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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}

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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.

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在数学及科学中，'''非线性系统 Nonlinear System'''是一种输出的变化与输入的变化不成比例的系统<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>。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、,<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>生物学家、物理学家<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>、数学家和许多其他科学家的兴趣。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉<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>。

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在数学及科学中，'''非线性系统 Nonlinear System'''是一种输出的变化与输入的变化不成比例的系统<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>。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、,<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>生物学家、物理学家<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>~~、数学家和许多其他科学家的兴趣~~<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>~~。描述变量随时间变化的非线性动力系统与较之简单得多的线性系统相比，可能显得混沌、不可预测或违反直觉。~~

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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.

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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被描述为一组非线性的联立方程组，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变量出现，或者作为一个非一次多项式函数的参数出现。

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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.

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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，在非线性方程系统中，待解的方程不能被写成未知变量或函数的线性组合。无论方程中是否有已知的线性函数，系统都可以被定义为非线性。特别是当一个微分方程的未知函数及其导数是线性的，即使其他变量是非线性的，也称该方程是线性的。

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由于非线性动力学方程难以求解，通常用线性方程来近似非线性系统（'''线性化 Linearization'''）。这种方法对于一定范围的输入和某些精度要求下的效果不错，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏<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>。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测、甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的整体影响。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。

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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[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.

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As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

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由于非线性动力学方程难以求解，通常用线性方程来近似非线性系统（'''线性化 Linearization'''）。这种方法对于一定范围的输入和某些精度要求下的效果不错，但一些有趣的现象如'''孤子 Soliton'''、'''混沌 Chaos'''和'''奇异性 Singularity'''在线性化后被隐藏。因此，非线性系统的动态行为在某些方面可能看起来违反直觉、不可预测、甚至混沌。尽管这种混沌行为可能感觉很像随机行为，但它实际上并不是随机的。例如，天气的某些方面被认为是混沌的，其系统某部分的微小扰动就会产生复杂的整体影响。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。

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~~Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:~~

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~~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一术语来研究非线性系统。这一术语引起了其他人的争议:

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{{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>}}

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{{quote|“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”|[[Stanislaw Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C}}</ref>}}

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~~“使用‘非线性科学’这样的术语，就如同把动物学里大部分对象称作‘非大象动物’研究一样可笑。”~~

——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''

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