  Here we consider stationary states for nonlinear Schrodinger equations with
symmetric double well potentials. These stationary states may bifurcate as the
strength of the nonlinear term increases and we observe two different pictures
depending on the value of the nonlinearity power: a simple pitch-fork
bifurcation, and a couple of saddle points which unstable branches collapse in
an inverse pitch-fork bifurcation. In this paper we show that in the
semiclassical limit, or when the barrier between the two wells is large enough,
the first kind of bifurcation always occurs when the nonlinearity power is less
than a critical value (2); in contrast, when the nonlinearity power is larger
than such a critical value then we always observe the second scenario. The
remarkable fact is that such a critical value is an universal constant in the
sense that it does not depends on the shape of the double well potential.
