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第1行： − 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。+ 第25行：
第25行： − 在数学上，在二维相空间动力系统的研究中，极限环是一个在相空间中具有至少一个其他轨迹在时间趋于无穷大或时间趋于负无穷大时螺旋进入的闭合轨迹。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 · 庞加尔(Henri poincar，1854-1912)提出的。+ 第41行：
第41行： − 数学 x’(t) v (x (t)) / 数学+ 第53行：
第53行： − 数学 v: mathbb { r } ^ 2 to mathbb { r } ^ 2 / math+ 第59行：
第59行： − 是一个平滑函数。这个系统的轨迹是一个平滑的函数数学 x (t) / math，它的数值是 mathbb { r } ^ 2 / math，满足这个微分方程。如果这种轨迹不是恒定的，而是返回到它的起点，那么这种轨迹称为闭合(或周期)轨迹。如果存在一些数学 t00 / math，比如 mathbb { r } / math 中的所有数学 t 的数学 x (t + t 0) x (t) / math。轨道是轨道的图像，是 math mathbb { r } ^ 2 / math 的子集。一个闭合轨道，或循环，是一个闭合轨迹的图像。极限环是一个循环，它是其他轨迹的极限集。+ 第77行：
第77行： − 给定一个极限环及其内部接近时间趋近于数学极限环的轨迹，然后在极限环周围有一个邻域，这样内部所有从邻域开始的轨迹都趋近时间趋近于数学极限环。相应的陈述适用于内部接近极限环的轨迹，接近时间接近数学推导的轨迹，也适用于外部接近极限环的轨迹。+ 第87行：
第87行： − 当时间趋于无穷大时，所有相邻轨迹都接近极限环时，称之为稳定或吸引极限环(- 极限环)。相反，当时间趋近于负无穷时，所有相邻轨线都趋近于它，则它是一个不稳定的极限环(- 极限环)。如果存在一个相邻的轨迹，当时间趋于无穷大时螺旋进入极限环，另一个轨迹在时间趋于负无穷大时螺旋进入极限环，那么它是一个半稳定的极限环。还有一些既不稳定、不稳定也不半稳定的极限环: 例如，一个相邻的轨迹可能从外部接近极限环，但是极限环的内部是由一系列其他的极限环逼近的(不会是极限环)。+ 第105行：
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第115行： − + 第121行：
第121行： − + − + 第133行：
第133行： − 极限环在许多具有自持振荡系统的科学应用中是重要的。一些例子包括:+ 第143行：
第143行： − + − − *Charged extended bodies describe limit cycle oscillations as a consequence of electromagnetic self-interactions.<ref>{{cite arxiv|last=López|first=Álvaro G.|date=2020-01-22|title=On an electrodynamic origin of quantum fluctuations|eprint=2001.07392|class=quant-ph}}</ref> 第171行：
第169行： − * {{cite web |url=https://planetmath.org/limitcycle |website=planetmath.org |title=limit cycle |access-date=2019-07-06}}+ + + + + 第186行：
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Moved page from wikipedia:en:Limit cycle (history)
此词条暂由彩云小译翻译，翻译字数共657，未经人工整理和审校，带来阅读不便，请见谅。
{{short description|Behavior in a nonlinear system}}
{{short description|Behavior in a nonlinear system}}
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).
在数学上，在二维相空间动力系统的研究中，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 Henri poincaré (1854-1912)提出的。
<math>x'(t)=V(x(t))</math>
<math>x'(t)=V(x(t))</math>
X’(t) = v (x (t)) </math >
where
where
<math>V:\mathbb{R}^2\to\mathbb{R}^2</math>
<math>V:\mathbb{R}^2\to\mathbb{R}^2</math>
2 to mathbb { r } ^ 2 </math >
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 (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.
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.
是一个平滑函数。这个系统的轨迹是满足这个微分方程的光滑函数。如果这个轨迹不是恒定的，而是返回到它的起始点，那么这个轨迹称为闭合(或周期)轨迹。如果存在一些 < math > t _ 0 > 0 </math > 这样的 < math > x (t + t _ 0) = x (t) </math > t 在 mathbb { r } </math > 。轨道是轨道的图像，是 < math > mathbb { r } ^ 2 </math > 的子集。一个闭合轨道，或循环，是一个闭合轨迹的图像。极限环是一个循环，它是其他轨迹的极限集。
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.
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.
给定一个极限环和它内部接近极限环的轨迹，在时间上接近 < math > + infty </math > ，然后在极限环周围有一个邻域，这样所有内部开始的轨迹在时间上接近 < math > + infty </math > 时都接近极限环。相应的陈述适用于接近极限环的内部轨道，接近极限环的时间接近 < math >-infty </math > ，也适用于接近极限环的外部轨道。
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).
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).
当所有相邻轨迹在时间趋于无穷大时都接近极限环时，称之为稳定或吸引极限环(ω- 极限环)。当时间趋近于负无穷时，若所有相邻轨线都逼近它，则它是一个不稳定的极限环(α- 极限环)。如果存在一个相邻轨迹，当时间趋于无穷大时螺旋进入极限环，另一个轨迹在时间趋于负无穷大时螺旋进入极限环，那么它是一个半稳定的极限环。还有一些既不稳定、不稳定也不半稳定的极限环: 例如，邻近轨迹可能从外部接近极限环，但极限环的内部是由一组其他的极限环逼近的(不会是极限环)。
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.
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.
每一个闭合轨迹在其内部都包含一个系统的驻点，即。A point math p / math where math v (p)0 / math.本迪克森-杜拉克定理和庞加莱-本迪克森定理分别预言了二维非线性动力系统极限环的缺失或存在。
每一个闭合轨迹在其内部都包含一个系统的驻点，即。A point < math > p </math > where < math > v (p) = 0.本迪克森-杜拉克定理和庞加莱-本迪克森定理分别预言了二维非线性动力系统极限环的缺失或存在。
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.
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.
一般来说，寻找极限环是一个非常困难的问题。平面上一个多项式微分方程的极限环的个数是 Hilbert 第十六题第二部分的主要对象。例如，在平面上是否存在系统数学 x’ v (x) / 数学，其中数学 v / 数学的两个组成部分都是两个变量的二次多项式，这样系统就有多于4个极限环。
一般来说，寻找极限环是一个非常困难的问题。平面上一个多项式微分方程的极限环的个数是 Hilbert 第十六题第二部分的主要对象。例如，在平面上是否存在一个系统 < math > x’ = v (x) </math > ，其中 < math > v </math > 的两个组成部分都是两个变量的二次多项式，因此该系统有多于4个极限环。
== Applications ==
== Applications ==
[[File:Hopfbifurcation.png|thumb|400px|Examples of limit cycles branching from fixpoints 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.]]
[[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.]]
Examples of limit cycles branching from fixpoints 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.]]
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.]]
极限环从[[霍普夫分岔]附近的不动点分支的例子。红色的轨迹，深蓝色的稳定结构，浅蓝色的不稳定结构。参数的选择决定了极限环的出现和稳定性
极限环从[[霍普夫分岔]附近的不动点分支的例子。红色的轨迹，深蓝色的稳定结构，浅蓝色的不稳定结构。参数的选择决定了极限环的出现和稳定性
Limit cycles are important in many scientific applications where systems with self-sustained oscillations are modelled. Some examples include:
Limit cycles are important in many scientific applications where systems with self-sustained oscillations are modelled. Some examples include:
极限环在许多自持振荡系统的科学应用中是重要的。一些例子包括:
* 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>
* 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>
* 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|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|issn=0960-9822}}</ref>
* 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>
* 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>
* 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>
* 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]].
* 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]].
{{Reflist}}
{{Reflist}}
==Further reading==
* {{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}}
* Arthur Mattuck, Limit Cycles: Existence and Non-existence Criteria, MIT Open Courseware http://videolectures.net/mit1803s06_mattuck_lec32/#
* Arthur Mattuck, Limit Cycles: Existence and Non-existence Criteria, MIT Open Courseware http://videolectures.net/mit1803s06_mattuck_lec32/#
==External links==
* {{cite web |url=https://planetmath.org/limitcycle |website=planetmath.org |title=limit cycle |access-date=2019-07-06}}