From what I understand, in bifurcation theory, one definition of the equivalence of two dynamical systems is that they are topologically equivalent. However if say proteins A, B start out as straight lines in terms of it's geometry and then fold into parabolas except A has a much larger focus than B (so effectively being much more curved).

This is clearly a case we want to distinguish between that topological equivalence won't. One might consider RMSD (root mean square deviation) but then one has to apply a cutoff which seems arbitrary.

My question is that is what are the equivalence notions usually used when talking about bifurcations of n-body systems in general and proteins in particular