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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 Nonlinear system ： ウィキペディア英語版 Nonlinear system In physics and other sciences, a nonlinear system, in contrast to a linear system, is a system which does not satisfy the superposition principle – meaning that the output of a nonlinear system is not directly proportional to the input. In mathematics, a nonlinear system of equations is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. It does not matter if nonlinear known functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it. Typically, the behavior of a ''nonlinear system'' is described by a ''nonlinear system of equations''. Nonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature. As nonlinear equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as chaos〔(Nonlinear Dynamics I: Chaos ) at (MIT's OpenCourseWare )〕 and singularities are hidden by linearization. It follows that some aspects of the behavior of a nonlinear system appear commonly to be chaotic, unpredictable or counterintuitive. Although such chaotic behavior may resemble random behavior, it is absolutely not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. ==Definition== In mathematics, a linear function (or map) $f(x)$ is one which satisfies both of the following properties: * Additivity or superposition: $\backslash textstyle\; f(x\; +\; y)\backslash \; =\; f(x)\backslash \; +\; f(y);$ * Homogeneity: $\backslash textstyle\; f(\backslash alpha\; x)\backslash \; =\; \backslash alpha\; f(x).$ Additivity implies homogeneity for any rational ''α'', and, for continuous functions, for any real ''α''. For a complex ''α'', homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in the superposition principle :$f(\backslash alpha\; x\; +\; \backslash beta\; y)\; =\; \backslash alpha\; f(x)\; +\; \backslash beta\; f(y)\; \backslash ,$ An equation written as :$f(x)\; =\; C\backslash ,$ is called linear if $f(x)$ is a linear map (as defined above) and nonlinear otherwise. The equation is called ''homogeneous'' if $C\; =\; 0$. The definition $f(x)\; =\; C$ is very general in that $x$ can be any sensible mathematical object (number, vector, function, etc.), and the function $f(x)$ can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If $f(x)$ contains differentiation with respect to $x$, the result will be a differential equation. 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「Nonlinear 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.