  Surface wave patterns are investigated experimentally in a system geometry
that has become a paradigm of quantum chaos: the stadium billiard. Linear waves
in bounded geometries for which classical ray trajectories are chaotic are
known to give rise to scarred patterns. Here, we utilize parametrically forced
surface waves (Faraday waves), which become progressively nonlinear beyond the
wave instability threshold, to investigate the subtle interplay between
boundaries and nonlinearity. Only a subset (three main types) of the computed
linear modes of the stadium are observed in a systematic scan. These correspond
to modes in which the wave amplitudes are strongly enhanced along paths
corresponding to certain periodic ray orbits. Many other modes are found to be
suppressed, in general agreement with a prediction by Agam and Altshuler based
on boundary dissipation and the Lyapunov exponent of the associated orbit.
Spatially asymmetric or disordered (but time-independent) patterns are also
found even near onset. As the driving acceleration is increased, the
time-independent scarred patterns persist, but in some cases transitions
between modes are noted. The onset of spatiotemporal chaos at higher forcing
amplitude often involves a nonperiodic oscillation between spatially ordered
and disordered states. We characterize this phenomenon using the concept of
pattern entropy. The rate of change of the patterns is found to be reduced as
the state passes temporarily near the ordered configurations of lower entropy.
We also report complex but highly symmetric (time-independent) patterns far
above onset in the regime that is normally chaotic.
