> Well, stable systems are can either be stationary or oscillatory.

In practice this is probably true, but I can see another possibility. The system could follow a trajectory that bounces around endlessly in some box without ever repeating or escaping the box.

You can treat that, and scientist often do treat it, as a stationary system with some error bounds.

For example, the concept of homeostasis in biology is like this. Lots of things are happening inside the living body, but it's still effectively at a functional equilibrium.

Similarly, lots of dynamic things are happening inside the Sun (or any star), but from the perspective of Earth, it is more or less stationary, because the behavior of the sun won't escape some bounds for billions of years.

If this box was of a bounded size then that trajectory would have interesting property - there are chunks of time you can edit out such that what remains will look as if they are converging on a point.

I suspect you will find ergodicity interesting.