In the case of Alpha Centauri A&B and Proxima Centauri, A&B orbit each other at elliptical orbits, getting almost as close as the Sun is to Saturn and about as far as the Sun is to Neptune. Proxima Centauri is about 1,000 times further out and it's a tiny compared to the inner 2, so for simulation purposes, it's essentially a 2 star system with a far distant 3rd orbiting the inner 2. The 3rd star is so far and comparatively small that it has essentially zero effect on the simulation.
Source of Photo
Now, in general, most 3 or more star systems operate in similar ways where you can get a 1 or 2 stars operating as the center of mass and then you need to extend a fair bit of distance for the 3rd star. 3 stars all about the same distance and about the same size can be set up on a computer, but it's unlikely to happen in reality cause instabilities tend to grow.
The Sun-Earth-Moon is effectively a stable 3 body system and not subject to 3-body chaos because the Moon is safely within the stable region of the Earth's Hill Sphere and those 3 objects have clear cut hierarchy in terms of gravitational influence.
Pluto, Neptune& the Sun, for example, is probably long term unstable. See here
This is Wikipedia's 3 body simulation, but this is probably not something you'd see in solar-systems because of the inherent instability. In such a 3 body orbital system, one of the 3 bodies would likely either be ejected or 2 of the 3 bodies would collide, creating a more stable 2 body system. It wouldn't have long term stability.
You could potentially get a 3 body system where the 2 outer bodies are in Trojan points to each other (L4 or L5 only). I'm not sure how common that is but I'd think it happens occasionally, provided the central star had significantly more mass than the 2 outer stars.
L4 and L5 Trojans