Let's just look at your Figure Eight. (Have you succeeded in publishing a paper where it is Fig. 8?) Are those perturbations which don't eject somebody periodic? Do all three bodies follow the same (slightly bent?) orbit, or is the orbit actually blurred?
If you change the masses, initial positions, and momenta slightly, you still get an approximate figure-8, sometimes with an overall precession. We think this can last forever, although I think we only know that to first order. A larger perturbation causes this precession to become more irregular, until a large enough perturbation makes the whole thing fall apart.
Does it tolerate small nonplanar perturbations?
Yes, those too.
Also, can you get enough decimal places of the aspect ratio of the unperturbed orbit to try an inverse symbolic calculator?
Good question. My own method only gets a few digits (and the fact that using the resulting initial conditions stays in an 8 for many revolutions is good evidence of stability in itself). But Carles Simo has much more precise results, with 10-20 digits of accuracy.
Neil B. showed me a picture captioned with a claim it was a quasiperiodic three-body solution, but I can't reconstruct its url. But here's a paper, QUASI-PERIODIC SOLUTIONS OF THE PLANE THREE-BODY PROBLEM NEAR EULER'S ORBITS<http://link.springer.com/article/10.1007%2FBF01230666?LI=true#page-1> that seems to affirm aperiodic immortality, which is what I was after.
Thanks for finding that paper! It mentions that Arnol'd proved the existence of some quasiperiodic orbits, presumably with the theory of KAM tori.
The idea that such a simple, bounded system can produce endless novelty is a little hard to swallow.
Well, it's highly chaotic, so as Henry says it pulls information up from the low-order bits. And chaotic systems often have very rich spectra of periodic orbits; this is true even for the logistic map in one dimension. Cris