Improving the Phase Plane Method to Study the Influence of the “Bifurcation Memory” Effect on Ship Dynamics

The study of the “bifurcation memory” effect plays an important role in the study of dynamic features of real systems. Practical interest lies in studying the possibility of predicting a temporary decrease in response to control, which can significantly improve navigation safety. The effect of “bifurcation memory” is a temporary decrease in the phase velocity of the imaging point when passing through a certain area (“phase spot”) on the phase plane. A “phase spot” appears near the equilibrium state that disappeared during bifurcation. Over the almost half-century history of studying this dynamic feature, very few methods have been proposed that make it possible to unambiguously and with sufficient accuracy identify the “bifurcation memory” effect. This article proposes an improved phase plane method, which consists in constructing a phase velocity hodograph. A distinctive feature of the proposed method is not only that it surpasses previously developed methods in accuracy, but also covers both phase coordinates, and also gives an adequate result for any initial conditions. The method is quite universal and can be used to study the effect of “bifurcation memory” in various dynamic systems. Information about the boundary values of the parameter - the rudder angle, at which the effect of “bifurcation memory” begins (ends) to manifest itself can be used, for example, in the problem of optimizing the design of the hull and rudders or when creating control algorithms.