  We conduct a global, weakly nonlinear analysis of the magnetorotational
instability (MRI) in a Taylor-Couette flow. This is a multiscale perturbative
treatment of the nonideal, axisymmetric MRI near threshold, subject to
realistic radial boundary conditions and cylindrical geometry. We analyze both
the standard MRI, initialized by a constant vertical background magnetic field,
and the helical MRI, with an azimuthal background field component. This is the
first weakly nonlinear analysis of the MRI in a global Taylor-Couette geometry,
as well as the first weakly nonlinear analysis of the helical MRI. We find that
the evolution of the amplitude of the standard MRI is described by a real
Ginzburg-Landau equation (GLE), while the amplitude of the helical MRI takes
the form of a complex GLE. This suggests that the saturated state of the
helical MRI may itself be unstable on long spatial and temporal scales.
