A nonlinear single-degree-of freedom governing equation with a cubic nonlinearity to represent the yaw motion of an autopilot ship integrated with a PD controller under the influence of an external wave force was derived. The model is conditionally equivalent to the Abkowitz model, a three-degree-of-freedom nonlinear steering model. The model provides an accessible nonlinear model for steering motion of the ship adequately representing the dynamics of real ships. The nonlinearity in the model is introduced by a nonlinear function of the yaw rate into Nomoto’s second-order model which is obtained by comparing the Abkowitz model. Using the parameters of an example container ship from the literature, the nonlinear dynamics was investigated analytically and numerically.The above pictures shows marine vehicle and typical coordinate reference frames.

The harmonic balance method (HBM) was employed to study the nonlinear frequency response under the influence of external wave force, linear, nonlinear stiffness, and damping. Further, a comprehensive numerical analysis of the container ship reveals bifurcation structures that comprise multiple distinct regions with different behaviors. The governing parameters of the bifurcation structure are linear and nonlinear stiffness, damping, and excitation due to external wave force. For lower values of linear stiffness, higher values of nonlinear stiffness, and higher values of external wave force corresponding to very rough, high, and very high sea states, the bifurcation structure reveals a mixed dynamic response in which periodic solutions evolve into period doubled, period tripled, and chaotic solutions [Ref].The above pictures shows top row (a) and (b) frequency response moderate and very rough sea states, top row (c) and (d) bifurcation diagram of yaw motion for very rough and high sea states, bottom row (a), (b), (c) and (d) phase space with Poincare points of yaw motion for a very rough sea state.

[Ref] Prashant N. Kambali, C. Nataraj, Nonlinear dynamics of yaw motion of surface vehicles. Nonlinear Dyn 111, 5047–5062 (2023).