  We consider the first order differential equation with a sinusoidal
nonlinearity and periodic time dependence, that is, the periodically driven
overdamped pendulum. The problem is studied in the case that the explicit
time-dependence has symmetries common to pure ac-driven systems. The only
bifurcation that exists in the system is a degenerate pitchfork bifurcation,
which describes an exchange of stability between two symmetric nonlinear modes.
Using a type of Prufer transform to a pair of linear differential equations, we
derive an approximate condition of the bifurcation. This approximation is in
very good agreement with our numerical data. In particular, it works well in
the limit of large drive amplitudes and low external frequencies. We
demonstrate the usefulness of the theory applying it to the models of pure
ac-driven semiconductor superlattices and Josephson junctions. We show how the
knowledge of bifurcations in the overdamped pendulum model can be utilized to
describe effects of rectification and amplification of electric fields in these
microstructures.
