> 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).
> There are stable solutions. See: Earth’s Moon (or any other planetary moon in the solar system).
Those are not stable solutions. Remember that Earth's moon only came into existence because of a collision with a protoplanet in the past, and if a large enough body passed close by in the future, we might lose our moon -- all because of the complexity of orbital systems with more than two members.
> (or any other planetary moon in the solar system)
There are any number of examples of planets gaining and/or losing moons because of multi-body orbital complexity.
If you are presupposing external perturbations or collisions, it's not an N=3 system... we're talking about the three body problem. A tidally locked system with periodic resonance is permanently stable in the absence of external forces.