So by chaotic you just mean that we can expect that a million years from now the Earth will be in a very similar orbit to what it is in now, but we couldn’t predict exactly where it will be in that orbit, or it might be a bit higher orbit or a bit lower orbit. And the greater the masses of the other bodies besides the sun the more chaotic the orbit will get…to the point were eventually the planet will be ejected or impact one of the stars. And “eventually” means over 10s or 100s of millions of years.
This doesn’t follow. Planets have to maintain constant angular momentum in a single-star system because the potential is central (the force is directed radially and so cannot change the angular momentum). For a binary system this is no longer true; the forces are not all radially directed and so the planet’s angular momentum is not conserved. This means that the centrifugal pseudopotential is not a very useful model (the height of the centrifugal barrier is no longer constant).
The problem with figure-8 orbits is not one of angular momentum conservation; it’s just one of stability.
Here (PDF) is an interesting three-body orbit, with three equal masses following each other around a single figure-8 trajectory. I haven’t played with it lately, but the authors claim that it is stable.
I believe this is an animation of the result described in your link.
Oops, missed your earlier link. Yeah, same result.
That’s a pretty good sketch. If the Earth starts in one place with one momentum, everything’s cool, but move it a centimeter to the left and things go haywire in a hundred million years. This is the true meaning of the “Butterfly Effect”: two systems that differ by a small amount eventually diverge in their behaviors to differ by a huge amount.