  We investigate the buckling and post-buckling properties of a hyperelastic
half-space coated by two hyperelastic layers when the composite structure is
subjected to a uniaxial compression. In the case of a half-space coated with a
{\it single} layer, it is known that when the shear modulus $\mu_f$ of the
layer is larger than the shear modulus $\mu_s$ of the half-space, a linear
analysis predicts the existence of a critical stretch and wave number, whereas
a weakly nonlinear analysis predicts the existence of a threshold value of the
modulus ratio $\mu_s/\mu_f\approx 0.57$ below which the buckling is
super-critical and above which the buckling is sub-critical. It is shown in
this paper that when another layer is added, a larger variety of behaviour can
be observed. For instance, buckling can occur at a preferred wavenumber
super-critically even if both layers are softer than the half-space although
the top layer would need to be harder than the bottom layer. When the shear
modulus of the bottom layer lies in a certain interval, the super-critical to
sub-critical transition can happen a number of times as the shear modulus of
the top layer is increased gradually. Thus, an extra layer imparts more
flexibility in producing wrinkling patterns with desired properties, and our
weakly nonlinear analysis provides a road map on the parameter regimes where
this can be achieved.
