On Mar 15, 2013, at 10:42 PM, Dan Asimov wrote:
Does the approximate figure-8 behave like the original one (up to homeomorphism of a neighborhood of its orbit in phase space)?
I guess a homeomorphism here means that the topology remains the same... that is, if you take (R^2)^3 and remove the points where two masses collide, the resulting space has a fundamental group, and a periodic orbit is an element in that group. (By removing a few degrees of freedom this space looks like a sphere with three holes.) When you perturb it, though, there is often an overall rotation, so you have to go into a rotating frame to make the topology the same again. (It's the difference between a 3-straid braid, and a 3-strand braid with some overall twists to it.)
If so, can it be that it belongs to some kind of hyperbolic set with at least local structural stability?
I _think_ so. But like I said, I don't thing anything is known rigorously beyond the first-order stability. Cris
--Dan
On 2013-03-15, at 5:41 PM, Cris Moore wrote:
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.
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