Description of attractors and repellers
| Attractor/repellor | Description | Intuition |
|---|---|---|
| Stable fixed point | A point in a dynamical system where, if the system is slightly perturbed in any direction, it will return to that point | Like a marble at the bottom of a bowl—no matter where you nudge it, it will roll back to the bottom. Stable because it always returns to the same spot |
| Unstable fixed point | An unstable fixed point is a point in a system that repels nearby trajectories. If the system is at that point and is perturbed, it will not return to that point | Like a marble balanced on top of a dome—any tiny push will cause it to roll away. Unstable because even the smallest push will move it away |
| Saddle fixed point | A point where the system may be attracted in some directions but repelled in others | Like a marble on a horse saddle—it stays put if moved front-to-back but rolls off side-to-side. Saddle because stable in one direction but unstable in another |
| Stable limit cycle | A stable limit cycle is a repeating pattern or cycle in a system that attracts nearby trajectories. If the system is slightly perturbed, it returns to this cycle | Like a ceiling fan spinning at steady speed—even if slowed slightly, it returns to original rhythm. Stable because it settles into repeating motion |
| Unstable limit cycle | An unstable limit cycle is a repeating pattern or cycle in a system that repels nearby trajectories | If you spin a coin perfectly, slight wobble eventually causes it to fall. Unstable because the spinning motion is hard to maintain |