斜め追波中の船の転覆について : 第4報強制マシュー型転覆方程式におけるカオスとフラクタル

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A forced Mathieu type capsize equation is examined numerically on the basis of the nonlinear dynamical systems theory. It is shown that the capsize boundary in the control parameter plane, which consists of the forcing amplitude and the forcing frequency, is varied in a complicated and fractal like way as the amplitude of the parametric excitation increases. It is also shown that the Mathieu's first (principal) unstable region is formed from a part of the second (fundamental) unstable region and is developed as the parametric excitation increases. The capsize boundaries in the other control parameter planes and the variation of the safe basin in the initial value plane are also examined. Qualitative characteristics of the solution are examined by means of the bifurcation diagrams, which are obtained by slowly increasing one of the parameters, the forcing amplitude in this paper. It is clarified that there are two kinds of the period bifurcation in the Mathieu type capsize equation. One is the ordinary type, which leads to the capsize through a cascade of the period bifurcation including the chaos. In this case the period bifurcation and the chaos shoud be regarded as the precursor of the imminent danger of the capsize. Another is the parametric type period bifurcation, which appears suddenly as a period doubling bifurcation and somtimes goes to the period quadrupling bifurcation but goes back to the fundamental solution of the period one as the forcing amplitude increases. In the latter case the period bifurcation never leads to the capsize as far as judging from the bifurcation diagram. Therefore the parametric type period bifurcation can not be regarded as the precursor of the capsize, though it is confirmed that the Mathieu's first unstable region is dangerous if the parametric excitation is large enough. Some examples of the solution including the superharmonics and super-subharmonics, which appear in the third and sixth unstable regions, are illustrated in the form of the time histories and the phase portraits. By these numerical examinations some aspects of the solution of the forced Mathieu type capsize equation are clarified. In order to clarify the totality of the global and local characteristics of the solution of the forced Mathieu type capsize equation, further numerical as well as analytical examinations should be continued.
社団法人日本船舶海洋工学会の論文

斜め追波中の船の転覆について : 第4報強制マシュー型転覆方程式におけるカオスとフラクタル

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斜め追波中の船の転覆について : 第4報 強制マシュー型転覆方程式におけるカオスとフラクタル

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斜め追波中の船の転覆について : 第4報強制マシュー型転覆方程式におけるカオスとフラクタル