Now, each orbit is confined to a plane, so the plots will be easier to interpret. The number under the energy is the mean radius of particles in that slice (but in the first plot, the radius didn't print out). The fiducial number of timesteps is 40000, but, each energy slice is integrated for a longer time than the previous energy slice, and since the energy slices are small in this set, you'll see more orbit crossings for less bound orbits. The number of orbit crossings in each slice should be more evenly distributed when we increase the depth of the slices.

By simply increasing the particle number in the spherical model, you can start to see loops, though with significant scatter.

The most startling change in the accuracy of the orbits results from 8-folding the coefficients. See below:

Notice how clean all the orbits are...and they're all loops like we should expect.

How accurate are these orbits? Compare the previous plot of an 512K, 8-folded spherical model to a perfect sphere (setting the first coefficient to 1 and the others to 0), below:

There is very little difference in these surfaces of section. It looks like coefficients made from a 512K sim, 8-folded contain suprisingly little noise...at least in a spherical model.