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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'''<font color="#ff8000">稳定 Stable</font>'''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)。
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'''<font color="#ff8000">吸引子 Attractor</font>'''的例子。它们意味着自我维持的'''<font color="#ff8000">振荡 Oscillation</font>''': 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，使系统保持极限环。
==Finding limit cycles==
==Finding limit cycles==
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'''<font color="#ff8000">驻点 Stationary Point</font>'''，即点<math>p</math>当有<math>V(p)=0</math>.'''<font color="#ff8000">本迪克森-杜拉克定理 Bendixson–Dulac theorem</font>'''和'''<font color="#ff8000">庞加莱-本迪克森定理 Poincaré–Bendixson theorem </font>'''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。
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'''<font color="#ff8000">希尔伯特第十六题 Hilbert's sixteenth problem</font>'''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。