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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 Pullback attractor ： ウィキペディア英語版 Pullback attractor In mathematics, the attractor of a random dynamical system may be loosely thought of as a set to which the system evolves after a long enough time. The basic idea is the same as for a deterministic dynamical system, but requires careful treatment because random dynamical systems are necessarily non-autonomous. This requires one to consider the notion of a pullback attractor or attractor in the pullback sense. ==Set-up and motivation== Consider a random dynamical system $\backslash varphi$ on a complete separable metric space $(X,\; d)$, where the noise is chosen from a probability space $(\backslash Omega,\; \backslash mathcal,\; \backslash mathbb)$ with base flow $\backslash vartheta\; :\; \backslash mathbb\; \backslash times\; \backslash Omega\; \backslash to\; \backslash Omega$. A naïve definition of an attractor $\backslash mathcal$ for this random dynamical system would be to require that for any initial condition $x\_\; \backslash in\; X$, $\backslash varphi(t,\; \backslash omega)\; x\_\; \backslash to\; \backslash mathcal$ as $t\; \backslash to\; +\; \backslash infty$. This definition is far too limited, especially in dimensions higher than one. A more plausible definition, modelled on the idea of an omega-limit set, would be to say that a point $a\; \backslash in\; X$ lies in the attractor $\backslash mathcal$ if and only if there exists an initial condition $x\_\; \backslash in\; X$, there is a sequence of times $t\_\; \backslash to\; +\; \backslash infty$ such that :$d\; \backslash left(\; \backslash varphi(t\_,\; \backslash omega)\; x\_,\; a\; \backslash right)\; \backslash to\; 0$ as $n\; \backslash to\; \backslash infty$. This is not too far from a working definition. However, we have not yet considered the effect of the noise $\backslash omega$, which makes the system non-autonomous (i.e. it depends explicitly on time). For technical reasons, it becomes necessary to do the following: instead of looking $t$ seconds into the "future", and considering the limit as $t\; \backslash to\; +\; \backslash infty$, one "rewinds" the noise $t$ seconds into the "past", and evolves the system through $t$ seconds using the same initial condition. That is, one is interested in the pullback limit :$\backslash lim\_\; \backslash varphi\; (t,\; \backslash vartheta\_\; \backslash omega)$. So, for example, in the pullback sense, the omega-limit set for a (possibly random) set $B(\backslash omega)\; \backslash subseteq\; X$ is the random set :$\backslash Omega\_\; (\backslash omega)\; :=\; \backslash left\backslash \; \backslash in\; B(\backslash vartheta\_\; \backslash varphi\; (t\_,\; \backslash vartheta\_\; \backslash to\; x\; \backslash mathrm\; n\; \backslash to\; \backslash infty\; \backslash right.\; \backslash right\backslash \}.$ Equivalently, this may be written as :$\backslash Omega\_\; (\backslash omega)\; =\; \backslash bigcap\_\; \backslash overline\; \backslash omega)\; B(\backslash vartheta\_\; \backslash omega)\}.$ Importantly, in the case of a deterministic dynamical system (one without noise), the pullback limit coincides with the deterministic forward limit, so it is meaningful to compare deterministic and random omega-limit sets, attractors, and so forth. 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「Pullback attractor」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.