A warning: The surfaces of section on this page are generated by taking the entire inital phase space of the particle...so the particle's subsequent orbit is not confined to a plane. That's why these sos's look weird; the orbits overlap and aren't symmetric about the velocity axis. Unless you're better than I am at deciphering a 2-d projection of a 3-d orbit in phase space, you might as well just look at the huge scatter in each orbit.

Now, each section represents a different slice in energy space. The energy of the most bound particle in each slice is given in the panel. Each orbit is a different color, and there are only 7 orbits in each of the 9 sections. The orbits are integrated for 10000 timesteps.

Here's the 128K, non-8-folded sos integrated with dteps=0.01.

Notice how it's basically crap. We wondered if the timestep might be a problem, so we set the timestep in the largest energy slice to be dteps=0.01, and then cut dteps by a factor of 2 for the more bound slices. It made no difference...see below.

Here's the 128K, non-8-folded sos integrated with a variable dteps:

So, we can conclude that the timestep is not the problem here.