  Stability and dynamical behavior of binary Bose-Einstein condensed mixtures
trapped on the surface of a rigid spherical shell are investigated in the
mean-field level, exploring the miscibility with and without vortex charges,
considering repulsive and attractive interactions. In order to compute the
critical points for the stability, we follow the Bogoliubov-de Gennes method
for the analysis of perturbed solutions, with the constraint that initially the
stationary states are in a complete miscible configuration. For the perturbed
equal density mixture, of a homogeneous uniform gas and when hidden vorticity
is verified, with the species having opposite azimuthal circulation, we
consider small perturbation analysis for each unstable mode, providing a
complete diagram with the intra- and inter-species interaction role on the
stability of the miscible system. Finally, beyond small perturbation analysis,
we explore the dynamics of some repulsive and attractive inter-species states
by full numerical solutions of the time-dependent Gross-Pitaevskii equation.
