  We report on the numerical and theoretical study of the subcritical
bifurcation of parametrically amplified waves appearing at the interface
between two immiscible incompressible fluids when the layer of the lower fluid
is very shallow. As a critical control parameter is surpassed, small amplitude
surface waves bifurcate towards highly nonlinear ones, with twice their
amplitude. We propose a simple phenomenological model which can describe the
observed bifurcation. We relate this hysteresis with the change of shear stress
using a simple stress balance, in agreement with numerical results.
