  The large deviation functional of the density field in the weakly asymmetric
exclusion process with open boundaries is studied using a combination of
numerical and analytical methods. For appropriate boundary conditions and bulk
drives the functional becomes non-differentiable. This happens at
configurations where instead of a single history, several distinct histories of
equal weight dominate their dynamical evolution. As we show, the structure of
the singularities can be rather rich. We identify numerically analogues in
configuration space of first order phase transition lines ending at a critical
point and analogues of tricritical points. First order lines terminating at a
critical point appear when there are configurations whose dynamical evolution
is controlled by two distinct histories with equal weight. Tricritical point
analogues emerge when there are configurations whose dynamical evolution is
controlled by three distinct histories with equal weight. A numerical analysis
suggests that the structure of the singularities can be described by a Landau
like theory. Finally, in the limit of an infinite bulk bias we identify
singularities which arise from a competition of s histories, with s arbitrary.
In this case we show that all the singularities can be described by a Landau
like theory.
