Through symmetry breaking to higher dimensional chaotic scattering

The goal of this thesis is to gain some insight into chaotic scattering dynamics in
systems with more than two effective degrees of freedom. We use the idea of "gradually"
introducing another degree of freedom by perturbating slightly a symmetry
that has reduced a three–dimensional system to two effective dimensions.
We furthermore specifically study the case of a system where a particle was scattered
by a hard disc moving on a circle which will be slightly deformed to an
ellipse. This leads to a slight violation of the Jacobi integral which causes the increase
of dimension of phase space.
We first complement previous studies of the problem with symmetry by showing
elliptic areas, obtaining linear response results and the bifurcation scenario.
Then we show how analyzing spaces of initial conditions in terms of the number
of the scattering events before ejection yields information about periodic orbits,
stable or longlived islands and bifurcations. This allows us to gain considerable
insight into the scattering process, that can be of use in qualitative understanding
of planetary rings in celestial mechanics and seems to be valid far beyond the toy
model we use.