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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}
{{Complex systems}}
{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}
In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.
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.
在数学和科学中，非线性是一种输出的变化与输入的变化不成比例的系统。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。描述变量随时间变化的非线性动力系统，与简单的线性系统相比，可能显得混沌、不可预测或违反直觉。
Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|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，一个非线性的行为在数学上是由一个非线性的方程组来描述的，这是一组方程组，其中未知数(或者在微分方程的情况下未知函数)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。
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 variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
换句话说，在一个非线性的方程式中，要求解的方程式不能被写成未知变量或函数的线性组合。无论已知的线性函数是否出现在方程中，系统都可以被定义为非线性。特别是，一个微分方程是线性的，如果它在未知函数及其导数方面是线性的，即使在其他变量方面是非线性的。
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 [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|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 [[randomness|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，非线性系统通常采用线性化方法来近似。这种方法在一定的精度和一定的范围内可以得到较好的输入值，但是一些有趣的现象，如孤子、混沌和奇异现象被线性化隐藏了起来。因此，非线性的动态行为的某些方面可能看起来是违反直觉的，不可预测的，甚至是混乱的。尽管这种混沌行为可能类似于随机行为，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乱的，系统的某个部分的简单变化会产生复杂的影响。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。
Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:
有些作者用非线性科学这个术语来研究非线性系统。这一术语引起了其他人的争议:
{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[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>}}
==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:
在数学中，线性映射(或线性函数)的数学 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
可加性意味着任何有理函数的齐次性，对于连续函数，意味着任何实函数的齐次性。对于复合体，均匀性并不遵循可加性。例如，反线性映射是可加映射，但不是齐次映射。可加性和均匀性的条件经常在叠加原理中被组合起来
:<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>
数学 f ( alpha x + beta y) alpha f (x) + beta f (y) / math
An equation written as
An equation written as
一个等式写成
:<math>f(x) = C</math>
<math>f(x) = C</math>
数学 f (x) c / 数学
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>.
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。
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.
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。
==Nonlinear algebraic equations==
==Nonlinear algebraic equations==
非线性代数方程
{{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,
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>
<math>x^2 + x - 1 = 0\,.</math>
数学 x ^ 2 + x-10 / 数学
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。
==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 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。
==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 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.
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。
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，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。
===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 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精确地求解。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>
数学问题-数学问题
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
:<math>\frac{du}{d x} + u^2=0</math>
<math>\frac{du}{d x} + u^2=0</math>
[数学][数学][数学]
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)。
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.
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。
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
非线性常微分方程定性分析的常用方法包括:
*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s
*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities
*Linearization via [[Taylor expansion]]
*Change of variables into something easier to study
*[[Bifurcation theory]]
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)
===Partial differential equations===
===Partial differential equations===
偏微分方程
{{main|Nonlinear partial differential equation}}
{{See also|List of nonlinear partial differential equations}}
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。
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.
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。
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.
其他方法包括研究常微分方程的特性和使用上述方法。
===Pendula===
===Pendula===
钟摆属
{{Main|Pendulum (mathematics)}}
[[File:PendulumLayout.svg|thumb|Illustration of a pendulum|right|200px]]
200px
200px
[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum|right|200px]]
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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
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{d^2 \theta}{d t^2} + \sin(\theta) = 0</math>
Math frac ^ 2 theta } d ^ 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
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{C_0 + 2 \cos(\theta)}}} = t + C_1</math>
Math int { frac { d theta }{ sqrt { c0 + 2 cos ( theta)}} t + c1 / 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)。
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{d^2 \theta}{d t^2} + \theta = 0</math>
Math frac ^ 2 theta } d ^ 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:
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{d^2 \theta}{d t^2} + \pi - \theta = 0</math>
Math frac ^ 2 theta } d ^ 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 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.
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。
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围绕着 math theta pi / 2 / math，围绕着 math sin ( theta) approx1 / math:
:<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>
<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>
Math frac ^ 2 theta } d t ^ 2} + 10. / 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。一个非常有用的定性图片的钟摆的动态可以得到一起拼凑这样的线性化，如图中所示。其他技术可用于寻找(精确的)相位图和近似周期。
==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
*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]
*[[Multistability]] – the presence of two or more stable states
*[[Soliton]]s – self-reinforcing solitary waves
*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted.
*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.
==Examples of nonlinear equations==
==Examples of nonlinear equations==
非线性方程的例子
{{Div col|colwidth=25em}}
*[[Algebraic Riccati equation]]
*[[Ball and beam]] system
*[[Bellman equation]] for optimal policy
*[[Boltzmann equation]]
*[[Colebrook equation]]
*[[General relativity]]
*[[Ginzburg–Landau theory]]
*[[Ishimori equation]]
*[[Kadomtsev–Petviashvili equation]]
*[[Korteweg–de Vries equation]]
*[[Landau–Lifshitz–Gilbert equation]]
*[[Liénard equation]]
*[[Navier–Stokes equations]] of [[fluid dynamics]]
*[[Nonlinear optics]]
*[[Nonlinear Schrödinger equation]]
*[[Power-flow study]]
*[[Richards equation]] for unsaturated water flow
*[[Self-balancing unicycle]]
*[[Sine-Gordon equation]]
*[[Van der Pol oscillator]]
*[[Vlasov equation]]
{{Div col end}}
==See also==
==See also==
参见
*[[Aleksandr Mikhailovich Lyapunov]]
*[[Dynamical system]]
*[[Feedback]]
*[[Initial condition]]
*[[Interaction]]
*[[Linear system]]
*[[Mode coupling]]
*[[Vector soliton]]
*[[Volterra series]]
==References==
==References==
参考资料
{{Reflist|35em}}
==Further reading==
==Further reading==
进一步阅读
{{Refbegin|35em}}
*{{cite book
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard
| author= Diederich Hinrichsen and Anthony J. Pritchard
作者 Diederich Hinrichsen 和 Anthony j. Pritchard
| year= 2005
| year= 2005
2005年
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性
| publisher= Springer Verlag
| publisher= Springer Verlag
出版商 Springer Verlag
| isbn=9783540441250
| isbn=9783540441250
9783540441250
}}
}}
}}
*{{cite book
| last= Jordan
| last= Jordan
最后一个乔丹
| first= D. W.
| first= D. W.
首先是 D.w。
| last2= Smith
| last2= Smith
最后2个史密斯
| first2= P.
| first2= P.
| first2 p.
| year= 2007
| year= 2007
2007年
| edition= fourth
| edition= fourth
第四版
| title= Nonlinear Ordinary Differential Equations
| title= Nonlinear Ordinary Differential Equations
非线性常微分方程
| publisher= Oxford University Press
| publisher= Oxford University Press
牛津大学出版社
| isbn= 978-0-19-920824-1
| isbn= 978-0-19-920824-1
[国际标准图书编号978-0-19-920824-1]
}}
}}
}}
*{{cite book
| last= Khalil
| last= Khalil
最后的哈利勒
| first= Hassan K.
| first= Hassan K.
首先是哈桑 · k。
| year= 2001
| year= 2001
2001年
| title= Nonlinear Systems
| title= Nonlinear Systems
非线性系统
| publisher= Prentice Hall
| publisher= Prentice Hall
出版商 Prentice Hall
| isbn= 978-0-13-067389-3
| isbn= 978-0-13-067389-3
[国际标准图书编号978-0-13-067389-3]
}}
}}
}}
*{{cite book
| last= Kreyszig
| last= Kreyszig
| last= Kreyszig
| first= Erwin
| first= Erwin
首先是欧文
| authorlink= Erwin Kreyszig
| authorlink= Erwin Kreyszig
作者 / 链接 Erwin Kreyszig
| year= 1998
| year= 1998
1998年
| title= Advanced Engineering Mathematics
| title= Advanced Engineering Mathematics
高等工程数学
| url= https://archive.org/details/advancedengineer0008krey
| url= https://archive.org/details/advancedengineer0008krey
Https://archive.org/details/advancedengineer0008krey
| url-access= registration
| url-access= registration
访问注册
| publisher= Wiley
| publisher= Wiley
出版商 Wiley
| isbn= 978-0-471-15496-9
| isbn= 978-0-471-15496-9
[国际标准图书编号978-0-471-15496-9]
}}
}}
}}
*{{cite book
| last= Sontag
| last= Sontag
最后一个桑塔格
| first= Eduardo
| first= Eduardo
首先是爱德华多
| authorlink= Eduardo D. Sontag
| authorlink= Eduardo D. Sontag
作者: Eduardo d. Sontag
| year= 1998
| year= 1998
1998年
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
数学控制理论: 确定性有限维系统。第二版
| publisher= Springer
| publisher= Springer
出版商斯普林格
| isbn= 978-0-387-98489-6
| isbn= 978-0-387-98489-6
[国际标准图书编号978-0-387-98489-6]
}}
}}
}}
{{Refend}}
==External links==
==External links==
外部链接
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]
*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]
*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]
{{Systems science}}
{{Differential equations topics}}
{{Authority control}}
[[Category:Nonlinear systems| ]]
[[Category:Dynamical systems]]
Category:Dynamical systems
类别: 动力系统
[[Category:Concepts in physics]]
Category:Concepts in physics
分类: 物理概念
<noinclude>
<small>This page was moved from [[wikipedia:en:Nonlinear system]]. Its edit history can be viewed at [[非线性系统/edithistory]]</small></noinclude>
[[Category:待整理页面]]
{{Complex systems}}
{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}
In [[mathematics]] and [[science]], a '''nonlinear system''' is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M}}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s because most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.
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.
在数学和科学中，非线性是一种输出的变化与输入的变化不成比例的系统。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。描述变量随时间变化的非线性动力系统，与简单的线性系统相比，可能显得混沌、不可预测或违反直觉。
Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|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，一个非线性的行为在数学上是由一个非线性的方程组来描述的，这是一组方程组，其中未知数(或者在微分方程的情况下未知函数)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。
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 variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
换句话说，在一个非线性的方程式中，要求解的方程式不能被写成未知变量或函数的线性组合。无论已知的线性函数是否出现在方程中，系统都可以被定义为非线性。特别是，一个微分方程是线性的，如果它在未知函数及其导数方面是线性的，即使在其他变量方面是非线性的。
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 [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|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 [[randomness|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，非线性系统通常采用线性化方法来近似。这种方法在一定的精度和一定的范围内可以得到较好的输入值，但是一些有趣的现象，如孤子、混沌和奇异现象被线性化隐藏了起来。因此，非线性的动态行为的某些方面可能看起来是违反直觉的，不可预测的，甚至是混乱的。尽管这种混沌行为可能类似于随机行为，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乱的，系统的某个部分的简单变化会产生复杂的影响。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。
Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:
有些作者用非线性科学这个术语来研究非线性系统。这一术语引起了其他人的争议:
{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of [[negation|non]]-elephant animals.|[[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>}}
==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:
在数学中，线性映射(或线性函数)的数学 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
可加性意味着任何有理函数的齐次性，对于连续函数，意味着任何实函数的齐次性。对于复合体，均匀性并不遵循可加性。例如，反线性映射是可加映射，但不是齐次映射。可加性和均匀性的条件经常在叠加原理中被组合起来
:<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>
数学 f ( alpha x + beta y) alpha f (x) + beta f (y) / math
An equation written as
An equation written as
一个等式写成
:<math>f(x) = C</math>
<math>f(x) = C</math>
数学 f (x) c / 数学
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>.
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。
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.
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。
==Nonlinear algebraic equations==
==Nonlinear algebraic equations==
非线性代数方程
{{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,
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>
<math>x^2 + x - 1 = 0\,.</math>
数学 x ^ 2 + x-10 / 数学
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。
==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 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。
==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 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.
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。
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，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。
===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 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精确地求解。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>
数学问题-数学问题
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
:<math>\frac{du}{d x} + u^2=0</math>
<math>\frac{du}{d x} + u^2=0</math>
[数学][数学][数学]
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)。
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.
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。
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
Common methods for the qualitative analysis of nonlinear ordinary differential equations include:
非线性常微分方程定性分析的常用方法包括:
*Examination of any [[conserved quantities]], especially in [[Hamiltonian system]]s
*Examination of dissipative quantities (see [[Lyapunov function]]) analogous to conserved quantities
*Linearization via [[Taylor expansion]]
*Change of variables into something easier to study
*[[Bifurcation theory]]
*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)
===Partial differential equations===
===Partial differential equations===
偏微分方程
{{main|Nonlinear partial differential equation}}
{{See also|List of nonlinear partial differential equations}}
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。
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.
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。
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.
其他方法包括研究常微分方程的特性和使用上述方法。
===Pendula===
===Pendula===
钟摆属
{{Main|Pendulum (mathematics)}}
[[File:PendulumLayout.svg|thumb|Illustration of a pendulum|right|200px]]
200px
200px
[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum|right|200px]]
200px
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<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
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{d^2 \theta}{d t^2} + \sin(\theta) = 0</math>
Math frac ^ 2 theta } d ^ 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
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{C_0 + 2 \cos(\theta)}}} = t + C_1</math>
Math int { frac { d theta }{ sqrt { c0 + 2 cos ( theta)}} t + c1 / 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)。
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{d^2 \theta}{d t^2} + \theta = 0</math>
Math frac ^ 2 theta } d ^ 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:
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{d^2 \theta}{d t^2} + \pi - \theta = 0</math>
Math frac ^ 2 theta } d ^ 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 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.
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。
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围绕着 math theta pi / 2 / math，围绕着 math sin ( theta) approx1 / math:
:<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>
<math>\frac{d^2 \theta}{d t^2} + 1 = 0.</math>
Math frac ^ 2 theta } d t ^ 2} + 10. / 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。一个非常有用的定性图片的钟摆的动态可以得到一起拼凑这样的线性化，如图中所示。其他技术可用于寻找(精确的)相位图和近似周期。
==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
*[[Chaos theory|Chaos]] – values of a system cannot be predicted indefinitely far into the future, and fluctuations are [[aperiodic]]
*[[Multistability]] – the presence of two or more stable states
*[[Soliton]]s – self-reinforcing solitary waves
*[[Limit cycle|Limit cycles]] – asymptotic periodic orbits to which destabilized fixed points are attracted.
*[[Self-oscillation|Self-oscillations]] - feedback oscillations taking place in open dissipative physical systems.
==Examples of nonlinear equations==
==Examples of nonlinear equations==
非线性方程的例子
{{Div col|colwidth=25em}}
*[[Algebraic Riccati equation]]
*[[Ball and beam]] system
*[[Bellman equation]] for optimal policy
*[[Boltzmann equation]]
*[[Colebrook equation]]
*[[General relativity]]
*[[Ginzburg–Landau theory]]
*[[Ishimori equation]]
*[[Kadomtsev–Petviashvili equation]]
*[[Korteweg–de Vries equation]]
*[[Landau–Lifshitz–Gilbert equation]]
*[[Liénard equation]]
*[[Navier–Stokes equations]] of [[fluid dynamics]]
*[[Nonlinear optics]]
*[[Nonlinear Schrödinger equation]]
*[[Power-flow study]]
*[[Richards equation]] for unsaturated water flow
*[[Self-balancing unicycle]]
*[[Sine-Gordon equation]]
*[[Van der Pol oscillator]]
*[[Vlasov equation]]
{{Div col end}}
==See also==
==See also==
参见
*[[Aleksandr Mikhailovich Lyapunov]]
*[[Dynamical system]]
*[[Feedback]]
*[[Initial condition]]
*[[Interaction]]
*[[Linear system]]
*[[Mode coupling]]
*[[Vector soliton]]
*[[Volterra series]]
==References==
==References==
参考资料
{{Reflist|35em}}
==Further reading==
==Further reading==
进一步阅读
{{Refbegin|35em}}
*{{cite book
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard
| author= Diederich Hinrichsen and Anthony J. Pritchard
作者 Diederich Hinrichsen 和 Anthony j. Pritchard
| year= 2005
| year= 2005
2005年
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
数学系统理论 i-建模，状态空间分析，稳定性和鲁棒性
| publisher= Springer Verlag
| publisher= Springer Verlag
出版商 Springer Verlag
| isbn=9783540441250
| isbn=9783540441250
9783540441250
}}
}}
}}
*{{cite book
| last= Jordan
| last= Jordan
最后一个乔丹
| first= D. W.
| first= D. W.
首先是 D.w。
| last2= Smith
| last2= Smith
最后2个史密斯
| first2= P.
| first2= P.
| first2 p.
| year= 2007
| year= 2007
2007年
| edition= fourth
| edition= fourth
第四版
| title= Nonlinear Ordinary Differential Equations
| title= Nonlinear Ordinary Differential Equations
非线性常微分方程
| publisher= Oxford University Press
| publisher= Oxford University Press
牛津大学出版社
| isbn= 978-0-19-920824-1
| isbn= 978-0-19-920824-1
[国际标准图书编号978-0-19-920824-1]
}}
}}
}}
*{{cite book
| last= Khalil
| last= Khalil
最后的哈利勒
| first= Hassan K.
| first= Hassan K.
首先是哈桑 · k。
| year= 2001
| year= 2001
2001年
| title= Nonlinear Systems
| title= Nonlinear Systems
非线性系统
| publisher= Prentice Hall
| publisher= Prentice Hall
出版商 Prentice Hall
| isbn= 978-0-13-067389-3
| isbn= 978-0-13-067389-3
[国际标准图书编号978-0-13-067389-3]
}}
}}
}}
*{{cite book
| last= Kreyszig
| last= Kreyszig
| last= Kreyszig
| first= Erwin
| first= Erwin
首先是欧文
| authorlink= Erwin Kreyszig
| authorlink= Erwin Kreyszig
作者 / 链接 Erwin Kreyszig
| year= 1998
| year= 1998
1998年
| title= Advanced Engineering Mathematics
| title= Advanced Engineering Mathematics
高等工程数学
| url= https://archive.org/details/advancedengineer0008krey
| url= https://archive.org/details/advancedengineer0008krey
Https://archive.org/details/advancedengineer0008krey
| url-access= registration
| url-access= registration
访问注册
| publisher= Wiley
| publisher= Wiley
出版商 Wiley
| isbn= 978-0-471-15496-9
| isbn= 978-0-471-15496-9
[国际标准图书编号978-0-471-15496-9]
}}
}}
}}
*{{cite book
| last= Sontag
| last= Sontag
最后一个桑塔格
| first= Eduardo
| first= Eduardo
首先是爱德华多
| authorlink= Eduardo D. Sontag
| authorlink= Eduardo D. Sontag
作者: Eduardo d. Sontag
| year= 1998
| year= 1998
1998年
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
数学控制理论: 确定性有限维系统。第二版
| publisher= Springer
| publisher= Springer
出版商斯普林格
| isbn= 978-0-387-98489-6
| isbn= 978-0-387-98489-6
[国际标准图书编号978-0-387-98489-6]
}}
}}
}}
{{Refend}}
==External links==
==External links==
外部链接
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]
*[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]
*[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]
*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]
{{Systems science}}
{{Differential equations topics}}
{{Authority control}}
[[Category:Nonlinear systems| ]]
[[Category:Dynamical systems]]
Category:Dynamical systems
类别: 动力系统
[[Category:Concepts in physics]]
Category:Concepts in physics
分类: 物理概念
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