Every path leads to the same loop.

What you're seeing

Two dots, same equation, same rule, nothing else touching either one. The amber dot starts almost exactly at the center. The cyan dot starts wherever you drag its slider to — right next to the amber one, or way out at the corner. At zero pull, that is the whole story forever: each dot moves along its own closed circle, keeping whatever size it started at, and the two never come any closer together. Nothing pulls a big loop down to a small one, or a small loop out to a big one. That is not a limit cycle — it's just the plain harmonic oscillator, one closed orbit for every possible starting size, all equally stable, all permanently different.

Now turn the pull up. A single rule switches on: push outward whenever a dot is inside a certain curve, brake it whenever it's outside that same curve. Inside the curve the "damping" term is negative — it adds energy; outside it, the same term flips sign and removes energy. A dot that starts small gets pushed out. A dot that starts huge gets reeled in. There is exactly one curve where the two effects balance over a full lap, and every dot, from every starting point, ends up tracing it. Watch the amber and cyan trails long enough and they stop being two different shapes — they become the same shape, retraced by two dots that simply arrived from opposite directions.

Push the pull dial further and the loop itself changes character: from a near-circle at low pull into a stretched, lopsided racetrack — a slow crawl along one side, then a fast snap across to the other. That slow-crawl-fast-snap shape is a relaxation oscillation, and it's why this equation turns up everywhere from vacuum-tube circuits to heartbeats: it describes anything that charges up slowly and discharges in a rush, over and over, always finding its way back to the same rhythm no matter how it was disturbed.