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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 Hartman–Grobman theorem ： ウィキペディア英語版 Hartman–Grobman theorem In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearization theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearization---our first resort in applications---is unreasonably effective in predicting qualitative patterns of behaviour. The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point, provided that no eigenvalue of the linearization has its real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyze its behaviour around equilibria. == Main theorem == Consider a system evolving in time with state $u(t)\backslash in\backslash mathbb\; R^n$ that satisfies the differential equation $du/dt=f(u)$ for some smooth map $f:\; \backslash mathbb^n\; \backslash to\; \backslash mathbb^n$. Suppose the map has a hyperbolic equilibrium state $u^$ *\in\mathbb R^n: that is, $f(u^$ *)=0 and the Jacobian matrix $A=(f\_i/\backslash partial\; x\_j)$ of $f$ at state $u^$ * has no eigenvalue with real part equal to zero. Then there exists a neighborhood $N$ of the equilibrium $u^$ * and a homeomorphism $h\; :\; N\; \backslash to\; \backslash mathbb^n$, such that $h(u^$ *)=0 and such that in the neighbourhood $N$ the flow of $du/dt=f(u)$ is topologically conjugate by the smooth map $U=h(u)$ to the flow of its linearization $dU/dt=AU$. 〔C. Chicone. Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, 2006.〕 Even for infinitely differentiable maps $f$, the homeomorphism $h$ need not to be smooth, nor even locally Lipschitz. However, it turns out to be Hölder continuous, with an exponent depending on the constant of hyperbolicity of $A$. 〔Genrich Belitskii and Victoria Rayskin, On the Grobman--Hartman theorem in $\backslash alpha$-Holder class for Banach spaces. Report, http://www.ma.utexas.edu/pub/mp_arc/html/c/11/11-134.pdf〕 The Hartman-Grobman theorem has been extended to infinite dimensional Banach spaces, non-autonomous systems $du/dt=f(u,t)$ (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part. 〔B. Aulbach and T. Wanner. Integral manifolds for Caratheodory type differential equations in Banach spaces. In B. Aulbach and F. Colonius, editors, Six Lectures on Dynamical Systems, pages 45--119. World Scientific, Singapore, 1996.〕 〔 B. Aulbach and T. Wanner. Invariant foliations for Caratheodory type differential equations in Banach spaces. In V. Lakshmikantham and A. A. Martynyuk, editors, Advances of Stability Theory at the End of XX Century. Gordon & Breach Publishers, 1999. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.5229&rep=rep1&type=pdf〕 〔B. Aulbach and T. Wanner. The Hartman–Grobman theorem for Caratheodory-type differential equations in Banach spaces. Non-linear Analysis, 40:91--104, 2000. http://dx.doi.org/10.1016/ S0362-546X(00)85006-3〕 〔A. J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. Physica A, 387:12--38, 2008. http://dx.doi.org/10.1016/j.physa.2007.08.023〕 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「Hartman–Grobman 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.