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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 Kaplan–Yorke conjecture ： ウィキペディア英語版 Kaplan–Yorke conjecture In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents.〔J. Kaplan and J. Yorke, "Chaotic behavior of multidimensional difference equations," in: Functional Differential Equations and the Approximation of Fixed Points, Lecture Notes in Mathematics, vol. 730, H.O. Peitgen and H.O. Walther, eds. (Springer, Berlin), p. 228.〕〔P. Frederickson, J. Kaplan, E. Yorke and J. Yorke, "The Lyapunov Dimension of Strange Attractors," J. Diff. Eqs. 49 (1983) 185.〕 By arranging the Lyapunov exponents in order from largest to smallest $\backslash lambda\_1\backslash geq\backslash lambda\_2\backslash geq\backslash dots\backslash geq\backslash lambda\_n$, let ''j'' be the index for which :$\backslash sum\_^j\; \backslash lambda\_i\; >\; 0$ and : Then the conjecture is that the dimension of the attractor is :$D=j+\backslash frac.$ == Examples == Especially for chaotic systems, the Kaplan–Yorke conjecture is a useful tool in order to determine the fractal dimension of the corresponding attractor.〔A. Wolf, A. Swift, B. Jack, H. L. Swinney and J.A. Vastano "Determining Lyapunov Exponents from a Time Series," Physica 16D, 1985, 16, pp. 285–317.〕 * The Hénon map with parameters ''a'' = 1.4 and ''b'' = 0.3 has the ordered Lyapunov exponents $\backslash lambda\_1=0.603$ and $\backslash lambda\_2=-2.34$. In this case, we find ''j'' = 1 and the dimension formula reduces to :: $D=j+\backslash frac=1+\backslash frac=1.26.$ * The Lorenz system shows chaotic behavior at the parameter values $\backslash sigma=16$, $\backslash rho=45.92$ and $\backslash beta=4.0$. The resulting Lyapunov exponents are . Noting that ''j'' = 2, we find :: $D=2+\backslash frac=2.07.$ 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「Kaplan–Yorke conjecture」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.