Chaotic motion of the parametrically excited roll motion for a class of ships in regular longitudinal waves

Employed both analytical and numerical methods, chaotic motions for the parametrically excited roll motion of a ship in regular longitudinal waves are investigated in this paper. It is presented that the mechanism for chaos is the intersection of the stable and unstable manifolds of the homoclinic orbits or heteroclinic cycle. The parameter conditions for chaos are obtained rigorously with the Melnikov method. The chaotic regions of the system parameters are illustrated. The chaotic feature on the system parameters is discussed in detail. It is obtained that the critical values of chaos for homoclinic orbits increase monotonously as the increasing of the nonlinear roll attenuation coefficient, while the critical values of chaos for the heteroclinic cycle decrease monotonously as the increasing of the nonlinear roll attenuation coefficient. On the other hand, critical values of chaos for both homoclinic orbits and heteroclinic cycle increase monotonously as the increasing of the excitation amplitude. It is also demonstrated that there exist chaotic bands for this model. Numerical simulations verify the analytical results.