Stochastic Hopf bifurcation and random chaos of the ship rolling dynamic system in random longitudinal wave induced by GWN

Abstract

In this paper, the dynamic system of ship rolling in random longitudinal waves induced by Gaussian white noise (GWN) is studied analytically and numerically. The dynamic equation of this model which is a quintic extended Duffing system with cubic damping as well as parametric and random excitations is obtained with the Li’s modeling method. The basic dynamic properties including equilibria, potential wells, and classical solutions are derived using analytical methods and their parameter bifurcations are discussed. Stochastic Hopf bifurcation is investigated by using the three-indexes method and the stochastic dynamic theory. It is found that there is a security threshold and a risk factor affecting the traveling of the ship in random waves. Random chaos is studied by random Melnikov method. The chaotic thresholds of some applicable homoclinic orbits and heteroclinic orbits are given. Numerical simulations including phase portraits, time histories, basin of attraction, Poincaré maps, Lyapunov spectra and bifurcation diagrams are given, which verify the analytical results. Utilizing numerical methods, it is also found that the system can experience random chaotic states through random Hopf bifurcation.