The effect of second-order slowly varying wave drift (SVWD) forces on the horizontal plane motions of moored floating vessels has been studied for nearly 30 years. Large amplitude oscillations of moored vessels have been observed in the field or predicted numerically. Often, those have been incorrectly attributed to resonance or time-varying excitation from current/wind. In previous work, the authors have shown that resonance is only one of numerous interaction phenomena, and that large amplitude oscillations can be induced by SVWD forces or even time-independent excitation. Currently, there is no mathematical theory to study stability and bifurcations of mooring systems subjected to nonautonomous spectral excitation. Thus, in this paper, bifurcation boundaries are approximated by analyzing simulation data from a grid of points in the design space. These boundaries are plotted in the catastrophe sets of the corresponding autonomous system, for which a design methodology has been developed at the University of Michigan since 1985. This approach has revealed a wealth of dynamics phenomena, characterized by static (pitchfork) and dynamic (Hopf) bifurcations. Interaction of SVWD forces with the Hopf bifurcations may result in motions with amplitudes 2–3 orders of magnitude larger than those due to resonance. On the other hand, in other cases the SVWD/Hopf interaction may reduce or even eliminate limit cycles.