Authors: Sergey V. Ershkov
The main motivation of the current research is the analytical exploring of the dynamics of satellite rotation during the motion on the elliptic orbit around the planet. We should discuss the revisited results of J.Wisdom, et al. (1984). By elegant change of variables (considering the true anomaly f as the independent variable), the governing equation of satellite rotation is presented in a form of the Abel ODE of the 2-nd type, a kind of generalization of Riccati ODE. We should also note that for the reason of a special character of the solutions of Riccati-type ODE, there exists a possibility for sudden jumping of the magnitude of a solution at some moment of time-parameter. In physical sense, such the jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration of the satellite rotation around the chosen principle axis at definite moment of parametric time. It means that there exists not only a chaotic regime of rotation of satellite (according to the results of J.Wisdom, et al. (1984)), but a kind of gradient catastrophe Arnold 1992 could occur during the process of satellite rotation. We should especially note that if gradient catastrophe could occur, it does not mean that it must occur: such a possibility depends on the initial conditions. Besides, the asymptotical solutions have been obtained, manifesting a quasi-periodic character of the solution even at a strong simplifying assumptions e → 0, p = 1, which reduces the governing equation of J.Wisdom, et al. (1984) to a kind of the Beletskii’s equation.
Comments: 15 Pages. AMS Subject Classification: 70F15, 70F07 (Celestial mechanics); Keywords: Beletskii’s equation, satellite rotation, Abel ODE, gradient catastrophe.
[v1] 2016-10-21 09:32:59
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