Well, they can't, because the system might still fly apart at some future point. It is possible that these are close to periodic solutions, but to make it rigorous what they need to do is show that the principle of least action is satisfied for some configuration close to the approximations. This was done by Greg Minton well before the posted paper (abstract [1]).

[1] http://jointmathematicsmeetings.org/amsmtgs/2141_abstracts/1...