极限环
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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
Stable limit cycle (shown in bold) for the Van der Pol oscillator
范德波尔振荡器的稳定极限环(粗体显示)
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。
Definition
We consider a two-dimensional dynamical system of the form
We consider a two-dimensional dynamical system of the form
我们考虑一个形式的二维动力系统
- [math]\displaystyle{ x'(t)=V(x(t)) }[/math]
[math]\displaystyle{ x'(t)=V(x(t)) }[/math]
数学 x’(t) v (x (t)) / 数学
where
where
在哪里
- [math]\displaystyle{ V:\mathbb{R}^2\to\mathbb{R}^2 }[/math]
[math]\displaystyle{ V:\mathbb{R}^2\to\mathbb{R}^2 }[/math]
数学 v: mathbb { r } ^ 2 to mathbb { r } ^ 2 / math
is a smooth function. A trajectory of this system is some smooth function [math]\displaystyle{ x(t) }[/math] with values in [math]\displaystyle{ \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]\displaystyle{ t_0\gt 0 }[/math] such that [math]\displaystyle{ x(t+t_0)=x(t) }[/math] for all [math]\displaystyle{ t\in\mathbb{R} }[/math]. An orbit is the image of a trajectory, a subset of [math]\displaystyle{ \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]\displaystyle{ x(t) }[/math] with values in [math]\displaystyle{ \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]\displaystyle{ t_0\gt 0 }[/math] such that [math]\displaystyle{ x(t+t_0)=x(t) }[/math] for all [math]\displaystyle{ t\in\mathbb{R} }[/math]. An orbit is the image of a trajectory, a subset of [math]\displaystyle{ \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。
Properties
By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
通过若尔当曲线定理，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。
Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching [math]\displaystyle{ + \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]\displaystyle{ + \infty }[/math]. The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching [math]\displaystyle{ - \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]\displaystyle{ + \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]\displaystyle{ + \infty }[/math]. The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching [math]\displaystyle{ - \infty }[/math], and also for trajectories in the exterior approaching the limit cycle.
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。
Stable, unstable and semi-stable 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).
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)。
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.
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.
稳定极限环是吸引子的例子。它们意味着自我维持的振荡: 闭合轨迹描述了系统的完美周期行为，任何来自这个闭合轨迹的微小扰动都会导致系统返回到它，使系统坚持到极限环。
Finding limit cycles
Every closed trajectory contains within its interior a stationary point of the system, i.e. a point [math]\displaystyle{ p }[/math] where [math]\displaystyle{ 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]\displaystyle{ p }[/math] where [math]\displaystyle{ 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。
Open problems
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]\displaystyle{ x'=V(x) }[/math] in the plane where both components of [math]\displaystyle{ 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]\displaystyle{ x'=V(x) }[/math] in the plane where both components of [math]\displaystyle{ 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。
Applications
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.
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:
极限环在许多具有自持振荡系统的科学应用中是重要的。一些例子包括:
- Aerodynamic limit-cycle oscillations[1]
- The Hodgkin–Huxley model for action potentials in neurons.
- The Sel'kov model of glycolysis.[2]
- The daily oscillations in gene expression, hormone levels and body temperature of animals, which are part of the circadian rhythm.[3][4]
- The migration of cancer cells in confining micro-environments follows limit cycle oscillations.[5]
- Some non-linear electrical circuits exhibit limit cycle oscillations,[6] which inspired the original Van der Pol model.
- Charged extended bodies describe limit cycle oscillations as a consequence of electromagnetic self-interactions.[7]
See also
References
- ↑ Thomas, Jeffrey P.; Dowell, Earl H.; Hall, Kenneth C. (2002), "Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations" (PDF), AIAA Journal, American Institute of Aeronautics and Astronautics, 40 (4): 638, Bibcode:2002AIAAJ..40..638T, doi:10.2514/2.1720, retrieved December 9, 2019
- ↑ Sel'kov, E. E. (1968). "Self-Oscillations in Glycolysis 1. A Simple Kinetic Model". European Journal of Biochemistry (in English). 4 (1): 79–86. doi:10.1111/j.1432-1033.1968.tb00175.x. ISSN 1432-1033. PMID 4230812.
- ↑ 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. ISSN 0748-7304.
- ↑ Roenneberg, Till; Chua, Elaine Jane; Bernardo, Ric; Mendoza, Eduardo (2008-09-09). "Modelling Biological Rhythms". Current Biology. 18 (17): R826–R835. doi:10.1016/j.cub.2008.07.017. ISSN 0960-9822.
- ↑ Brückner, David B.; Fink, Alexandra; Schreiber, Christoph; Röttgermann, Peter J. F.; Rädler, Joachim; Broedersz, Chase P. (2019). "Stochastic nonlinear dynamics of confined cell migration in two-state systems". Nature Physics (in English). 15 (6): 595–601. Bibcode:2019NatPh..15..595B. doi:10.1038/s41567-019-0445-4. ISSN 1745-2481.
- ↑ Ginoux, Jean-Marc; Letellier, Christophe (2012-04-30). "Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept". Chaos: An Interdisciplinary Journal of Nonlinear Science. 22 (2): 023120. arXiv:1408.4890. Bibcode:2012Chaos..22b3120G. doi:10.1063/1.3670008. ISSN 1054-1500. PMID 22757527.
- ↑ López, Álvaro G. (2020-01-22). "On an electrodynamic origin of quantum fluctuations". arXiv:2001.07392 [quant-ph].
- "limit cycle". planetmath.org. Retrieved 2019-07-06.
- Steven H. Strogatz (2014). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Avalon. ISBN 9780813349114.
- M. Vidyasagar (2002). Nonlinear Systems Analysis (Second ed.). SIAM. ISBN 9780898715262.
- Philip Hartman, "Ordinary Differential Equation", Society for Industrial and Applied Mathematics, 2002.
- Witold Hurewicz, "Lectures on Ordinary Differential Equations", Dover, 2002.
- Solomon Lefschetz, "Differential Equations: Geometric Theory", Dover, 2005.
- Lawrence Perko, "Differential Equations and Dynamical Systems", Springer-Verlag, 2006.
- Arthur Mattuck, Limit Cycles: Existence and Non-existence Criteria, MIT Open Courseware http://videolectures.net/mit1803s06_mattuck_lec32/#
Category:Limit sets
类别: 极限集
Category:Nonlinear systems
类别: 非线性系统
Category:Dynamical systems
类别: 动力系统
This page was moved from wikipedia:en:Limit cycle. Its edit history can be viewed at 极限环/edithistory