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第19行： − + − − − − {{quote|“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。”|[[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）<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> − − *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 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: − − − − − − − − An equation written as − − − − − − − − 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>. − − − − − − 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 [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|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. − 第105行：
第49行： − + − − − − {{Main|Algebraic equation|System of polynomial equations}} − − − − Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (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, − + + + + − For a single polynomial equation, [[root-finding algorithm]]s 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.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref>+ − − 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。 − − − − − − ==Nonlinear recurrence relations 非线性递推关系== − − − − 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 sequence]]s. 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.<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> 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。 − − − − − ==Nonlinear differential equations 非线性微分方程== − − − − A [[simultaneous equations|system]] of [[differential equation]]s 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 equation]]s 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. 第174行：
第72行： − + − − First order [[ordinary differential equation]]s 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（自主）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+ − − − − − − has <math>u=\frac{1}{x+C}</math> as a general solution (and also ''u'' = 0 as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as − − has <math>u=\frac{1}{x+C}</math> as a general solution (and also u = 0 as a particular solution, corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as − − − − − − − 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). − − − − − − Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield [[closed-form expression|closed-form]] solutions, though implicit solutions and solutions involving [[nonelementary integral]]s are encountered. − − Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered. − − − − − − Common methods for the qualitative analysis of nonlinear ordinary differential equations include: − − Common methods for the qualitative analysis of nonlinear ordinary differential equations include: + + − + − − − − *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 − − *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]] − − *基于'''泰勒展开 Taylor Expansion'''的线性化 − − *Change of variables into something easier to study − + − + − *[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too) − *'''摄动理论 Perturbation Theory'''（也可应用于代数方程） + + − ===Partial differential equations 偏微分方程===+ − − − {{main|Nonlinear partial differential equation}} − − − − {{See also|List of nonlinear partial differential equations}} − − 参见：非线性偏微分列表 − − The most common basic approach to studying nonlinear [[partial differential equation]]s is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more [[ordinary differential equation]]s, as seen in [[separation of variables]], which is always useful whether or not the resulting ordinary differential equation(s) is solvable. − − The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the equation may be transformed into one or more ordinary differential equations, as seen in separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is solvable. − − 研究非线性偏微分方程最常用的基本方法是变量代换（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，对研究问题总是有用的。 − − − − − Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use [[scale analysis (mathematics)|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. − − − − − − − − Other methods include examining the [[method of characteristics|characteristics]] and using the methods outlined above for ordinary differential equations. − − Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations. 第299行：
第117行： − + − − − {{Main|Pendulum (mathematics)}} − − − + + − [[File:PendulumLayout.svg.png|thumb|图一：Illustration of a pendulum 摆图解|right|200px]] − + − [[File:PendulumLinearizations.png|thumb|图二：Linearizations of a pendulum 摆的线性化|right|200px]] − − − − − − − 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+ − − − − − − − − 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 Integrating Factor'''，最终得+ + + + + − 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）。 − − − − − − Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through [[Taylor expansion]]s. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is − − Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the various points of interest through Taylor expansions. For example, the linearization at <math>\theta = 0</math>, called the small angle approximation, is − − 另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些感兴趣的点线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为 − − <math>\frac{d^2 \theta}{d t^2} + \theta = 0</math> − − − − − − 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 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: − − − − − − − − since <math>\sin(\theta) \approx \pi - \theta</math> for <math>\theta \approx \pi</math>. The solution to this problem involves [[hyperbolic sinusoid]]s, 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. − − − − − − 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>: − + − − − − − − 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 portrait]]s 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 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 − − *'''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]] − − *'''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 − − *'''多稳态 Multistability'''——两或多个稳态的存在 − − *[[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. − − *'''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. − *'''自激振荡 Self-oscillations'''——开放耗散物理系统中的反馈振荡+ 第441行：
第159行： − + + + − {{Div col|colwidth=25em}}+ + + − + + + + − *[[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 + − + − + − + − + − + − + − + − + − + 第536行：
第261行： + + − + − ==See also 参见== − − − − − − − *[[Dynamical system]] 动态系统 − − − − *[[Feedback]] 反馈 − − *[[Initial condition]] 初始条件 + − *[[Interaction]] 相互作用 + − *[[Linear system]] 线性系统 + − *[[Mode coupling]] 模式耦合 + − *[[Vector soliton]] 矢量孤子 − + − − + + − − − + − − − − − − −
无编辑摘要
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:
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:
“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。”<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>
——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''
——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''
==Definition 定义==
==Definition 定义==
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:
*'''可加性 Additivity'''（'''叠加性 Superposition principle'''）: <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 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math>
*'''齐次性 Homogeneity''': <math>\textstyle f(\alpha x) = \alpha f(x).</math>
当α是有理数或实数，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：
当α是有理数或实数，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>
对一个写成
对一个写成
<math>f(x) = C</math>
<math>f(x) = C</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。
定义 <math>f(x) = C</math> 是非常具有一般性的，因为 <math>x</math> 可以是任意合理的数学对象（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> 可以是任意合理的数学对象（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。
===相关概念辨析===
===相关概念辨析===
在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。
在线性系统中，整体等于部分和，描述线性系统的方程满足叠加原理，作用的总和正好等于每一部分作用相加的代数和，这意味着每一部分作用都是独立的、互不相关的；而在普遍存在的非线性系统中，作用的总和不等于每一部分作用相加的代数和，因为系统内部要素之间存在着复杂的非线性相互作用。
==非线性代数方程==
==Nonlinear algebraic equations 非线性代数方程==
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：
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>
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<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref>。
==非线性递推关系==
[[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<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref>。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。
==非线性微分方程==
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'''。
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，其解可以写成（随时间变化）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。
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，其解可以写成（随时间变化）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。
===Ordinary differential equations 常微分方程===
===常微分方程===
一阶[[常微分方程]]，尤其是自治（自主）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>
将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成
将 <math>u=\frac{1}{x+C}</math> 作为一般解（也有特解 ''u'' = 0，对应于 ''C'' 趋于无穷时的一般解的极限)。该方程是非线性的，因为它可以改写成
<math>\frac{du}{d x} + u^2=0</math>
<math>\frac{du}{d x} + u^2=0</math>
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）。
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）。
二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。
二阶和高阶常微分方程（更一般地说，非线性方程组）很少能产生封闭解，而隐式解和非初等函数积分形式的解较为常见。
非线性常微分方程定性分析的常用方法包括:
非线性常微分方程定性分析的常用方法包括:
*检查是否有任意'''守恒量 Conserved Quantities'''（特别是在'''[[哈密顿]]系统 Hamiltonian System'''中）
*检查是否有类似守恒量的耗散量（见'''[[李亚普诺夫函数]] Lyapunov Function'''）
*基于'''[[泰勒展开] Taylor Expansion'''的线性化
*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s
*将变量进行代换以便更好的进行研究
*将变量进行代换以便更好的进行研究
*[[Bifurcation theory]]
*'''[[分岔理论]] Bifurcation Theory'''
*'''分岔理论 Bifurcation Theory'''
*'''[[摄动理论]] Perturbation Theory'''（也可应用于代数方程）
===偏微分方程===
研究非线性偏微分方程最常用的基本方法是变量代换（或转换问题），使变换后的问题更简单（甚至可能变为线性的）。有时可以将此类方程转化成一或多个常微分方程（如同分离变量法所示），此时不论得到的常微分方程是否可解，但是对研究问题总是有用的。
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'''将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瞬态时，非线性的[[纳维－斯托克斯方程]] Navier-Stokes 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。
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瞬态时，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。
其他方法包括检查特征线法及前面所述研究常微分方程的方法。
其他方法包括检查特征线法及前面所述研究常微分方程的方法。
===Pendula 摆===
===单摆===
[[File:PendulumLayout.svg.png|thumb|图一：单摆图解|right|200px]]
[[File:PendulumLinearizations.png|thumb|图二：单摆的线性化|right|200px]]
A classic, extensively studied nonlinear problem is the dynamics of a [[pendulum (mathematics)|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 (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> 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
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单摆的动力学。利用'''[[拉格朗日力学]] Lagrangian Mechanics'''，可以证明单摆的运动<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref>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>
描述，其中重力指向“下”，<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>
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）。
另一种解决这个问题的方法是利用泰勒展开将任意非线性项（此时为正弦函数项)在某些点进行线性化。例如，在<math>\theta = 0</math> 的点附近线性化（称为小角度近似）为
:<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math>
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>
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。
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。
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>:
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>
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。
==Examples of nonlinear equations 非线性方程的例子==
==非线性动力学行为的类型==
*'''[[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
*'''[[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
*'''[[多稳态]] Multistability'''——两或多个稳态的存在
*'''[[孤波]] Soliton'''s——自增强的孤立波
*'''[[极限环]] Limit Cycles'''——吸引不稳定不动点的渐近周期轨道
*[[Algebraic Riccati equation]] 代数黎卡提方程
*'''[[自激振荡]] Self-oscillations'''——开放耗散物理系统中的反馈振荡
==非线性方程的例子==
{{colbegin||25em}}
*[[代数黎卡提方程]] Algebraic Riccati equation
*[[球杆系统]] Ball and beam system
*最佳策略的[[贝尔曼方程]] Bellman equation for optimal policy
*[[玻尔兹曼方程 ]] Boltzmann equation
*[[科尔布鲁克方程]] Colebrook equation
*[[广义相对论]] General relativity
*[[金兹堡-朗道方程]] Ginzburg–Landau theory
*[[石森方程]] Ishimori equation
*[[卡东穆塞夫-彼得韦亚斯维利方程]] Kadomtsev–Petviashvili equation
*[[KDV方程]] Korteweg–de Vries equation
*[[朗道-利夫希兹-吉尔伯特方程]] Landau–Lifshitz–Gilbert equation
*[[林纳德方程]] Liénard equation
*[[Navier–Stokes equations]] of [[fluid dynamics]] 流体力学的纳维－斯托克斯方程
*[[流体力学]] 的 [[纳维－斯托克斯]] 方程Navier–Stokes equations of fluid dynamics
*[[Nonlinear optics]] 非线性光学
*[[非线性光学]] Nonlinear optics
*[[Nonlinear Schrödinger equation]] 非线性薛定谔方程
*[[非线性薛定谔方程]] Nonlinear Schrödinger equation
*[[Power-flow study]] 功率流研究
*[[功率流研究]] Power-flow study
*[[Richards equation]] for unsaturated water flow 未饱和层水流的理查氏方程
*未饱和层水流的[[理查氏方程]]Richards equation for unsaturated water flow
*[[Self-balancing unicycle]] 自平衡独轮车
*[[自平衡独轮车]] Self-balancing unicycle
*[[Sine-Gordon equation]] 正弦-戈尔登方程
*[[正弦-戈尔登方程]] Sine-Gordon equation
*[[Van der Pol oscillator]] 范德波尔振荡器
*[[范德波尔振荡器]] Van der Pol oscillator
*[[Vlasov equation]] 弗拉索夫方程
*[[弗拉索夫方程]] Vlasov equation
{{Div col end}}
{{Div col end}}
==参见==
*[[亚历山大·李亚普诺夫]] Aleksandr Mikhailovich Lyapunov
*[[动态系统]] Dynamical system
*[[Aleksandr Mikhailovich Lyapunov]] 亚历山大·李亚普诺夫
*[[反馈]] Feedback
*[[初始条件]] Initial condition
*[[相互作用]] Interaction
*[[线性系统]] Linear system
*[[模式耦合]] Mode coupling
*[[Volterra series]] 沃尔泰拉级数
*[[矢量孤子]] Vector soliton
*[[沃尔泰拉级数]] Volterra series
==References 参考资料==
==References 参考资料==
{{Reflist|35em}}
{{Reflist|35em}}
==进一步阅读==
==Further reading 进一步阅读==