  Within random matrix theory, the statistics of the eigensolutions depend
fundamentally on the presence (or absence) of time reversal symmetry. Accepting
the Bohigas-Giannoni-Schmit conjecture, this statement extends to quantum
systems with chaotic classical analogs. For practical reasons, much of the
supporting numerical studies of symmetry breaking have been done with billiards
or maps, and little with simple, smooth systems. There are two main
difficulties in attempting to break time reversal invariance in a continuous
time system with a smooth potential. The first is avoiding false symmetry
breaking. The second is locating a parameter regime in which the symmetry
breaking is strong enough to transition the fluctuation properties fully toward
the broken symmetry case, and yet remain weak enough so as not to regularize
the dynamics sufficiently that the system is no longer chaotic. We give an
example of a system of two-coupled quartic oscillators whose energy level
statistics closely match those of the Gaussian unitary ensemble, and which
possesses only a minor proportion of regular motion in its phase space.
