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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 Slow manifold ： ウィキペディア英語版 Slow manifold In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of the slow manifold—center manifold theory rigorously justifies the modelling.〔J. Carr, ''Applications of centre manifold theory'', Applied Math. Sci. 35, 1981, Springer-Verlag〕〔Y. A. Kuznetsov, ''Elements of applied bifurcation theory'', Applied Mathematical Sciences 112, 1995, Springer-Verlag〕 For example, some global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics,〔R. Camassa, On the geometry of an atmospheric slow manifold, ''Physica D'', 84:357–397, 1995.〕 and is thus crucial to forecasting with a climate model. ==Definition== Consider the dynamical system :$\backslash frac\; =\; \backslash vec\; f(\backslash vec)$ for an evolving state vector $\backslash vec\; x(t)$ and with equilibrium point $\backslash vec\; x^$ *. Then the linearization of the system at the equilibrium point is :$\backslash frac\; =\; A\backslash vec\; \backslash quad\; \backslash text\; A\; =\; \backslash frac(\backslash vec\; x^$ *). The matrix $A$ defines four invariant subspaces characterized by the eigenvalues $\backslash lambda$ of the matrix: as described in the entry for the center manifold three of the subspaces are the stable, unstable and center subspaces corresponding to the span of the eigenvectors with eigenvalues $\backslash lambda$ that have real part negative, positive, and zero, respectively; the fourth subspace is the slow subspace given by the span of the eigenvectors, and generalized eigenvectors, corresponding to the eigenvalue $\backslash lambda=0$ precisely. The slow subspace is a subspace of the center subspace, or identical to it, or possibly empty. Correspondingly, the nonlinear system has invariant manifolds, made of trajectories of the nonlinear system, corresponding to each of these invariant subspaces. There is an invariant manifold tangent to the slow subspace and with the same dimension; this manifold is the slow manifold. Stochastic slow manifolds also exist for noisy dynamical systems (stochastic differential equation), as do also stochastic center, stable and unstable manifolds.〔Ludwig Arnold, ''Random Dynamical Systems'', Springer Monographs in Mathematics, 2003.〕 Such stochastic slow manifolds are similarly useful in modeling emergent stochastic dynamics, but there are many fascinating issues to resolve such as history and future dependent integrals of the noise.〔A. J. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, ''Physica A'' 387:12–38, 2008.〕〔Ludwig Arnold and Peter Imkeller, Normal forms for stochastic differential equations, ''Probab. Theory Relat. Fields'', 110:559–588, 1998.〕 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「Slow 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.