When monitoring the wave-driven motions of a compliant offshore facility, be it an articulated mooring tower or a vessel, the engineer would like to be able to predict, in real time, any incipient jump to resonance that might be imminent due to the slowly evolving sea-state. We explore in this paper a study of some new possible prediction techniques for both a jump to a main fundamental resonance leading to capsize and a flip bifurcation to a subharmonic resonance. Stroboscopic Poincaré mapping techniques based on discrete time sampling are used to give information about the approach to instability. The first application of these techniques is in the prediction of the jump in resonance and consequent capsize at a cyclic fold in the roll response of a vessel in regular beam seas. Secondly, the techniques are shown to work extremely well in a variety of computational situations when applied to the simulation of an articulated mooring tower during the approach to the potentially dangerous oscillations produced by the onset of subharmonic resonance at a flip bifurcation, in both regular and irregular ocean waves.