Periodic orbits of a billiard ball bouncing in a square have been well-studied.
I am seeking similar analysis of what is sometimes called a *rough ball*, one
whose high friction causes it to pick up spin when it hits a wall.
Assuming no slip at the point of contact, that kinetic energy is conserved (a "superball"), and
that gravity is not relevant, there is a definite dynamics, dependent upon the
moment of inertia of the ball. For example, a solid ball (e.g., a lacrosse ball) has
moment $I= \alpha m r^2$ where $\alpha=\frac{2}{5}$ (and $m$ and $r$ are mass and radius).
<br /> ![SuperBall][1]<br />
As an example, if such a ball is thrown against the bottom side of a square, entering
(along the red vector) with zero spin at $45^\circ$, horizontal velocity $1$,
it exits at about $68^\circ$, with a clockwise spin
resulting in a horizontal ball-rim velocity of $\frac{-10}{7}$.
I've tried to track above the collision equations, without at all being certain
that I am exactly correct. In my calculation, six bounces *almost* completes a cycle,
but not quite.
Regardless of the accuracy of these calculations, my question is whether or not periodic orbits of such rough, elastic balls have been explored. Thanks!
<hr />
(Added *10 Dec 2013*.) Just found this paper by Aston & Shail: "The dynamics of a bouncing superball with spin." *Dynamical Systems: An International Journal*, Volume 22, Issue 3, 2007. [Journal link](http://www.tandfonline.com/doi/pdf/10.1080/14689360701198142#.UqdMBmRDurd).
<br /> ![Fig9d][2]<br />
Blue trajectories have back-spin, red forward-spin. Obviously including dissipation.
One cool aspect of this paper is that they analyze a ball bouncing up and down a staircase:
<br /> ![Fig13][3]<br />
The dashed line is the initial "throw" that initiates the illustrated sequence of bounces.
[1]: https://i.sstatic.net/NFKS3.jpg
[2]: https://i.sstatic.net/IAVLy.png
[3]: https://i.sstatic.net/k7AAT.png