更改

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which 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.

Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which 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.

通常，一个非线性的行为在数学上是由一个非线性的方程组来描述的，这是一组方程组，其中未知数(或者在微分方程的情况下未知函数)作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。

通常，非线性系统的行为在数学上是由一组非线性联立方程来描述的，其中未知数(或微分方程中的未知函数)作为一个高于一次的多项式的变量出现，或者作为一个不是一次多项式的函数的论元出现。

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 [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear 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.

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 [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear 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.

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. Systems can be defined as nonlinear, regardless of whether known linear 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.

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. Systems can be defined as nonlinear, regardless of whether known linear 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.

As nonlinear dynamical 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 solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact 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.

As nonlinear dynamical 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 solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact 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.

Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:

Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:

有些作者用非线性科学这个术语来研究非线性系统。这一术语引起了其他人的争议:

有些作者用非线性科学这一术语来研究非线性系统。这一术语引起了其他人的争议:

{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}

{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}

In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:

In mathematics, a linear map (or linear function) <math>f(x)</math> is one which satisfies both of the following properties:

在数学中，线性映射(或线性函数)的数学 f (x) / math 满足以下两个性质:

在数学中，线性映射(或线性函数)的数学<math>f (x)</math>满足以下两个性质:

*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math>

*Additivity or [[superposition principle]]: <math>\textstyle f(x + y) = f(x) + f(y);</math>

*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math>

*Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math>

Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|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

Additivity implies homogeneity for any [[rational number|rational]] ''α'', and, for [[continuous function]]s, for any [[real number|real]] ''α''. For a [[complex number|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

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

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:

 

:<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math>

α是有理数，或α是实数且<math>f(x)</math>是连续函数时，由可加性可以推出齐次性。但当α是复数时，可加性不能导出齐次性。例如，反线性映射是可加的，但不是齐次的。可加性和齐次性条件经常组合，称为叠加原理：

<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math>

<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math>

An equation written as

An equation written as

:<math>f(x) = C</math>

<math>f(x) = C</math>

is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.

is called linear if <math>f(x)</math> is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if <math>C = 0</math>.

如果 math f (x) / math 是线性映射(如上所定义的) ，则称为线性映射，否则称为非线性映射。如果用数学的话，这个方程叫齐次方程。

的一个等式称为是线性的，如果 <math> f (x) </math> 是线性映射(如上所定义的) ，则称为线性映射，否则称为非线性映射。如果用数学的话，这个方程叫齐次方程。