash frac\; \backslash right)\; dxdy\; \}\; =\backslash oint\; \_^$ :$=\backslash oint\; \_^\; dy\; \backslash right)\; \}.$ But on $C$, $dx=\backslash dot\; dt$ and $dy=\backslash dot\; dt$, so the integral evaluates to 0. This is a contradiction, so there can be no such closed trajectory $C$. 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「Bendixson–Dulac theorem」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.