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

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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.

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作者简介

A. Chernyshov

Nizhny Novgorod State Technical University n. a. R.E. Alekseev

编辑信件的主要联系方式.
Email: andrey.chernyshov5@gmail.com
俄罗斯联邦, Nizhny Novgorod, 603155

S. Chernyshova

Nizhny Novgorod State Technical University n. a. R.E. Alekseev

Email: sofya.chernyshova99@gmail.com
俄罗斯联邦, Nizhny Novgorod, 603155

参考

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  7. A.V. Chernyshov and S.A. Chernyshova, “A method of investigating the phenomenon of bifurcation memory in the dynamics of river ships,” Russ. J. Nonlin. Dyn. 18 (2), 171–181 (2022). https://doi.org/10.20537/nd220202
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2. Fig. 1. “Phase spots" (V) on the static-dynamic plane OωU. The presence of “phase spots” is a criterion for the manifestation of the “bifurcation memory” effect; UM is the minimum value of the parameter at which the image point no longer falls into the “phase spot”, UBIF is the bifurcation value of the parameter.

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3. Fig. 2. The absence of the "bifurcation memory" effect: т.Fm – the point of intersection of the trajectory l of the accelerated motion area (A) towards the slow motion area (D).

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4. Fig. 3. The effect of “bifurcation memory”: t.F1 and t.Fm are the points of intersection of the trajectory l with the boundary of the region of accelerated motions (A) towards the region of slow motions (D); Ui ∈[UBIF,Umax]. Region C is the region of initial conditions that limit the application of the method [٧].

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5. Fig. 4. The phase velocity hodograph built for the vessel Volgoneft-71 at U = 5°, . The arrows show the direction of movement of the radius vector over time. Initial conditions: point.

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6. Fig. 5. The phase velocity hodograph built for the Volgoneft-71 vessel at U = UM (UM = 22°). The arrows show the direction of movement of the radius vector over time. The initial conditions correspond to the point .

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7. Fig. 6. The phase velocity hodograph built for the vessel Volgoneft-71 at U = 5°, . The arrows show the direction of movement of the radius vector over time. Initial conditions: point.

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8. Fig. 7. Object: Volgoneft-71. A phase plane with regions (A) and (D) (method [7]), isolines of phase velocities , , (hodograph method of phase velocity); l is the phase trajectory. Initial conditions: point ; U = 5°, .

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