What shape do satellites trace as they wobble in orbit around another body?

What shape does the center of gravity of a satellite trace in 3-d space as it wobbles in orbit around another body? How is this shape modeled geometrically? Is it a topological torus? Is it feasible/useful to cross-section this shape at a right angle to the orbital path and graph the probability distribution of pathways in 2-d within the cross-section? Has this been done for the Moon around the Earth for example? That's a lot of questions but I think they're all closely related and the point is to obtain a conceptual model of orbit paths.

EDIT: The answers diverge so let me specify the question further. I have a vague recollection of the application of Lorenz Attractors to planet satellite paths. (This problem set is by the way one of the historical bases of chaos theory.) However, Lorenz Attractors are generally represented as full orbits in 3-d space, like so:

I'm trying to get to the 2d representation, if it exists, i.e. a 2d cross-section of the orbit paths, which probably do not fit in a disc, but in a torus, unlike the above image generated by a simple model. Might this cross-section look like a Hénon map?

• The shape depends on the reference system of choice. Which are you referring to? The satellite, another space craft, the moon, the sun, Earth, galactic centre,...? Each makes sense to some degree. – planetmaker Oct 23 '20 at 7:48
• Are you thinking of something like this ? en.wikipedia.org/wiki/… – usernumber Oct 23 '20 at 8:27
• @usernumber Yes something like this, but more detailed about the distributional patterns in 2-d sections over time. I remember reading about 30 years ago that these patterns were fractal but I may be mistaken. – syre Oct 23 '20 at 8:47
• @planetmaker Basically moons around planets and planets around stars. – syre Oct 23 '20 at 8:48
• I don't think there is a generic answer, since different uneven objects cause different kinds of wobbles. You can get torus-shaped orbits around torus-shaped heavy objects, but for more plausible convex bodies the answer is basically Keplerian motion plus precession terms due to the different spherical harmonics, making the orbit a rosette, a wase or a chaotic bundle (maybe what you would call a torus). – Anders Sandberg Oct 23 '20 at 10:11