  We study the closed Hamiltonian dynamics of a free particle moving on a ring,
over one section of which it interacts linearly with a single harmonic
oscillator. On the basis of numerical and analytical evidence, we conjecture
that at small positive energies the phase space of our model is completely
chaotic except for a single region of complete integrability with a smooth
sharp boundary showing no KAM-type structures of any kind. This results in the
cleanest mixed phase space structure possible, in which motions in the
integrable region and in the chaotic region are clearly separated and
independent of one another. For certain system parameters, this mixed phase
space structure can be tuned to make either of the two components disappear,
leaving a completely integrable or completely chaotic phase space. For other
values of the system parameters, additional structures appear, such as KAM-like
elliptic islands, and one parameter families of parabolic periodic orbits
embedded in the chaotic sea. The latter are analogous to bouncing ball orbits
seen in the stadium billiard. The analytical part of our study proceeds from a
geometric description of the dynamics, and shows it to be equivalent to a
linked twist map on the union of two intersecting disks.
