Editing Limit cycle (section)

==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 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 would not be limit cycles).

Stable limit cycles are examples of [[attractor]]s. 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.