Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation

• Main Author: Sergey V. Ershkov
• Format: Article
• Language: English
• Published: Elsevier 2017-05-01
• Series: Theoretical and Applied Mechanics Letters
• Subjects: Beletskii’s equation; Satellite rotation; Abel ODE; Gradient catastrophe
• Online Access: http://www.sciencedirect.com/science/article/pii/S2095034917300776

The main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet. First, we revisit the results of J. Wisdom et al. (1984), in which, by the elegant change of variables (considering the true anomaly f as the independent variable), the governing equation of satellite rotation takes the form of an Abel ordinary differential equation (ODE) of the second kind, a sort of generalization of the Riccati ODE. We note that due to the special character of solutions of a Riccati-type ODE, there exists the possibility of sudden jumping in the magnitude of the solution at some moment of time. In the physical sense, this jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration in the satellite rotation around the chosen principle axis at a definite moment of parametric time. This means that there exists not only a chaotic satellite rotation regime (as per the results of J. Wisdom et al. (1984)), but a kind of gradient catastrophe (Arnold, 1992) could occur during the satellite rotation process. We especially note that if a gradient catastrophe could occur, this does not mean that it must occur: such a possibility depends on the initial conditions. In addition, we obtained asymptotical solutions that manifest a quasi-periodic character even with the strong simplifying assumptions e→0, p=1, which reduce the governing equation of J. Wisdom et al. (1984) to a kind of Beletskii’s equation.