  We study local instabilities of a differentially rotating viscous flow of
electrically conducting incompressible fluid subject to an external azimuthal
magnetic field. In the presence of the magnetic field the hydrodynamically
stable flow can demonstrate non - axisymmetric azimuthal magnetorotational
instability (AMRI) both in the diffusionless case and in the double-diffusive
case with viscous and ohmic dissipation. Performing stability analysis of
amplitude transport equations of short-wavelength approximation, we find that
the threshold of the diffusionless AMRI via the Hamilton-Hopf bifurcation is a
singular limit of the thresholds of the viscous and resistive AMRI
corresponding to the dissipative Hopf bifurcation and manifests itself as the
Whitney umbrella singular point. A smooth transition between the two types of
instabilities is possible only if the magnetic Prandtl number is equal to
unity, $\rm Pm=1$. At a fixed $\rm Pm\ne 1$ the threshold of the
double-diffusive AMRI is displaced by finite distance in the parameter space
with respect to the diffusionless case even in the zero dissipation limit. The
complete neutral stability surface contains three Whitney umbrella singular
points and two mutually orthogonal intervals of self-intersection. At these
singularities the double-diffusive system reduces to a marginally stable system
which is either Hamiltonian or parity-time (PT) symmetric.
