Capsizing of a Ship in Quartering Seas

In the first paper, the authors pointed out that a new mode of capsizing, which accompanied the period doubling bifurcation phenomenon, had been observed and that about 25 % of the total capsizing had been classified into such a new capsizing mode. Since the period doubling bifurcation phenomenon was regarded as a precursor of the chaos in the nonlinear dynamical systems, and the chaos was regarded as a precursor of the unconditional capsizing, which was equivalent to the blue sky catastrophe or the boundary crisis in the softening spring system, the chaos and fractal in the symmetrical capsize equation was examined in the authors' second paper. Although the mysterious nonlinear phenomena appeared in the symmetrical capsize equation were examined numerically, a cascade of the period bifurcations was not observed in the numerical study.In the present paper, the authors have examined the asymmetrical capsize equation as well as the symmetrical one. As a result, the cascade of the Feigenbaum bifurcation, which begins from the period doubling bifurcation and ends at the unconditional capsizing through the chaos, has been observed for both asymmetrical and symmetrical capsize equations. Further it has been clarified that in the symmetrical equation the cascade of the period bifurcations begins after the break of the symmetry of solution, and that the period bifurcation with the distinct difference of the adjoining amplitude, such as observed in the model tests, appears only in the asymmetrical equation. The differences of the fractal capsize boundaries in the initial value plane and the control space between the symmetrical and asymmetrical equations have been also examined, and the danger of the biased ship is emphasized by the numerical results. In the examination of the safe basin in the initial value plane, a cell-to-cell mapping method developed by Hsu has been atempted to confirm its effectiveness and accuracy. A numerical analysis of the fold bifurcation which approximates the capsizing boundary in the control space, and also a numerical Melnikov analysis which approximates the beginning of the fractal metamorphosis in the safe basin in the initial value plane have been carried out. An approximate analytical expression for the flip bifurcation is obtained.The idea that the appearance of the period bifurcation in the roll motion of a ship implies the imminent danger of the capsizing has proved to be correct.