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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 Saddle-node bifurcation ： ウィキペディア英語版 Saddle-node bifurcation In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often instead called a fold bifurcation. Another name is blue skies bifurcation in reference to the sudden creation of two fixed points. If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node). Saddle-node bifurcations may be associated with hysteresis loops and catastrophes. ==Normal form== A typical example of a differential equation with a saddle-node bifurcation is: :$\backslash frac=r+x^2.$ Here $x$ is the state variable and $r$ is the bifurcation parameter. *If there are two equilibrium points, a stable equilibrium point at $-\backslash sqrt$ and an unstable one at $+\backslash sqrt$. *At $r=0$ (the bifurcation point) there is exactly one equilibrium point. At this point the fixed point is no longer hyperbolic. In this case the fixed point is called a saddle-node fixed point. *If $r>0$ there are no equilibrium points. In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation $\backslash tfrac\; =\; f(r,x)$ which has a fixed point at $x\; =\; 0$ for $r\; =\; 0$ with $\backslash tfrac(0,0)\; =\; 0$ is locally topological equivalent to $\backslash frac\; =\; r\; \backslash pm\; x^2$, provided it satisfies $\backslash tfrac(0,0)\; \backslash ne\; 0$ and $\backslash tfrac(0,0)\; \backslash ne\; 0$. The first condition is the nondegeneracy condition and the second condition is the transversality condition. 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「Saddle-node bifurcation」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.