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

## TL;DR

This paper analytically explores the dynamics of planar satellite rotation, revealing potential for sudden jumps and chaos in solutions of Abel equations, and introduces a revolving scheme to solve a cascade of these equations.

## Contribution

It presents a novel revolving scheme for solving a cascade of Abel equations in satellite rotation dynamics, extending previous work and analyzing solution behaviors including chaos and gradient catastrophes.

## Key findings

- Potential for sudden jumps in satellite rotation solutions.
- Existence of chaotic regimes and gradient catastrophes.
- Asymptotic solutions showing quasi-periodic behavior.

## Abstract

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 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 equation.

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Source: https://tomesphere.com/paper/1706.02990