{{short description|Behavior in a nonlinear system}}
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[[Image:VanDerPolPhaseSpace.png|right|250px|thumb|'''<font color="#ff8000">范德波尔振荡器 Van der Pol oscillator</font>'''的稳定极限环 ]]
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{{Use American English|date=April 2020}}
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[[File:Limit cycle Poincare map.svg|thumb|250px|right|Stable limit cycle (shown in bold) and two other trajectories spiraling into it 稳定的极限环(粗体显示)和另外两个螺旋进入它的轨迹 ]]
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Stable limit cycle (shown in bold) and two other trajectories spiraling into it
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'''稳定极限环'''以及另外两个轨迹以螺旋方式进入
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[[Image:VanDerPolPhaseSpace.png|right|250px|thumb|Stable limit cycle (shown in bold) for the [[Van der Pol oscillator]] '''<font color="#ff8000">范德波尔振荡器 Van der Pol oscillator</font>'''的稳定极限环(粗体显示) ]]
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Stable limit cycle (shown in bold) for the [[Van der Pol oscillator]]
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'''<font color="#ff8000">范德波尔振荡器 Van der Pol oscillator</font>'''中的'''稳定极限环'''。
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In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
在数学中，二维'''<font color="#ff8000">相空间 phase space</font>''' 动力系统的研究中，'''极限环'''是一个在相空间中的闭合轨迹，它具有当时间趋于无穷大或时间趋于负无穷大时至少有一条其他轨迹螺旋进入的性质。这种行为在一些非线性系统中表现出来。极限环已经被用来模拟许多实际振动系统的行为。对极限环的研究是由'''<font color="#ff8000">亨利·庞加莱 Henri poincaré</font>'''提出的。
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==Definition==
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==定义==
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定义
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我们认为一个二维动力系统的形式如下：
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We consider a two-dimensional dynamical system of the form
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We consider a two-dimensional dynamical system of the form
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我们考虑一个二维动力系统的形式
:<math>x'(t)=V(x(t))</math>
:<math>x'(t)=V(x(t))</math>
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:<math>x'(t)=V(x(t))</math>
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where
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where
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:<math>V:\mathbb{R}^2\to\mathbb{R}^2</math>
:<math>V:\mathbb{R}^2\to\mathbb{R}^2</math>
:<math>V:\mathbb{R}^2\to\mathbb{R}^2</math>
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is a smooth function. A ''trajectory'' of this system is some smooth function <math>x(t)</math> with values in <math>\mathbb{R}^2</math> which satisfies this differential equation. Such a trajectory is called ''closed'' (or ''periodic'') if it is not constant but returns to its starting point, i.e. if there exists some <math>t_0>0</math> such that <math>x(t+t_0)=x(t)</math> for all <math>t\in\mathbb{R}</math>. An [[orbit (dynamics)|orbit]] is the [[image (mathematics)|image]] of a trajectory, a subset of <math>\mathbb{R}^2</math>. A ''closed orbit'', or ''cycle'', is the image of a closed trajectory. A ''limit cycle'' is a cycle which is the [[limit set]] of some other trajectory.
is a smooth function. A trajectory of this system is some smooth function <math>x(t)</math> with values in <math>\mathbb{R}^2</math> which satisfies this differential equation. Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some <math>t_0>0</math> such that <math>x(t+t_0)=x(t)</math> for all <math>t\in\mathbb{R}</math>. An orbit is the image of a trajectory, a subset of <math>\mathbb{R}^2</math>. A closed orbit, or cycle, is the image of a closed trajectory. A limit cycle is a cycle which is the limit set of some other trajectory.
By the [[Jordan curve theorem]], every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
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By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
通过'''<font color="#ff8000">若尔当曲线定理 Jordan Curve Theorem</font>'''，每一个封闭的轨迹将平面分成两个区域，内部和外部的曲线。
通过'''<font color="#ff8000">若尔当曲线定理 Jordan Curve Theorem</font>'''，每一个封闭的轨迹将平面分成两个区域，内部和外部的曲线。
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Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching <math>+ \infty</math>, then there is a neighborhood around the limit cycle such that ''all'' trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching <math> + \infty</math>. The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching <math>- \infty</math>, and also for trajectories in the exterior approaching the limit cycle.
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Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching <math>+ \infty</math>, then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching <math> + \infty</math>. The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching <math>- \infty</math>, and also for trajectories in the exterior approaching the limit cycle.
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a ''[[stable manifold|stable]]'' or ''attractive'' limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an ''unstable'' limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a ''semi-stable'' limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which wouldn't be limit cycles).
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a stable or attractive limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an unstable limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a semi-stable limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which wouldn't be limit cycles).
Stable limit cycles are examples of [[attractor]]s. They imply self-sustained [[oscillations]]: the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.
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==寻找极限环==
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Stable limit cycles are examples of attractors. They imply self-sustained oscillations: the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.
Every closed trajectory contains within its interior a [[stationary point]] of the system, i.e. a point <math>p</math> where <math>V(p)=0</math>. The [[Bendixson–Dulac theorem]] and the [[Poincaré–Bendixson theorem]] predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.
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Every closed trajectory contains within its interior a stationary point of the system, i.e. a point <math>p</math> where <math>V(p)=0</math>. The Bendixson–Dulac theorem and the Poincaré–Bendixson theorem predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.
Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of [[Hilbert's sixteenth problem]]. It is unknown, for instance, whether there is any system <math>x'=V(x)</math> in the plane where both components of <math>V</math> are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
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Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of Hilbert's sixteenth problem. It is unknown, for instance, whether there is any system <math>x'=V(x)</math> in the plane where both components of <math>V</math> are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
[[File:Hopfbifurcation.png|thumb|400px|Examples of limit cycles branching from fixed points near [[Hopf bifurcation]]. Trajectories in red, stable structures in dark blue, unstable structures in light blue. The parameter choice determines the occurrence and stability of limit cycles.]]
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Examples of limit cycles branching from fixed points near [[Hopf bifurcation. Trajectories in red, stable structures in dark blue, unstable structures in light blue. The parameter choice determines the occurrence and stability of limit cycles.]]
* The Sel'kov model of [[glycolysis]].<ref>{{Cite journal|last=Sel'kov|first=E. E.|date=1968|title=Self-Oscillations in Glycolysis 1. A Simple Kinetic Model|journal=European Journal of Biochemistry|language=en|volume=4|issue=1|pages=79–86|doi=10.1111/j.1432-1033.1968.tb00175.x|pmid=4230812|issn=1432-1033}}</ref>
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'''<font color="#ff8000">糖酵解 glycolysis</font>'''的塞尔科夫模型<ref>{{Cite journal|last=Sel'kov|first=E. E.|date=1968|title=Self-Oscillations in Glycolysis 1. A Simple Kinetic Model|journal=European Journal of Biochemistry|language=en|volume=4|issue=1|pages=79–86|doi=10.1111/j.1432-1033.1968.tb00175.x|pmid=4230812|issn=1432-1033}}</ref>。
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* The daily oscillations in gene expression, hormone levels and body temperature of animals, which are part of the [[circadian rhythm]].<ref>{{Cite journal|last=Leloup|first=Jean-Christophe|last2=Gonze|first2=Didier|last3=Goldbeter|first3=Albert|date=1999-12-01|title=Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora|journal=Journal of Biological Rhythms|language=en|volume=14|issue=6|pages=433–448|doi=10.1177/074873099129000948|pmid=10643740|issn=0748-7304}}</ref><ref>{{Cite journal|last=Roenneberg|first=Till|last2=Chua|first2=Elaine Jane|last3=Bernardo|first3=Ric|last4=Mendoza|first4=Eduardo|date=2008-09-09|title=Modelling Biological Rhythms|journal=Current Biology|volume=18|issue=17|pages=R826–R835|doi=10.1016/j.cub.2008.07.017|pmid=18786388|issn=0960-9822}}</ref>
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动物基因表达、激素水平和体温的日常变化，这些都是'''<font color="#ff8000">昼夜节律 circadian rhythm</font>'''的一部分<ref>{{Cite journal|last=Leloup|first=Jean-Christophe|last2=Gonze|first2=Didier|last3=Goldbeter|first3=Albert|date=1999-12-01|title=Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora|journal=Journal of Biological Rhythms|language=en|volume=14|issue=6|pages=433–448|doi=10.1177/074873099129000948|pmid=10643740|issn=0748-7304}}</ref><ref>{{Cite journal|last=Roenneberg|first=Till|last2=Chua|first2=Elaine Jane|last3=Bernardo|first3=Ric|last4=Mendoza|first4=Eduardo|date=2008-09-09|title=Modelling Biological Rhythms|journal=Current Biology|volume=18|issue=17|pages=R826–R835|doi=10.1016/j.cub.2008.07.017|pmid=18786388|issn=0960-9822}}</ref>。
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* The [[Cell migration|migration]] of [[cancer cell]]s in confining micro-environments follows limit cycle oscillations.<ref>{{Cite journal|last=Brückner|first=David B.|last2=Fink|first2=Alexandra|last3=Schreiber|first3=Christoph|last4=Röttgermann|first4=Peter J. F.|last5=Rädler|first5=Joachim|last6=Broedersz|first6=Chase P.|date=2019|title=Stochastic nonlinear dynamics of confined cell migration in two-state systems|journal=Nature Physics|language=en|volume=15|issue=6|pages=595–601|doi=10.1038/s41567-019-0445-4|issn=1745-2481|bibcode=2019NatPh..15..595B}}</ref>
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'''<font color="#ff8000">癌细胞 cancer cell</font>'''在局限微环境中的'''<font color="#ff8000">迁移 migration</font>'''遵循极限环振荡<ref>{{Cite journal|last=Brückner|first=David B.|last2=Fink|first2=Alexandra|last3=Schreiber|first3=Christoph|last4=Röttgermann|first4=Peter J. F.|last5=Rädler|first5=Joachim|last6=Broedersz|first6=Chase P.|date=2019|title=Stochastic nonlinear dynamics of confined cell migration in two-state systems|journal=Nature Physics|language=en|volume=15|issue=6|pages=595–601|doi=10.1038/s41567-019-0445-4|issn=1745-2481|bibcode=2019NatPh..15..595B}}</ref> 。
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* Some non-linear [[Electrical Circuit|electrical circuits]] exhibit limit cycle oscillations,<ref>{{Cite journal|last=Ginoux|first=Jean-Marc|last2=Letellier|first2=Christophe|date=2012-04-30|title=Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=22|issue=2|pages=023120|doi=10.1063/1.3670008|pmid=22757527|issn=1054-1500|arxiv=1408.4890|bibcode=2012Chaos..22b3120G}}</ref> which inspired the original [[Van der Pol oscillator|Van der Pol model]].
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一些非线性'''<font color="#ff8000">电路 electrical circuits</font>'''表现出极限环振荡<ref>{{Cite journal|last=Ginoux|first=Jean-Marc|last2=Letellier|first2=Christophe|date=2012-04-30|title=Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=22|issue=2|pages=023120|doi=10.1063/1.3670008|pmid=22757527|issn=1054-1500|arxiv=1408.4890|bibcode=2012Chaos..22b3120G}}</ref>，这启发了最初的'''<font color="#ff8000">范德波尔模型 Van der Pol model</font>'''。
* 糖酵解的'''塞尔科夫模型 Sel'kov model'''<ref>{{Cite journal|last=Sel'kov|first=E. E.|date=1968|title=Self-Oscillations in Glycolysis 1. A Simple Kinetic Model|journal=European Journal of Biochemistry|language=en|volume=4|issue=1|pages=79–86|doi=10.1111/j.1432-1033.1968.tb00175.x|pmid=4230812|issn=1432-1033}}</ref>。
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* 动物基因表达、激素水平和体温的日常变化，这些都是'''<font color="#ff8000">昼夜节律 circadian rhythm</font>'''的一部分<ref>{{Cite journal|last=Leloup|first=Jean-Christophe|last2=Gonze|first2=Didier|last3=Goldbeter|first3=Albert|date=1999-12-01|title=Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora|journal=Journal of Biological Rhythms|language=en|volume=14|issue=6|pages=433–448|doi=10.1177/074873099129000948|pmid=10643740|issn=0748-7304}}</ref><ref>{{Cite journal|last=Roenneberg|first=Till|last2=Chua|first2=Elaine Jane|last3=Bernardo|first3=Ric|last4=Mendoza|first4=Eduardo|date=2008-09-09|title=Modelling Biological Rhythms|journal=Current Biology|volume=18|issue=17|pages=R826–R835|doi=10.1016/j.cub.2008.07.017|pmid=18786388|issn=0960-9822}}</ref>。
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== 编者推荐 ==
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* 癌细胞在局限微环境中的迁移遵循极限环振荡<ref>{{Cite journal|last=Brückner|first=David B.|last2=Fink|first2=Alexandra|last3=Schreiber|first3=Christoph|last4=Röttgermann|first4=Peter J. F.|last5=Rädler|first5=Joachim|last6=Broedersz|first6=Chase P.|date=2019|title=Stochastic nonlinear dynamics of confined cell migration in two-state systems|journal=Nature Physics|language=en|volume=15|issue=6|pages=595–601|doi=10.1038/s41567-019-0445-4|issn=1745-2481|bibcode=2019NatPh..15..595B}}</ref> 。
* 一些非线性电路表现出极限环振荡<ref>{{Cite journal|last=Ginoux|first=Jean-Marc|last2=Letellier|first2=Christophe|date=2012-04-30|title=Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=22|issue=2|pages=023120|doi=10.1063/1.3670008|pmid=22757527|issn=1054-1500|arxiv=1408.4890|bibcode=2012Chaos..22b3120G}}</ref>，这启发了最初的'''<font color="#ff8000">范德波尔模型 Van der Pol model</font>'''。
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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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== 另见 ==
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* [[双曲集 Hyperbolic set]]
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* [[周期点 Periodic point]]
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* [[自我振荡 Self-oscillation]]
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* [[ 稳定流型 Stable manifold]]
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==References==
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==参考文献==
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参考资料
{{Reflist}}
{{Reflist}}
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==Further reading==
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==进一步阅读==
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延伸阅读
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* {{cite book |author=Steven H. Strogatz |title=Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering |publisher=Avalon |date=2014 |isbn=9780813349114}}
* {{cite book |author=Steven H. Strogatz |title=Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering |publisher=Avalon |date=2014 |isbn=9780813349114}}
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==External links==
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==其他链接==
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外部链接
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* {{cite web |url=https://planetmath.org/limitcycle |website=planetmath.org |title=limit cycle |access-date=2019-07-06}}
* {{cite web |url=https://planetmath.org/limitcycle |website=planetmath.org |title=limit cycle |access-date=2019-07-06}}
<small>This page was moved from [[wikipedia:en:Limit cycle]]. Its edit history can be viewed at [[极限环/edithistory]]</small></noinclude>
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[[Category:待整理页面]]
2021年1月10日 (日) 21:12的版本
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稳定极限环和另外两个螺旋进入它的轨迹
范德波尔振荡器 Van der Pol oscillator的稳定极限环
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
在数学中，二维相空间 phase space 动力系统的研究中，极限环是一个在相空间中的闭合轨迹，它具有当时间趋于无穷大或时间趋于负无穷大时至少有一条其他轨迹螺旋进入的性质。这种行为在一些非线性系统中表现出来。极限环已经被用来模拟许多实际振动系统的行为。对极限环的研究是由亨利·庞加莱 Henri poincaré提出的。
↑Leloup, Jean-Christophe; Gonze, Didier; Goldbeter, Albert (1999-12-01). "Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora". Journal of Biological Rhythms (in English). 14 (6): 433–448. doi:10.1177/074873099129000948. ISSN0748-7304. PMID10643740.
