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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.
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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。
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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。__[[用户:Dorr|Dorr]]([[用户讨论:Dorr|讨论]])+后半句不知如何翻译
    
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.
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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.
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换句话说,在非线性方程系统中,要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数,系统都可以被定义为非线性。特别是,如果一个微分方程的未知函数及其导数是线性的,即使其他变量是非线性的,也称其是线性的。
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换句话说,在非线性方程系统中,要求解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数,系统都可以被定义为非线性。特别是,如果一个微分方程的未知函数及其导数是线性的,即使其他变量是非线性的,也称该方程是线性的。
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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.
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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。
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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。
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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>.
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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。
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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。
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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.
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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。
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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。
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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,
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非线性代数方程,又称多项式方程,由某多项式(次数大于1)等于零定义。例如:
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非线性'''代数方程 Algebraic equation''',又称'''多项式方程 Polynomial equation''',由某多项式(次数大于1)等于零定义。例如:
    
<math>x^2 + x - 1 = 0\,.</math>
 
<math>x^2 + x - 1 = 0\,.</math>
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[[File:PendulumLayout.svg.png|thumb|Illustration of a pendulum 摆图解|right|200px]]
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[[File:PendulumLayout.svg.png|thumb|图一:Illustration of a pendulum 摆图解|right|200px]]
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[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum 摆的线性化|right|200px]]
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[[File:PendulumLinearizations.png|thumb|图二:Linearizations of a pendulum 摆的线性化|right|200px]]
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==编者推荐==
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集智学园:[https://campus.swarma.org/course/697 非线性动力学与混沌]
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非线性动力学和混沌理论是系统发展的,从一阶微分方程及其分岔开始,然后是相平面分析,极限环和它们的分岔,最终得到Lorenz方程,混沌,迭代映射,周期倍增,重整化,分形和奇怪吸引。
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此系列课程由[[斯蒂文·斯特罗加茨 Steven H. Strogatz]]主持,内容包括机械振动,激光,生物节律,超导电路,昆虫爆发,化学振荡器,遗传控制系统,混沌水轮,甚至是使用混乱发送秘密信息的技术。在每种情况下,科学背景都在初级阶段进行解释,并与数学理论紧密结合。
    
[[Category:Nonlinear systems| ]]
 
[[Category:Nonlinear systems| ]]
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