In this work, we utilize bifurcation diagrams to study the role of mathematical artifacts in deteriorating the physical behavior in statistical mechanically based equations of state of pure fluids. We study the impact of common empirical approximations usually employed to overcome some of the mathematical and physical challenges such as the parametrization of mean field models or pair correlations functions at contact. The proposed diagrams elucidate how the reduced molar volume bifurcates with the variation of temperature at constant pressure. We generate bifurcation diagrams for the modified van der Waals equation of state (EOS) of Poole et al, SAFT-VR Mie, Soft-SAFT, CK-SAFT, and the original SAFT EOSs for spherical and nonspherical molecules. We find that the bifurcation diagram can serve as a useful schematic tool to reveal the unphysical PVT behavior, demonstrate the existence of physical and spurious two-phase separation regions, and illustrate how the number of molar volume roots vary with temperatures. Our method shows that the presence of unphysical branches can cause spurious two-phase separation regions in the stability limit of vapor-liquid equilibrium. We demonstrate that the existence of customary and spurious phase envelopes is accompanied by S-shaped behavior in the volume-temperature bifurcation diagrams. The study reveals that none of the SAFT models is free from producing unphysical behavior. While the SAFT-VR Mie EOS exhibits solid-liquid-like behavior for nonspherical molecules, the CK-SAFT EOS shows liquid-liquid demixing behavior for spherical and nonspherical compounds. For the soft-SAFT EOS, three different two-phase separation regions are observed in addition to the common vapor-liquid phase separation region.