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第61行：
第61行： − ——斯塔尼斯拉夫．乌拉姆+ 第67行：
第67行： − ==Definition== − + − 定义 第81行：
第79行： + − − *Homogeneity: <math>\textstyle f(\alpha x) = \alpha f(x).</math> − − 第111行：
第106行： − 写成+ 第125行：
第120行： − 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。+ 第135行：
第130行： − + 第141行：
第136行： − + − ==Nonlinear algebraic equations== − − 非线性代数方程 第155行：
第147行： − 非线性代数方程，也称为多项式方程，是用多项式(次数大于1)等于零来定义的。比如说,+ − − :<math>x^2 + x - 1 = 0\,.</math> − − 数学 x ^ 2 + x-10 / 数学 第171行：
第159行： − 对于一个单一的多项式方程，求根算法可用于求解该方程(即满足该方程的变量的值集)。然而，代数方程组更加复杂; 他们的研究是代数几何领域的一个动力，这是现代数学的一个难以理解的分支。甚至很难判断一个给定的代数系统是否有复解(见希尔伯特的零点定律)。然而，对于具有有限个复解的系统，这些多项式方程组现在已经被很好地理解，并且存在有效的方法来解决它们。+ 第177行：
第165行： − + − ==Nonlinear recurrence relations== − 非线性递推关系 第187行：
第173行： − 非线性递回关系式将序列的连续项定义为前面项的非线性函数。非线性递归关系的例子有 logistic 映射和定义各种 Hofstadter 序列的关系。非线性离散模型是一类广泛的非线性递归关系，包括 NARMAX (非线性自回归移动平均加外生输入)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一类广泛的复杂非线性行为的时间，频率和时空域。+ 第193行：
第179行： − + − ==Nonlinear differential equations== − 非线性微分方程 第203行：
第187行： − 如果一个微分方程组不是一个线性系统，我们就说它是非线性的。涉及非线性微分方程的问题是非常多样的，解决或分析的方法是依赖于问题的。非线性微分方程的例子有流体力学中的 Navier-Stokes 方程和生物学中的 Lotka-Volterra 方程。+ 第213行：
第197行： − 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，一族线性无关的解可以通过叠加原理来构造通解。一个很好的例子是一维带有狄利克雷边界条件的热传输，其解可以写成频率不同的正弦波随时间变化的线性组合，这使得解非常灵活。通常可以找到几个非常具体的非线性方程的解，但是由于缺乏叠加原理，无法构造新的解。+ 第219行：
第203行： − + − − ===Ordinary differential equations=== − − 常微分方程 第229行：
第209行： − 一阶常微分方程，尤其是自治方程，通常可以用分离变量法精确地求解。例如，非线性方程+ − − :<math>\frac{d u}{d x} = -u^2</math> − − 数学问题-数学问题 第245行：
第221行： − + − − :<math>\frac{du}{d x} + u^2=0</math> − [数学][数学][数学] 第261行：
第234行： − + 第271行：
第244行： − 二阶和高阶常微分方程(更一般地说，非线性方程组)很少能产生闭合形式的解，尽管遇到了隐式解和涉及非初等积分的解。+ 第289行：
第262行： − + − + − + − + − + + 第314行：
第288行： + − ===Partial differential equations=== − − ===Partial differential equations=== − − 偏微分方程 第327行：
第297行： − + 第333行：
第303行： − 研究非线性偏微分方程最常用的基本方法是改变变量(或转换问题) ，使得问题更简单(甚至可能是线性的)。有时候，这个方程可以转化成一个或多个常微分方程，就像在分离变量法中看到的那样，不管得到的常微分方程是否可解，这个常微分方程都是有用的。+ 第343行：
第313行： − 另一个常见的策略(虽然不是数学上的) ，通常见于流体力学和热力学，是使用尺度分析来简化某一特定边值问题的一般自然方程。例如，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，在圆管中的瞬态、层流、一维流动; 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。+ 第353行：
第323行： − 其他方法包括研究常微分方程的特性和使用上述方法。+ − − ===Pendula=== − + − 钟摆属 第373行：
第340行： − + − − 200px − − 200px − − [[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum|right|200px]] − 200px − + 第393行：
第353行： − 一个经典的，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，可以证明摆的运动可以用无量纲的非线性方程来描述+ − − :<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math> − − Math frac ^ 2 theta } d ^ 2} + sin ( theta)0 / math 第409行：
第365行： − 其中重力指向“下方”和 math theta / math 是摆与其静止位置形成的角度，如右图所示。“解决”这个方程式的一个方法是用 math d theta / dt / math 作为积分因子，它最终会产生+ − − :<math>\int{\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1</math> − − Math int { frac { d theta }{ sqrt { c0 + 2 cos ( theta)}} t + c1 / math 第425行：
第377行： − 这是一个隐式的解决方案，涉及到一个椭圆积分。这个“解”通常没有多少用处，因为这个解的大部分性质都隐藏在非初等积分中(除非是数学 c02 / math)。+ 第435行：
第387行： − 另一种解决这个问题的方法是通过泰勒展开将任何非线性(在这种情况下是正弦函数项)在各个感兴趣的点线性化。例如，math theta 0 / math 的线性化，称为小角度近似+ − − :<math>\frac{d^2 \theta}{d t^2} + \theta = 0</math> − − Math frac ^ 2 theta } d ^ 2} + theta 0 / math 第451行：
第399行： − 从 math sin theta approx theta / math theta approx 0 / math 开始。这是一个简单的谐振子，对应于摆在其路径底部附近的摆动。另一种线性化方法是 math theta pi / math，对应于钟摆是笔直向上的:+ − − :<math>\frac{d^2 \theta}{d t^2} + \pi - \theta = 0</math> − − Math frac ^ 2 theta } d ^ 2} + pi- theta 0 / math 第467行：
第411行： − 自从 math sin ( theta) approx pi theta / math 对 math theta π / math。这个问题的解决方案涉及到双曲正弦曲线，并且注意到这个近似不同于小角度近似，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。+
无编辑摘要
“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
==Definition==
==Definition 定义==
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:
*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>
*可加性（叠加性）: <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>
*齐次性: <math>\textstyle f(\alpha x) = \alpha f(x).</math>
*齐次性: <math>\textstyle f(\alpha x) = \alpha f(x).</math>
An equation written as
An equation written as
对一个写成
<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>.
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。
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 是非常通用的，因为 math x / math 可以是任何可感知的数学对象(数字、向量、函数等等)。) ，函数 math f (x) / math 实际上可以是任何映射，包括与相关约束(如边界值)的集成或微分。如果 math f (x) / math 包含了与 math x / math 相关的微分，那么结果将是一个微分方程。
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==
==Nonlinear algebraic equations 非线性代数方程==
{{Main|Algebraic equation|System of polynomial equations}}
{{Main|Algebraic equation|System of polynomial equations}}
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,
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：
<math>x^2 + x - 1 = 0\,.</math>
<math>x^2 + x - 1 = 0\,.</math>
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（见希尔伯特的零点定律）。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==
==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 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.
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==
==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 [[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.
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。
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 Dirichlet 边界条件的一维热传导问题，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。
===Ordinary differential equations===
===Ordinary differential equations 常微分方程===
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 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
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>
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
将 math u frac { x + c } / math 作为一般解(也将 u 0作为特殊解，对应于 c 趋于无穷时的一般解的极限)。这个方程是非线性的，因为它可以写成
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>
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，如果 u sup 2 / sup 项替换为 u，问题将是线性的(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(指数衰减问题)。
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.
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.
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。
*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）
*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）
*Linearization via [[Taylor expansion]]
*Linearization via [[Taylor expansion]]
*基于泰勒展开的线性化
*Change of variables into something easier to study
*Change of variables into something easier to study
*将变量转化得更易于研究
*[[Bifurcation theory]]
*[[Bifurcation theory]]
*分岔理论
*[[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）
===Partial differential equations 偏微分方程===
{{main|Nonlinear partial differential equation}}
{{main|Nonlinear partial differential equation}}
{{See also|List of nonlinear partial differential equations}}
{{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 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.
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换变量（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 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。
Other methods include examining the 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.
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===
===Pendula 摆===
{{Main|Pendulum (mathematics)}}
{{Main|Pendulum (mathematics)}}
[[File:PendulumLayout.svg|thumb|Illustration of a pendulum|right|200px]]
[[File:PendulumLayout.svg.png|thumb|Illustration of a pendulum 摆图解|right|200px]]
200px
[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum 摆的线性化|right|200px]]
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，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
<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 expansions. 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>
<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，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。