Reconnection involving magnetic separators is known to lead to the spontaneous generation of new separator pairs. In this work, we explore the bifurcation process for a system composed of a pair of null points with a joining separator. We begin with a simplified analytical model to derive the basic principles of bifurcation in this system and then consider models with more general separator curve geometry and generic localised null structure. We demonstrate that the maximum pairwise linking (net-winding) of the separator and the local fan plane always approaches a multiple of 0.25 just before bifurcation. Additionally, we show the integrated twisting along the separator (the field strength normalised parallel current) can be used to determine when this limit will definitely lead to bifurcation. We present step-by-step algorithms to assess how close such systems are to bifurcation.