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In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

   --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。  这里调整语序很棒，但是因果关系是不是

   --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）大多数系统在本质上是非线性的，因而非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。  这里调整语序很棒，但是因果关系是不是

因为大多数系统在本质上是非线性的，所以非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。

因为大多数系统在本质上是非线性的，所以非线性问题引起了工程师、生物学家、物理学家、数学家和许多其他科学家的兴趣。

 

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.

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

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.

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。

==Nonlinear algebraic equations 非线性代数方程==

==Nonlinear algebraic equations 非线性代数方程==

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,

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>

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。

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-斯托克斯方程 Navier-Stokes equations'''和生物学中的'''洛特卡－沃尔泰拉方程 Lotka-Volterra equations'''。

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。

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

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 u² 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 u² 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''² 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''² 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 Conserved quantities'''（特别是在'''哈密顿系统 Hamiltonian system'''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 Lyapunov function'''）

*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 Taylor expansion'''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

*将变量转化得更易于研究

*将变量进行代换以便更好的进行研究

*[[Bifurcation theory]]

*[[Bifurcation theory]]

*'''分岔理论 Bifurcation theory'''

*'''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 Perturbation theory'''（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。

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

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 Integrating factor'''，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）。

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 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:

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>

*[[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

*'''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]]

*[[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

*'''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.

*[[Self-balancing unicycle]]

*[[Self-balancing unicycle]]自平衡独轮车