Chaos thru period doubling

Hi, Im on something that associates with period doubling bifurcation and wanted to know: The idea of a system that demonstrates pdb, is that the period-time doubles as a function of some parameter (call it ), until the length of a single period reaches "infinity". Bascially, if I double a period-time which was initially finite, i will always get a finite time, and so in-order to reach an infinite period, the system needs to undergo infinite amount of bifurcations. My question is- which one of these options is correct for pdb: (a) Is there a certain finite value, before-which the system has an inifinite amount of bifurcations? or- (b) For every sampling resolution R, there exists a *finite* value of the parameter , such that the period that is measured in resolution R is "infinite" ? Thanks