I wonder if the periodic paths of a lightray trapped between two nonconcentric circles,
each perfectly reflecting, are known?
The behavior of such rays seems chaotically complicated. For example, left below the highlighted
initial ray has slope $\frac{1}{12}$, while that to the right has slope $\frac{1}{9}$.
(The green circle has radius $\frac{3}{4}$.) After 500 reflections, neither ray has become periodic.

I suspect the combination of dispersive and focusing reflection
(from the inner and outer circles respectively)
leads to this complex behavior.
But perhaps the periodic paths are known?

I only know the 2-cycles from rays collinear with the circle centers, and those that Noam Elkies kindly identified in his comment: "regular $n$-gons (and $(n/k)$-gons, i.e. stars) that stay so close to the outer circle that they never hit the inner one."

(Related MO question: "Trapped rays bouncing between two convex bodies.")

**Update 1**. I found one! :-) No doubt among the "simple short cycles" that Noam had in mind:

**Update 2**. With the benefit of the search terms helpfully provided by Ian Agol and Igor Riven, I found a useful

*Physical Review*paper by G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Grehan, "Periodic orbits in Hamiltonian chaos of the annular billiard," Phys. Rev. E 65, 016212 (2001). Their approach is more experimental than theoretical:

Periodic orbits embedded in the phase space are systematically investigated, with a focus on inclusion-touching periodic orbits, up to symmetrical orbits of period 6. Candidates for periodic orbits are detected by investigating grayscale distance charts and, afterward, each candidate is validated (or rejected) by using analytical and/or numerical methods.

The (unstable) 3-cycle I found they label "2(1)1" in their
(barely discernable) Fig.5 inventory below: