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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>。
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>。
——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''
——'''斯塔尼斯拉夫·乌拉姆 Stanislaw Ulam'''
==Definition 定义==
==定义==
在数学中，[[线性映射]]（或线性函数）<math>f(x)</math>满足以下两个性质:
在数学中，[[线性映射]]（或线性函数）<math>f(x)</math>满足以下两个性质:
*'''[[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）
===偏微分方程===
===偏微分方程===
其他方法包括检查特征线法及前面所述研究常微分方程的方法。
其他方法包括检查特征线法及前面所述研究常微分方程的方法。
[[File:PendulumLinearizations.png|thumb|图二：单摆的线性化|right|200px]]
[[File:PendulumLinearizations.png|thumb|图二：单摆的线性化|right|200px]]
一个经典的被广泛研究的非线性问题是重力影响下的单摆的动力学。利用'''[[拉格朗日力学]] Lagrangian Mechanics'''，可以证明单摆的运动<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref>可以用无量纲的非线性方程
一个经典的被广泛研究的非线性问题是重力影响下的单摆的动力学。利用'''[[拉格朗日力学]] Lagrangian Mechanics'''，可以证明单摆的运动<ref>[http://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref>可以用无量纲的非线性方程
这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''[[相图]] Phase Portrait'''和估计周期。
这相当于一个自由落体问题。把这样线性化的结果合在一起看，就能得到有关摆的运动的非常有用的图像，如右图所示。利用其他方法寻找（精确的）'''[[相图]] Phase Portrait'''和估计周期。
==非线性动力学行为的类型==
==非线性动力学行为的类型==
==非线性方程的例子==
==非线性方程的例子==
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{{Div col|colwidth=25em}}
*[[代数黎卡提方程]] Algebraic Riccati equation
*[[代数黎卡提方程]] Algebraic Riccati equation
*[[球杆系统]] Ball and beam system
*[[球杆系统]] Ball and beam system
*最佳策略的[[贝尔曼方程]] Bellman equation for optimal policy
*最佳策略的[[贝尔曼方程]] Bellman equation for optimal policy
*[[玻尔兹曼方程 ]] Boltzmann equation
*[[玻尔兹曼方程 ]] Boltzmann equation
*[[科尔布鲁克方程]] Colebrook equation
*[[科尔布鲁克方程]] Colebrook equation
*[[广义相对论]] General relativity
*[[广义相对论]] General relativity
*[[金兹堡-朗道方程]] Ginzburg–Landau theory
*[[金兹堡-朗道方程]] Ginzburg–Landau theory
*[[石森方程]] Ishimori equation
*[[石森方程]] Ishimori equation
*[[卡东穆塞夫-彼得韦亚斯维利方程]] Kadomtsev–Petviashvili equation
*[[卡东穆塞夫-彼得韦亚斯维利方程]] Kadomtsev–Petviashvili equation
*[[KDV方程]] Korteweg–de Vries equation
*[[KDV方程]] Korteweg–de Vries equation
*[[朗道-利夫希兹-吉尔伯特方程]] Landau–Lifshitz–Gilbert equation
*[[朗道-利夫希兹-吉尔伯特方程]] Landau–Lifshitz–Gilbert equation
*[[林纳德方程]] Liénard equation
*[[林纳德方程]] Liénard equation
*[[流体力学]] 的 [[纳维－斯托克斯]] 方程Navier–Stokes equations of fluid dynamics
*[[流体力学]] 的 [[纳维－斯托克斯]] 方程Navier–Stokes equations of fluid dynamics
*[[非线性光学]] Nonlinear optics
*[[非线性光学]] Nonlinear optics
*[[非线性薛定谔方程]] Nonlinear Schrödinger equation
*[[非线性薛定谔方程]] Nonlinear Schrödinger equation
*[[功率流研究]] Power-flow study
*[[功率流研究]] Power-flow study
*未饱和层水流的[[理查氏方程]]Richards equation for unsaturated water flow
*未饱和层水流的[[理查氏方程]]Richards equation for unsaturated water flow
*[[自平衡独轮车]] Self-balancing unicycle
*[[自平衡独轮车]] Self-balancing unicycle
*[[正弦-戈尔登方程]] Sine-Gordon equation
*[[正弦-戈尔登方程]] Sine-Gordon equation
*[[范德波尔振荡器]] Van der Pol oscillator
*[[范德波尔振荡器]] Van der Pol oscillator
*[[弗拉索夫方程]] Vlasov equation
*[[弗拉索夫方程]] Vlasov equation
{{Div col end}}
{{Div col end}}
==参见==
==参见==
*[[亚历山大·李亚普诺夫]] Aleksandr Mikhailovich Lyapunov
*[[亚历山大·李亚普诺夫]] Aleksandr Mikhailovich Lyapunov
*[[动态系统]] Dynamical system
*[[动态系统]] Dynamical system
*[[反馈]] Feedback
*[[反馈]] Feedback
*[[初始条件]] Initial condition
*[[初始条件]] Initial condition
*[[相互作用]] Interaction
*[[相互作用]] Interaction
*[[线性系统]] Linear system
*[[线性系统]] Linear system
*[[模式耦合]] Mode coupling
*[[模式耦合]] Mode coupling
*[[矢量孤子]] Vector soliton
*[[矢量孤子]] Vector soliton
*[[沃尔泰拉级数]] Volterra series
*[[沃尔泰拉级数]] Volterra series
==进一步阅读==
==进一步阅读==
{{Refbegin|35em}}
{{Refbegin|35em}}
*{{cite book
*{{cite book
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard
| author= [[Diederich Hinrichsen]] and Anthony J. Pritchard
| year= 2005
| year= 2005
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
| title= Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness
| publisher= Springer Verlag
| publisher= Springer Verlag
| isbn=9783540441250
| isbn=9783540441250
}}
}}
*{{cite book
*{{cite book
| last= Jordan
| last= Jordan
| first= D. W.
| first= D. W.
| last2= Smith
| last2= Smith
| first2= P.
| first2= P.
| year= 2007
| year= 2007
| edition= fourth
| edition= fourth
| title= Nonlinear Ordinary Differential Equations
| title= Nonlinear Ordinary Differential Equations
| publisher= Oxford University Press
| publisher= Oxford University Press
| isbn= 978-0-19-920824-1
| isbn= 978-0-19-920824-1
}}
}}
*{{cite book
*{{cite book
| last= Khalil
| last= Khalil
| first= Hassan K.
| first= Hassan K.
| year= 2001
| year= 2001
| title= Nonlinear Systems
| title= Nonlinear Systems
| publisher= Prentice Hall
| publisher= Prentice Hall
| isbn= 978-0-13-067389-3
| isbn= 978-0-13-067389-3
}}
}}
*{{cite book
*{{cite book
| last= Kreyszig
| last= Kreyszig
| first= Erwin
| first= Erwin
| authorlink= Erwin Kreyszig
| authorlink= Erwin Kreyszig
| year= 1998
| year= 1998
| title= Advanced Engineering Mathematics
| title= Advanced Engineering Mathematics
| url= https://archive.org/details/advancedengineer0008krey
| url= https://archive.org/details/advancedengineer0008krey
| url-access= registration
| url-access= registration
| publisher= Wiley
| publisher= Wiley
| isbn= 978-0-471-15496-9
| isbn= 978-0-471-15496-9
}}
}}
*{{cite book
*{{cite book
| last= Sontag
| last= Sontag
| first= Eduardo
| first= Eduardo
| authorlink= Eduardo D. Sontag
| authorlink= Eduardo D. Sontag
| year= 1998
| year= 1998
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
| title= Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
| publisher= Springer
| publisher= Springer
| isbn= 978-0-387-98489-6
| isbn= 978-0-387-98489-6
}}
}}
{{Refend}}
{{Refend}}
==外部链接==
==External links 外部链接==
*[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]
*[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://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://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://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]
*[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]
{{Systems science}}
{{Systems science}}
{{Differential equations topics}}
{{Differential equations topics}}
{{Authority control}}
{{Authority control}}