> No physics expert but isn't this unpredictable (based on what I saw in series) ?

A three-body orbital problem is an example of a chaotic system, meaning a system extraordinarily sensitive to initial conditions. So no, not unpredictable in the classical sense, because you can always get the same result for the same initial conditions, but it's a system very sensitive to initial settings.

> Amd this does seem predictable, I saw this for almost a minute

The fact that it remains calculable indefinitely isn't evidence that it's predictable in advance -- consider the solar system, which technically is also a chaotic system (as is any orbital system with more than two bodies).

For example, when we spot a new asteroid, we can make calculations about its future path, but those are just estimates of future behavior. Such estimates have a time horizon, after which we can no longer offer reliable assurances about its future path.

You mentioned the TV series. The story is pretty realistic about what a civilization would face if trapped in a three-solar-body system, because the system would have a time horizon past which predictions would become less and less reliable.

I especially like the Three Body Problem series because, unlike most sci-fi, it includes accurate science -- at least in places.

There are stable solutions. See: Earth’s Moon (or any other planetary moon in the solar system).