Consideration of nonlinear effects in the dynamic analysis of offshore structures leads to complex phenomena even for relatively simple deterministic mathematical models. Nonlinearity in the stiffness or restoring force of a structure can be conveniently incorporated into a differential equation of motion, which can then be solved either analytically, using perturbation techniques, or numerically for various parameter ranges and forcing conditions. A particular feature of nonlinear dynamics is the appearance of multiple, competing steady-state oscillations which depend crucially on initial conditions. These coexisting stable solutions can be thought of as attractors for transient motion. A central question is therefore: How does the final long-term behavior of a dynamical system depend on the starting conditions? This paper concentrates on the computation of the domains of attraction or catchment regions using numerical techniques based on Poincare mapping ideas. The three specific examples illustrated are the roll motion of a ship, the oscillations of an articulated column and the surge response of a moored semi-submersible. This work may be considered an extension of previous work by the same authors.