So the computer based solution proposed by Newton and Turing is rejected because the system is chaotic, meaning a general solution will always diverge from reality.
What I don't get is: this should still be good enough Run your solution every month or so, with updated measurements, and you'll have an ongoing "forecast" of conditions.
I'm referencing weather because that's we do. A weather forecast is a prediction of a chaotic system, but of one which changes every day or so. Prediction difficulty is dependent on local conditions and weather type, but we can still make predictions.
A gravitational system of 4 mutually interacting bodies is muuuuuch simpler than weather, and could be predicted far enough in advance to let a civilization adapt and persist!
Well, alpha centauri is a 3 star system in reality. 2 sun like stars orbit each other and a red dwarf orbits 13000 AU away from them, which is really far away, but happens because the red dwarf is so much less massive than the other two. The two stars will be orbiting close to their center of mass and barely feel the gravity of the red dwarf. So the center of the star system is the pair and planets would orbit them. Nasa says the biggest worry fo life in this system would be the x-rays emissions by proxima centauri (the red dwarf).