A Nonlinear Model for the Rings of Saturn

This is a nonlinear mapping for a qualitative simulation of the rings of Saturn. The model is due to Fröyland, from considerations of the Roche limit for the case of the moon Mimas.

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Converting the differential equation of the model to a pair of finite-difference equations, we find
,
,
where is the radial distance from the center of Saturn to Mimas, and are the radial and angular positions of a ring particle after revolutions, and is a gravitation parameter derived from Newton's law of gravitation and Kepler's third law.
Reference
[1] R. H. Enns and G. C. McGuire, Nonlinear Physics with Mathematica for Scientists and Engineers, Boston: Birkhäuser, 2001.
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