anyone willing to provide a math-proof like argument on why the shape seem to stick to the YY curve indefinitely as the "eternal" name suggests?
Should it always be this way or is there at least one bad initial bouncing configuration for which chaos can take place and we loose the YY curve?
Does not seem that obvious to me.
I'm not even a dimwitted individual with an advanced degree in hyperbolic topology, but I can see what's happening intuitively. When one of the balls makes an indent large enough, that indent focusses the bounce from the circular edge which reinforces the indent further. This leads to a semi-stable shape where one of the balls is bouncing around a horseshoe and the other in a tunnel. However, if one side of the horseshoe becomes pinched small enough that ball is less likely to enter, that side of get eliminated, and you have a yin-yang.
More simply, the round edge seems to encourage tunnelling, and any asymmetry in the tunnelling is yin-yang-ish.