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.

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.