  We study a dyadic model for incompressible ideal magnetohydrodynamics.
Besides establishing global existence of weak solutions and local
well-posedness, a set of fixed points are found. Solutions starting from
positive data near a positive fixed point are shown to remain positive for all
the time. Solutions with positive data are also shown to develop blow-up at
finite time. In addition, we prove that the pure fluid steady state (with zero
magnetic field) attracts solutions with stronger perturbation in the fluid
component than that of the magnetic field; vice versa, the non-fluid steady
state attracts solutions with stronger perturbation in the magnetic field. The
solutions attracted by a fixed point dissipate energy.
