There's a name for objects in "toroidal orbits" - Hill's equations, in the "Hill's frame."
Burns, R., McLaughlin, C., Leitner, J., & Martin, M. (2000). TechSat 21: formation design, control, and simulation. In Aerospace conference proceedings, 2000 IEEE (Vol. 7, pp. 19-25).
shows the three equations below, and claims they are found here:
- Hill, George William. "Researches in the lunar theory." American journal of Mathematics 1, no. 1 (1878): 5-26.
but I don't see them in that three equation form. I suspect these are versions from a more recent paper.
\ddot x - 2 \omega \dot y - 3 \omega^2 x ~=~ f_x ~~~~ x is the radial direction
\ddot y + 2 \omega \dot x ~=~ f_y ~~~~~~~~~~~~~~~ y is the orbital direction
\ddot z + \omega^2 z ~=~ f_z ~~~~~~~~~~~~~~~~ z is perpendicular to the orbital plane. Right handed triad mentioned in Burns, the equation works with left triad, too.