Study On Multistable Control And Bifurcation Of The Ship’s Dynamic System

There are abundant and complex dynamic phenomena in ship’s dynamic system.Due to the nonlinear factors such as recovery moment and damping moment,the ship presents nonlinear dynamic characteristics when sailing.These nonlinear dynamic characteristics will directly or indirectly affect the stability and safety of navigation.Therefore,in order to improve the stability of ship motion and ensure the safety of ship sailing,it is necessary to study and analyze various nonlinear dynamic behaviors of ship’s dynamic system.In this article,the main work is as follows.Firstly,the motion of the ship in three-dimensional space is simplified,and only the angular motion of the ship is considered.By introducing nonlinear damping moment and recovery moment,the differential equation of rolling motion with one-degree-of-freedom is established by D’Alembert principle.In addition,considering the coupling of roll and pitch,the Tail-Bryan Angle was used to describe the angular motion of the ship,and a two-degreeof-freedom coupled differential equation of roll and pitch was established by Euler dynamics equation.Secondly,considering the ship roll model with single-degree-of-freedom,the multistable dynamics of the system is revealed by numerical methods such as bifurcation diagram,phase diagram and attraction basin.We use the intermittent control and the constrained intermittent control low to control the multistable phenomena of the system,and a positive definite Lyapunov function is constructed to prove the stability of the control method.The results show that the periodic solution of the system can be controlled to a relatively stable region without changing the main parameters of the system,so as to improve the stability of nonlinear rolling motion of the ship.Then,considering the ship’s two-degree-of-freedom coupled of roll and pitch model,the local Hopf bifurcation is studied.The perturbation of the system is analyzed by using the method of multiple scale and the stability of the equilibrium solution is discussed.Then we use the projection method to study the Hopf bifurcation of equilibrium and evaluate the stability of Hopf bifurcation by the first Lyapunov coefficient.Numerical simulation with Matcont software shows that there is a stable limit cycle in the neighborhood of the supercritical Hopf bifurcation point,and the two sides of the subcritical Hopf bifurcation point are stable focus and unstable focus respectively.Finally,the global bifurcation behavior of the ship’s two-degree-of-freedom coupled of roll and pitch model is studied.We obtain the average equation of the ship’s dynamic system by analytic method,and then transform it into an approximately integrable Hamiltonian system by regular transformation.We analyze the phase space structure of the unperturbed system,then derive the Melnikov function of the system according to the high dimensional Melnikov method.When Melnikov function has a simple zero,the system has chaos in the sense of Smale horseshoes,and it is verified by numerical simulation.