On the limits of stability of ships rolling in head seas

Limits of stability are a well-known and practical way of looking into the problem of parametric resonance. In the present paper an in-depth analysis of parametric rolling is presented employing the concept of limits of stability as a tool for improved understanding and assessment of the complex dynamics embedded in the problem of non-linearly coupled parametric rolling in extreme regular seas. A third-order coupled mathematical model is considered. The coupled modes of heave, roll, and pitch are contemplated. By means of the analysis of the linear variational equation derived from the extended third-order model, the appearance of super-harmonics and increased rigidity - proportional to wave amplitude squared - due to third-order terms is demonstrated. The mathematical model is then cast in the form of a Hill's equation.Two important new aspects are addressed: the possible appearance of upper limits to the unstable area of the diagram; and the computation of numerical limits of stability. These new limits are compared with the analytical results. Subsequently, by computing the bifurcation diagrams, the dependence of the limits of stability on the initial conditions is shown. Finally, the basins of attraction corresponding to two internal regions of the domain of unstable motions are computed.