This thesis is divided into three parts. In part I some theoretical and numerical processes are considered which arise when modelling the flow of a fluid through a pipe bend or deflector nozzle. These numerical processes include a new form of numerical integration and a finite element formulation which, it is suggested, could readily be extended to handle further realistic problems based on the pseudo three dimensional model chosen here. An introduction to nonlinear dynamics is included in part II leading towards a classification of bifurcational events in the light of recent advances in dynamics research. Most of the dynamical systems considered are dissipative such that the dynamic behaviour of the system decays onto a final steady state motion which may be modelled by a low order system of equations. In this way any resulting instability will adequately be described, qualitatively at least, by the low order bifurcation classified in part II. In part III the application of the geometrical theory of dynamical systems is used to study the wave driven motions of specified compliant offshore facilities with real data provided from structures currently in use in the offshore industry. In particular predictions are sought of any incipient jumps to resonance of the systems which might lead to potentially dangerous loads in the mooring lines or excessive displacements. Throughout the dynamics work stable steady state paths are closely followed and monitored so that any resulting bifurcation, including the possibility of chaotic behaviour, can be analysed with a view to its subsequent prediction.