swing spring, 1:1:2 resonance, etc

It can be argued that although the radial mode is not exactly double the doubly degenerate peripheral mode frequency, a substantial coupling term (manifested by the strong overtone peak) could make an analog to the 1:1:2 resonance of swing spring valid. It would be interesting to actually observe the pulsation and precession effects described in Lynch's paper.

Regarding the polygon (Sinai) billiard analog - basically we treat the whole motion as motion in 'triangles' that tiled the icosahedron, with a reflection rule at the boundaries. However, since the triangle is actually on the S² surface, special treatment is needed; for example, a rotation around the center of a pentagon should be mapped as a "bouncing" motion between adjacent two boundaries - which is quite ill-defined in the planar case. In fact, how much efforts are warranted to just carry out this simulation is unknown, as we can always do the whole simulation and mapped it back to the sub-region. The question is more interesting, only in analytical terms (since it reflects both the symmetry and/or interconnectivity).

Lastly, a measurement of 1/20 sub-region "revisit" frequency could be highly valuable. Say we have N trajectories originating from inside the boundaries, we want to calculate the number of trajectories that remain in the boundary as a function of time t. To check if this follows a polynomial or exponential decay law can be very enlightening on the stickiness of the phase space.

PS: assuming equal 'diffusion' toward the three adjacent tiles, in the long run, the decay should follow an exponential law of 0.95*exp(-0.255*r*t) + 0.05, in which r is the diffusion rate. (-0.254644 is the three leading exponents other than 0 of the eigenvalues of the diffusion matrix.) Update: preliminary simulations show a similar decay curve (the decay constant drops quickly) to an end value slightly above 0.05 and with a very fluctuating tail, even with 1000 trajectories. Despite being consistent with most of our speculations, the enormous fluctuation around the 0.05 tail is unaccounted for.