Ha. Lin et Scs. Yim, CHAOTIC ROLL MOTION AND CAPSIZE OF SHIPS UNDER PERIODIC EXCITATION WITH RANDOM NOISE, Applied ocean research, 17(3), 1995, pp. 185-204
A stochastic analysis procedure is developed to examine the properties
of chaotic roll motion and the capsize of ships subjected to periodic
excitation with a random noise disturbance. To take into account the
presence of randomness in the excitation and the response, a generaliz
ed Melnikov method is developed to provide an upper bound on the domai
n of the potential chaotic roll motion. The associated Fokker-Planck e
quation governing the evolution of the probability density function (P
DF) of the roll motion is derived and numerically solved by the path i
ntegral solution procedure to obtain joint probability density functio
ns (JPDFs) in state space. A chaotic response can be found in two regi
ons (near the homoclinic and heteroclinic orbits). The global behavior
of the roll motion can be depicted by the JPDF. It is found that the
presence of noise enlarges the boundary of the chaotic domains and bri
dges coexisting attracting basins in the local regimes. The attracting
domain of capsize is of the greatest strength. The probability of cap
size is considered in this paper as an extreme excursion problem with
the time-averaged PDF as an invariant measure. With this measure, the
heteroclinic region is identified as an 'unsafe' regime. Numerical res
ults indicate that, under the presence of noise, all roll motion traje
ctories of a ship that visit the regime near the heteroclinic orbit wi
ll eventually lead to capsize.