TITLE: Non-chaotic bouncing-ball curves
QUESTION [22 upvotes]: I was surprised to learn from two
Mathematica Demos by
Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel
exhibits chaotic behavior:

   
(The Poincaré map is shown above the red ball trajectories.)
I wonder:

Q. Are there curves (other than a horizontal line!)
  for which such a ball drop (from some range of horizontal
  placements $x$ and vertical heights $y$) is not chaotic? In other words, are the
  V and sine wave special? Or (more likely) is it that non-chaotic curves are special?

(I cannot now access Zeleny's citation to Chaotic Dynamics: An Introduction Based on Classical Mechanics.)

REPLY [7 votes]: This is really a collection of comments to Noam's wonderful parabola example, but a bit long...
The set of known integrable ordinary (non-gravitational) billiards consists of pieces of confocal conics, or their limits (straight lines).  The internal angles are all 90 degrees apart from some exceptions I will ignore henceforth: The 60-60-60 and 30-60-90 triangles and circular wedges.  The most common examples are the circle and ellipse, but also shapes such an elliptical and a hyperbolic arc, such that the two have the same foci.
In the case of a single conic, the equation to obtain the location of the next collision is linear: The intersection of a straight line and a conic is a quadratic equation, but we can divide by the known solution given by the initial point.

Remark: This feature also holds for similarly oriented parabolas, and hence also the parabola gravitational billiard.
Question: Is a gravitational billiard consisting of two confocal parabolic arcs also integrable?

Noam also conjectures that gravitational billiards with local minima cannot be chaotic: I have a recent preprint including some examples (circle, ellipse and oval gravitational billiards): http://www.arxiv.org/abs/1308.0362 Indeed, the low energy orbits appear regular.  There is also interesting higher energy dynamics: We also ask (with some numerical justification)

Conjecture: Are there any energies for which these gravitational billiards are ergodic?


Addendum: A generalisation of Noam's example is that of Newtonian gravity (ie inverse square force toward a centre), with the parabola replaced with a conic with a focus at
the centre.

Here the gravitational attraction is to the black dot, billiard boundary is one branch of
a hyperbola with this as focus, and the caustics are arcs of an ellipse and a hyperbola confocal with the boundary hyperbola.  As with the parabola, the caustics have a nice corner despite the smooth force field and billiard boundary.
Publishable?  Alas it is a trivial consequence of the conclusion of Horsch and Lang.  The second constant of motion for the two centre problem is preserved by billiard collisions with a confocal boundary.  So, I suspect not at present, though would love to be contradicted...