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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 center manifold ： ウィキペディア英語版 center manifold In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium points of dynamical systems is to linearize the system. The eigenvectors corresponding to eigenvalues with negative real part form the stable eigenspace, which gives rise to the stable manifold. Similarly, eigenvalues with positive real part yield the unstable manifold. This concludes the story if the equilibrium point is hyperbolic (i.e., all eigenvalues of the linearization have nonzero real part). However, if there are eigenvalues whose real part is zero, then these give rise to the center manifold. If the eigenvalues are precisely zero, rather than just real part being zero, then these more specifically give rise to a slow manifold. The behavior on the center (slow) manifold is generally not determined by the linearization and thus is more difficult to study. Center manifolds play an important role in: bifurcation theory because interesting behavior takes place on the center manifold; and multiscale mathematics because the long time dynamics often are attracted to a relatively simple center manifold. == Definition == Let $\backslash frac\; =\; \backslash textbf(\backslash textbf)$ be a dynamical system with equilibrium point $\backslash textbf^$ *. The linearization of the system near the equilibrium point is :$\backslash frac\; =\; A\backslash textbf,\; \backslash quad\; \backslash text\; A\; =\; \backslash frac\}(\backslash textbf^$ *). The matrix $A$ defines three main subspaces: * the stable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues $\backslash lambda$ with ; * the unstable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues $\backslash lambda$ with $\backslash operatorname\backslash lambda>0$; * the center subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues $\backslash lambda$ with $\backslash operatorname\backslash lambda=0$. Depending upon the application, other subspaces of interest include center-stable, center-unstable, sub-center, slow, and fast subspaces. These subspaces are all invariant subspaces of the linearized equation. Corresponding to the linearized system, the nonlinear system has invariant manifolds, each consisting of sets of orbits of the nonlinear system.〔, Section 3.2〕 * An invariant manifold tangent to the stable subspace and with the same dimension is the stable manifold. * The unstable manifold is of the same dimension and tangent to the unstable subspace. * A center manifold is of the same dimension and tangent to the center subspace. If, as is common, the eigenvalues of the center subspace are all precisely zero, rather than just real part zero, then a center manifold is often called a slow manifold. 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「center 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.