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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 Inertial manifold ： ウィキペディア英語版 Inertial manifold In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems. Inertial manifolds are finite-dimensional, smooth, invariant manifolds that contain the global attractor and attract all solutions exponentially quickly. Since an inertial manifold is finite-dimensional even if the original system is infinite-dimensional, and because most of the dynamics for the system takes place on the inertial manifold, studying the dynamics on an inertial manifold produces a considerable simplification in the study of the dynamics of the original system.〔R. Temam. Inertial manifolds. ''Mathematical Intelligencer'', 12:68–74, 1990〕 In many physical applications, inertial manifolds express an interaction law between the small and large wavelength structures. Some say that the small wavelengths are enslaved by the large (e.g. synergetics). Inertial manifolds may also appear as slow manifolds common in meteorology, or as the center manifold in any bifurcation. Computationally, numerical schemes for partial differential equations seek to capture the long term dynamics and so such numerical schemes form an approximate inertial manifold. ==Introductory Example== Consider the dynamical system in just two variables $p(t)$ and $q(t)$ and with parameter $a$:〔A. J. Roberts. Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations. ''J. Austral. Math. Soc. B'', 27:48--65, 1985.〕 :$\backslash frac=ap-pq\backslash ,,\backslash qquad\; \backslash frac=-q+p^2-2q^2.$ * It possesses the one dimensional inertial manifold $\backslash mathcal\; M$ of $q=p^2/(1+2a)$ (a parabola). * This manifold is invariant under the dynamics because on the manifold $\backslash mathcal\; M$ : $\backslash frac$ =\frac\frac =\frac} =\frac-\frac which is the same as : $-q+p^2-2q^2$ =-\frac+p^2-2\left(\frac\right)^2 =\frac-\frac. * The manifold $\backslash mathcal\; M$ attracts all trajectories in some finite domain around the origin because near the origin $\backslash frac\backslash approx\; -q$ (although the strict definition below requires attraction from all initial conditions). Hence the long term behavior of the original two dimensional dynamical system is given by the 'simpler' one dimensional dynamics on the inertial manifold $\backslash mathcal\; M$, namely $\backslash frac=ap-\backslash frac1p^3$. 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「Inertial 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.