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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'''<font color="#ff8000">数学 Mathematics</font>'''上，二维'''<font color="#ff8000">相空间 Phase Space</font>''' '''<font color="#ff8000">动力系统 Dynamical Systems</font>'''的研究中，'''极限环'''是一个在相空间中的闭合'''<font color="#ff8000">轨迹 Trajectory</font>'''，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'''<font color="#ff8000">非线性系统 Nonlinear Systems</font>'''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。

在'''<font color="#ff8000">数学 Mathematics</font>'''上，二维'''<font color="#ff8000">相空间 Phase Space</font>''' '''<font color="#ff8000">动力系统 Dynamical Systems</font>'''的研究中，'''极限环'''是一个在相空间中的闭合'''<font color="#ff8000">轨迹 Trajectory</font>'''，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'''<font color="#ff8000">非线性系统 Nonlinear Systems</font>'''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>'' (1854-1912)提出的。

where

where

:<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>

<math>V:\mathbb{R}^2\to\mathbb{R}^2</math>

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。

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>有<math>x(t+t_0)=x(t)</math>t对于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。