Relativistic orbits very close to black holes can produce stunning pictures, firstly because they precess around the black hole and secondly, they are are rather complex beasts.

The picture below was produced using the equations of Relativity 4 Engineers, plotted with a new color for every full orbit. The orbit 'whirls' around the black hole a few times and then 'zooms' out to the apoapsis and back again - a so-called whirl-zoom orbit. This happens when the periapsis is very close to the black hole, between two to three times the event horizon radius.

If the orbiting particle comes much closer, it will either fall into the black hole, or it will escape completely, depending on the total orbital energy of the particle.