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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.

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.

由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（线性化）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Solitons'''、'''混沌 Chaos'''和'''奇异性 Singularities'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。

由于非线性动力学方程难以求解，通常用线性化方程来近似非线性系统（'''线性化 Linearization'''）。这种方法在一定的精度和范围对输入值效果很好，但一些有趣的现象如'''孤子 Solitons'''、'''混沌 Chaos'''和'''奇异性 Singularities'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测，甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的影响。这种非线性是目前技术无法进行精确长期预测的原因之一。

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>.

的方程，若 <math> f (x) </math> 是线性映射(如上定义) ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。

的方程，若 <math> f (x) </math> 是线性映射（如上定义） ，则称其为'''线性的 Linear'''，否则称为'''非线性的 Nonlinear'''。若<math>C = 0</math>，该方程称为是齐次的。

The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.

The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any mapping, including integration or differentiation with associated constraints (such as boundary values). If <math>f(x)</math> contains differentiation with respect to <math>x</math>, the result will be a differential equation.

定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束(如给定边界值)的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。

定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意可感知的数学对象（数字、向量、函数等），函数 <math>f(x)</math> 实际上可以是任意映射，包括有相关约束（如给定边界值）的积分或微分。若 <math>f(x)</math> 包含对 <math>x</math> 的微分运算，则该方程为微分方程。

Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,

Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials (of degree greater than one) to zero. For example,

<math>x^2 + x - 1 = 0\,.</math>

<math>x^2 + x - 1 = 0\,.</math>

[[File:PendulumLayout.svg.png|thumb|Illustration of a pendulum 摆图解|right|200px]]

[[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]]

[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum 摆的线性化|right|200px]]

[[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]]

 

[[Category:Nonlinear systems| ]]

[[Category:Nonlinear systems| ]]