Perturbed circular billiard, chaos

[pomaranca]

Homework Statement

The center of a circular billiard is harmonically oscillating in horizontal direction with the amplitude a and frequency omega.

Describe the motion of elastic particle with mass m in this billiard. Use the proper phase space and Poincare map.
Under what conditions, dimensionless parameter eta=r/R, is energy of the system time limited and when time unlimited and when is the motion chaotic?

Does anyone has experience with this kind of problems?

 

[EnumaElish]

I'd suggest writing out relevant equations and attempted solution.

 

[pomaranca]

Dynamical system
This billiard is a dynamical system for which i should construct attractors and numerically find its fractal dimensions. (http://en.wikipedia.org/wiki/Dynamical_billiards)

Let's say that only the boundary is oscillating. Let mass of the particle be 1.

Phase space
What are the coordinates in the phase space?
Would coordinate pair [tex](s,v)[/tex] suffice to describe the motion of out particle? Here [tex]s[/tex] is the perimeter length, position of the collision on the circle, and [tex]v[/tex] speed of the particle.

Collision on the oscillating boundary
If the boundary would be fixed then the particle would reflect specularly (the angle of incidence equals to the angle of refection), with no change in the tangential component of speed and with instantaneous reversal of the speed component normal to the boundary.
But because our boundary is oscillating in horizontal direction, the angle of reflection is not the same to the angle of incidence because the particle gets another component of speed in the horizontal direction from the wall of billiard.
The collision is elastic, therefore the energy and momentum are conserved (http://en.wikipedia.org/wiki/Elastic_collision)
[tex]v_2' = \frac{v_2(m_2-m_1)+2m_1v_1}{m_1+m_2}[/tex]:
where [tex]v_2'[/tex] is the speed of particle after the collision and [tex]v_2[/tex] its speed before the collision, [tex]v_1[/tex] is the speed of the billiard boundary before the collision.

The potential of billiard is [tex]V(q)=\begin{cases} 0 \qquad q \in Q \\ \infty \qquad q \notin Q \end{cases}[/tex]
where [tex]Q[/tex] the region inside the circle. The particle can't affect the movement of the boundary: [tex]m_1\gg m_2[/tex] and thus from the above equation [tex]v_2'=2v_1-v_2[/tex].
We describe the oscillation of the boundary by [tex]x(t)=x_0\sin(\omega t)[/tex] for every point of boundary. The absolute value of speed in the moment after the collision is then [tex]v_2'=2x_0\omega\cos(\omega t)-v_2=f(t)-v_2[/tex], where [tex]t[/tex] is the moment of collision.

Poincaré return map
To find attractors in the phase space i should first construct the Poincaré map. For that i should know what will be the phase space and then discretize continuus time dynamics to discrete time dynamics. For the case of billiards that is of course dynamics between subsequent collisions: [tex](s_n,v_n)\to (s_{n+1},v_{n+1})[/tex].

I realize this is quite a geometric problem. But i don't know if this is at all the correct way to finding Poincaré return map.

Any suggestions?