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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 Linear dynamical system ： ウィキペディア英語版 Linear dynamical system Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems in general do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point. ==Introduction== In a linear dynamical system, the variation of a state vector (an $N$-dimensional vector denoted $\backslash mathbf$) equals a constant matrix (denoted $\backslash mathbf$) multiplied by $\backslash mathbf$. This variation can take two forms: either as a flow, in which $\backslash mathbf$ varies continuously with time :$$ \frac \mathbf(t) = \mathbf \cdot \mathbf(t) or as a mapping, in which $\backslash mathbf$ varies in discrete steps :$$ \mathbf_ = \mathbf \cdot \mathbf_ These equations are linear in the following sense: if $\backslash mathbf(t)$ and $\backslash mathbf(t)$ are two valid solutions, then so is any linear combination of the two solutions, e.g., $\backslash mathbf(t)\; \backslash \; \backslash stackrel\backslash \; \backslash alpha\; \backslash mathbf(t)\; +\; \backslash beta\; \backslash mathbf(t)$ where $\backslash alpha$ and $\backslash beta$ are any two scalars. The matrix $\backslash mathbf$ need not be symmetric. Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems. 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「Linear dynamical system」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.