Yes, I am the creator of the video. As shown in the first part of the video, these were created by combining two forms of harmonic oscillations - one for the particle around the "ring" fixed around the central axis (using a combination of sin and cos evolving in time together), and the second of the actual ring itself rotating about that central axis azimuthally.

After experimenting with different frequencies of motion (ratios between the period of the first ring and the second), I found that only half-integer and integer values (used in n*pi/2) of the azimuthal rotation gave rise to "stable" orbits. This is to say that any other number produced a trajectory that tended to stray away from perfectly overlapping itself repeatedly in time.

As stated above, this algorithm is only to show how complex behavior can emerge from a system utilizing incredibly simple rules. Given that, I'm sure an extra rule could be added to extend the distance of the particle from the nucleus - thus granting a radial distribution in line with spherical harmonics, etc. (or perhaps some other set of coupled rules not yet imagined).