  We study the stabilization of localized structures by discreteness in
one-dimensional lattices of diffusively coupled nonlinear sites. We find that
in an external driving field these structures may lose their stability by
either relaxing to a homogeneous state or nucleating a pair of oppositely
moving fronts. The corresponding bifurcation diagram demonstrates a cusp
singularity. The obtained analytic results are in good quantitative agreement
with numerical simulations.
