  Bistability generated via a pure noise-induced phase transition is reexamined
from the view of bifurcations in macroscopic cumulant dynamics. It allows an
analytical study of the phase diagram in more general cases than previous
methods. In addition using this approach we investigate patially-extended
systems with two degrees of freedom per site. For this system, the analytic
solution of the stationary Fokker-Planck equation is not available and a
standard mean field approach cannot be used to find noise induced phase
transitions. A new approach based on cumulant dynamics predicts a noise-induced
phase transition through a Hopf bifurcation leading to a macroscopic limit
cycle motion, which is confirmed by numerical simulation.
