Due to the movement of oil and gas exploration and production to deeper and deeper water depths, floating production systems are becoming ever more common. No longer are Column Stabilized Mobile Offshore Drilling Units (MODUs) being used only for site-temporary exploratory drilling; they are increasingly being considered for semipermanent production systems. As a result of this new purpose, analysis of their critical response when exposed to extreme winds and waves must be an ever more important aspect of their design. Unfortunately, vessel stability criteria are inadequate to study extremely large amplitude motions leading to capsizing. With respect to ships, much work has been done using dynamical systems approaches to study capsizing, and recently these same techniques have been extended and applied to study platform dynamics. This paper describes the global nonlinear dynamics of a typical MODU about its critical, approximately quartering axis. It is well-known that short and wide (i.e., small L/B ratio) twin-hull vessels such as MODUs have minimum righting moment about an approximately quartering axis. In a previous paper, Falzarano and Kota (1996) showed that this axis was also a critical rotational motion axis. In order to understand this problem, a global transient dynamical systems analysis was undertaken that compared the vessel's response at all heading angles. In the present work, an attempt is made to answer the question, What is the effect of the out-of-plane coupling? To answer this question advanced decoupling techniques known as nonlinear normal modes (Vakakis et al., 1996) are utilized. As an example, application of these techniques to the capsized MODU Ocean Ranger is also included.