  We study two dyadic models for incompressible ideal magnetohydrodynamics, one
with a uni-directional energy cascade and the other one with both forward and
backward energy cascades. Global existence of weak solutions and local
well-posedness are established for both models. In addition, solutions to the
model with uni-directional energy cascade associated with positive initial data
are shown to develop blow-up at a finite time. Moreover, a set of fixed points
is found for each model. Linear instability about some particular fixed points
is proved.
