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==Stable, unstable and semi-stable limit cycles==

==Stable, unstable and semi-stable limit cycles==

稳定、不稳定和半稳定极限环

稳定、不稳定和半稳定极限环

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 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).

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。

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==

==Finding limit cycles==