  The Faraday problem is an important pattern-forming system that provides some
middle ground between systems where the initial instability involves just a
single mode and in which complexity then results from mode interactions or
secondary bifurcations, and cases where a system is highly turbulent and many
spatial and temporal modes are excited. It has been a rich source of novel
patterns and of theoretical work aimed at understanding how and why such
patterns occur. Yet it is particularly challenging to tie theory to experiment:
the experiments are difficult to perform; the parameter regime of interest
(large box, moderate viscosity) along with the technical difficulties of
solving the free boundary Navier--Stokes equations make numerical solution of
the problem hard; and the fact that the instabilities result in an entire
circle of unstable wavevectors presents considerable theoretical difficulties.
  In principle, weakly nonlinear theory should be able to predict which
patterns are stable near pattern onset. In this paper we present the first
quantitative comparison between weakly nonlinear theory of the full
Navier-Stokes equations and (previously published) experimental results for the
Faraday problem with multiple frequency forcing. We confirm that three-wave
interactions sit at the heart of why complex patterns are stablised but also
highlight some discrepancies between theory and experiment. These suggest the
need for further experimental and theoretical work to fully investigate the
issues of pattern bistability and the role of bicritical/tricritical points in
determining bifurcation structure.
