Is the three-body problem always chaotic?

Question

I was reading this recent article in Forbes about the fact that relativistic problems can't be solved exactly. In it the author makes the argument "the two body problem has an exact solution, so all Newtonian mechanics are easy".

That immediately got me thinking about the three body problem, which then led me to wonder:

1. Is the orbital mechanics of three isolated bodies in Newtonian mechanics always chaotic? Have we proved that it either is always chaotic, or that there are definitely non-chaotic cases?
2. Even if chaotic, are there cases that are bounded-output stable in the sense that if I had god-like powers I could set up a system with a sun, planet, and moon that would continue orbiting in that relationship forever without ever suffering from a collision or rearranging the relationship (i.e., the planet and moon would continue to orbit one another, even if never quite the same way twice).
3. Once I've searched on "orbital mechanics", what additional keywords would I use to research the above topics myself?

Answer 1

Briefly:

1. a) Is the orbital mechanics of three isolated bodies in Newtonian mechanics always chaotic?

No, not always. While most initial conditions lead to chaotic orbits, there are (an infinite number of) initial conditions that lead to periodic trajectories (see. e.g., the section "Special-case solutions" of the Wikipedia entry you linked).

1. b) Have we proved that it either is always chaotic, or that there are definitely deterministic cases?

Important confusion here: "deterministic" is not the opposite of "chaotic". The typical chaotic system (including the three body one) is deterministic, meaning that there's no stochasticity involved, no randomness in the model - and the same initial conditions always lead to the same outcome.

Question 2 seems to be the same as question 1.b; as for question 3, you can also look for "celestial dynamics", "gravitational dynamics" and the keywords from the resources you already have.

If you're interested in chaos, then here at the Physics SE there are lots of question and also book indications. You can find a very readable and remarkably broad summary at the Stanford Encyclopedia of Philosophy and, of course, the web is also full of resources.