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Zbl 1092.34028
Arnold, Ludwig; Chueshov, Igor; Ochs, Gunter
Random dynamical systems methods in ship stability: a case study. (English)
[A] Deuschel, Jean-Dominique (ed.) et al., Interacting stochastic systems. Berlin: Springer. 409-433 (2005). ISBN 3-540-23033-5/hbk

Current regulations and criteria for assuring the stability of a ship and preventing it from capsizing are empirically taylored more or less. Infact, they are based on the properties of the righting lever of the ship, taking only hydrostatic forces into account. Therefore, researchers agree that these criteria have to be modified by using hydrodynamic models of the ship-sea system, by describing the sea as a random field and by analyzing the ship as a rigid body with 6 degrees of freedom using methods of nonlinear dynamics and the theory of random dynamical systems. In this paper, the authors explain how to derive the ``archetypal'' nonlinear differential equation describing the roll motion of a ship in random seaway from basic principles and simplifications. As their basis model, the roll motion $\Phi$ of a ship is modelled by ordinary stochastic differential equations $$ \ddot{\Phi} = - U'( \Phi ) - (\gamma+b\vert \dot{\Phi}\vert )\dot{\Phi} + (\delta_1+\delta_2 \Phi ) \sin (\alpha t) + (\varepsilon_1+\varepsilon_2 \Phi) \xi_t $$ with certain potentials $U$, where $\gamma, b, \alpha, \delta_1, \delta_2, \varepsilon_1, \varepsilon_2$ are parameters and $\xi$ is some sufficiently regular, stationary stochastic process (e.g., white noise). They also give a brief introduction into the theory of random dynamical systems. Their main contribution can be seen in the presentation of an analytic and numerical case study of two simplified nonlinear models of the roll motion using some concepts of the theory of random dynamical systems (i.e., the British and Brazilian models are treated). The authors also present a mathematical proof on existence of at least one compact invariant set in the case of additive white noise by using a random Conley index theory.
[Henri Schurz (Carbondale)]

MSC 2000:: *34F05 ODE with randomness
37H10 Random and stochastic difference and differential equations
37H15 Multiplicative ergodic theory, Lyapunov exponents
37H20 Bifurcation theory
60H10 Stochastic ordinary differential equations
70L05 Random vibrations (general mechanics)
93E15 Stochastic stability
34C60 Applications of qualitative theory of ODE

Keywords: stochastic differential equations; random dynamical systems; stability of ships; random seaway; ship capsizing; roll motion; stochastic stability; stochastic bifurcation; random attractor; random invariant sets; Conley index; numerical simulation

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