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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.
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。
“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”
“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”
——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''
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）<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>
*可加性（叠加性）: <math>\textstyle f(x + y) = f(x) + f(y);</math>
*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <math>\textstyle f(x + y) = f(x) + f(y);</math>
*齐次性: <math>\textstyle f(\alpha x) = \alpha f(x).</math>
*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</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>.
的方程，若 <math> f (x) </math> 是线性映射(如上定义) ，则称其为线性的，否则称为非线性的。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。
For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.
For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.
对于一个单一的多项式方程，'''求根算法 Root-finding algorithms'''可用于其求解（即找到满足该方程的变量的值集）。而代数方程组则相对复杂，其研究是现代数学的较难分支——'''代数几何 Algebraic geometry'''领域的动力之一。甚至很难判断一个给定的代数系统是否有复数解（见'''希尔伯特零点定律 Hilbert's Nullstellensatz'''）。不过，对于具有有限个复数解的系统的多项式方程组，我们现在已经有了充分的理解，并且找到了有效的求解方法。
A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.
A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences. Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and the related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains.
非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种霍夫斯塔特序列 Hofstadter sequences 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。
非线性递归关系中，序列的连续项被定义为其前项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种'''霍夫斯塔特序列 Hofstadter sequences''' 的关系。非线性离散模型代表了一类广泛的非线性递归关系，包括 NARMAX（外部输入非线性自回归移动平均）模型和相关的非线性系统辨识和分析程序。这些方法可用于研究时域、频域和时空域的广泛复杂非线性行为。
A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.
A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology.
若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 纳维-斯托克斯 Navier-Stokes 方程和生物学中的洛特卡－沃尔泰拉 Lotka-Volterra 方程。
若一个微分方程组不是线性系统，则称其为非线性的。涉及非线性微分方程的问题非常多样，对不同问题的解决或分析方法也不相同。非线性微分方程的例子有流体力学中的 '''纳维-斯托克斯方程 Navier-Stokes equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra equations'''。
One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.
One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle. A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions.
非线性问题最大的困难之一是通常不可能将已知的解组合成新的解。例如，在线性问题中，可以根据叠加原理以一族线性独立的解构造通解。一个很好的例子是带有'''狄利克雷边界条件 Dirichlet boundary conditions'''的一维热传导问题，其解（随时间变化）可以写成不同频率的正弦波的线性组合，这使得解非常灵活。而对非线性方程，通常可以找到几个非常特殊的解，但是此时叠加原理不适用，故无法构造新的解。
First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation
First order ordinary differential equations are often exactly solvable by separation of variables, especially for autonomous equations. For example, the nonlinear equation
一阶常微分方程，尤其是自治方程，通常可以用'''分离变量法 Separation of variables'''来精确求解。例如，非线性方程
<math>\frac{d u}{d x} = -u^2</math>
<math>\frac{d u}{d x} = -u^2</math>
and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).
and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u<sup>2</sup> term were replaced with u, the problem would be linear (the exponential decay problem).
方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的(指数衰减问题)。
方程的左边不是 ''u'' 及其导数的线性函数。注意，若将 ''u''<sup>2</sup> 项替换为''u''，该问题将变为线性的（'''指数衰减 Exponential decay'''问题）。
*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s
*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s
*检查是否有任意守恒量（特别是在哈密顿系统中）
*检查是否有任意'''守恒量 Conserved quantities'''（特别是在'''哈密顿系统 Hamiltonian system'''中）
*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities
*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities
*检查是否有类似守恒量的耗散量（见李亚普诺夫函数）
*检查是否有类似守恒量的耗散量（见'''李亚普诺夫函数 Lyapunov function'''）
*Linearization via [[Taylor expansion]]
*Linearization via [[Taylor expansion]]
*基于泰勒展开的线性化
*基于'''泰勒展开 Taylor expansion'''的线性化
*Change of variables into something easier to study
*Change of variables into something easier to study
*[[Bifurcation theory]]
*[[Bifurcation theory]]
*分岔理论
*'''分岔理论 Bifurcation theory'''
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)
*摄动理论（也可应用于代数方程）
*'''摄动理论 Perturbation theory'''（也可应用于代数方程）
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.
Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation.
另一个流体力学和热力学中常见的策略（虽然不是数学上的）是利用'''尺度分析 Scale analysis'''来在某一特定边界条件下简化一般自然方程。例如，在描述圆管内一维层流的暂态时，非线性的纳维－斯托克斯方程可以简化为一个线性的偏微分方程; 尺度分析提供了层流和一维流动的条件，也产生了简化的方程。
A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation
A classic, extensively studied nonlinear problem is the dynamics of a pendulum under the influence of gravity. Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation
一个经典的被广泛研究的非线性问题是重力影响下的摆的动力学。利用'''拉格朗日力学 Lagrangian mechanics'''，可以证明摆的运动可以用无量纲的非线性方程
<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math>
<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math>
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield
where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an integrating factor, which would eventually yield
描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为积分因子，最终得
描述，其中重力指向“下方”，<math>\theta</math> 是摆与其静止位置形成的角度，如右图所示。“解”这个方程的方法之一是用 <math>d\theta/dt</math> 作为'''积分因子 Integrating factor'''，最终得
<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math>
<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math>
which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).
which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless <math>C_0 = 2</math>).
这是一个含'''椭圆积分 Elliptic integral'''的隐式解。这个“解”通常没什么用，因为这个解的大部分性质都隐藏在非初等函数积分中（除非<math>C_0 = 2</math>，否则是非初等的）。
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:
since <math>\sin(\theta) \approx \theta</math> for <math>\theta \approx 0</math>. This is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at <math>\theta = \pi</math>, corresponding to the pendulum being straight up:
因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个简谐振子，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:
因为 <math>\theta \approx 0</math> 时，有 <math>\sin(\theta) \approx \theta</math>。这是一个'''简谐振子 Simple harmonic oscillator''' ，对应于摆在其路径底部附近的摆动。另一种线性化方法是在 <math>\theta = \pi</math>附近线性化，对应于运动到最高点的摆:
<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math>
<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math>
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.
since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves hyperbolic sinusoids, and note that unlike the small angle approximation, this approximation is unstable, meaning that <math>|\theta|</math> will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state.
因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长，尽管有界解是可能的。这相当于平衡一个直立的钟摆的难度，它实际上是一种不稳定的状态。
因为 <math>\theta \approx \pi</math> 时，有 <math>\sin(\theta) \approx \pi - \theta</math>。这个问题的解含双曲正弦曲线；注意到不同于小角度近似，它是不稳定的，这意味着 <math>|\theta|</math> 通常会无限增长（但解也有可能是有界的）。这就解释了摆在最高点达到平衡的困难，此时实际上是一种不稳定的状态。
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:
One more interesting linearization is possible around <math>\theta = \pi/2</math>, around which <math>\sin(\theta) \approx 1</math>:
一个更有趣的线性化可能是在 <math>\theta = \pi/2</math>附近，此时 <math>\sin(\theta) \approx 1</math>:
<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>
<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.
This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find (exact) phase portraits and approximate periods.
这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''相图 Phase portrait'''和估计周期。
==Types of nonlinear dynamic behaviors==
==Types of nonlinear dynamic behaviors 非线性动力学行为的类型==
*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system
*[[Amplitude death]] – any oscillations present in the system cease due to some kind of interaction with other system or feedback by the same system
*'''振幅死亡 Amplitude death'''——系统内的某振荡因系统的自回馈或与其他系统的某种相互作用而停止的现象
*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]
*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]
*'''混沌 Chaos'''——系统内的值无法无限期地预测到遥远的未来；波动是非周期性的
*[[Multistability]] – the presence of two or more stable states
*[[Multistability]] – the presence of two or more stable states
*'''多稳态 Multistability'''——两或多个稳态的存在
*[[Soliton]]s – self-reinforcing solitary waves
*[[Soliton]]s – self-reinforcing solitary waves
*'''孤波 Soliton'''s——自增强的孤立波
*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted.
*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted.
*'''极限环 Limit cycles'''——吸引不稳定不动点的渐近周期轨道
*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.
*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.
*'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡
==Examples of nonlinear equations 非线性方程的例子==
{{Div col|colwidth=25em}}
*[[Algebraic Riccati equation]] 代数黎卡提方程
*[[Ball and beam]] system 球杆系统
*[[Bellman equation]] for optimal policy 最佳策略的贝尔曼方程
*[[Boltzmann equation]] 玻尔兹曼方程
*[[Colebrook equation]] 科尔布鲁克方程
*[[General relativity]] 广义相对论
*[[Ginzburg–Landau theory]] 金兹堡-朗道方程
*[[Ishimori equation]] 石森方程
*[[Kadomtsev–Petviashvili equation]] 卡东穆塞夫-彼得韦亚斯维利方程
*[[Korteweg–de Vries equation]] kdV方程
*[[Landau–Lifshitz–Gilbert equation]] 朗道-利夫希兹-吉尔伯特方程
*[[Liénard equation]] 林纳德方程
*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程
*[[Nonlinear optics]] 非线性光学
*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程
*[[Power-flow study]]
*[[Power-flow study]] 功率流研究
*[[Richards equation]] for unsaturated water flow
*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程
*[[Sine-Gordon equation]]
*[[Sine-Gordon equation]] 正弦-戈尔登方程
*[[Van der Pol oscillator]]
*[[Van der Pol oscillator]] 范德波尔振荡器
*[[Vlasov equation]]
*[[Vlasov equation]] 弗拉索夫方程
==See also==
==See also 参见==
*[[Aleksandr Mikhailovich Lyapunov]]
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫
*[[Dynamical system]]
*[[Dynamical system]] 动态系统
*[[Feedback]]
*[[Feedback]] 反馈
*[[Initial condition]]
*[[Initial condition]] 初始条件
*[[Interaction]]
*[[Interaction]] 相互作用
*[[Linear system]]
*[[Linear system]] 线性系统
*[[Mode coupling]]
*[[Mode coupling]] 模式耦合
*[[Vector soliton]]
*[[Vector soliton]] 矢量孤子
*[[Volterra series]]
*[[Volterra series]] 沃尔泰拉级数
==References==
==References 参考资料==
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{{Reflist|35em}}
==Further reading==
==Further reading 进一步阅读==
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==External links==
==External links 外部链接==
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]