  Bifurcation equations, non-degeneracy and transversality conditions are
obtained for the fold, transcritical, pitchfork and flip bifurcations for
periodic points of one dimensional implicitly defined discrete dynamical
systems. The backward Euler method and the trapezoid method for numeric
solutions of ordinary differential equations fall in the category of implicit
dynamical systems. Examples of bifurcations are given for some implicit
dynamical systems including bifurcations for the backward Euler method when the
step size is changed.
