We consider an inhomogeneous complex Landau equation governing
the amplitude modulation of Taylor vortices between two rapidly
rotating concentric spheres, which bound a narrow gap and
almost co-rotate about a common axis of symmetry.
In this weakly nonlinear regime the latitudinal vortex
width is comparable to the gap between the shells.
The vortices are located close to the
equator and are modulated on a latitudinal length scale large compared to
the gap width but small compared to the shell radius.

We investigate the stability and subsequent evolution of the
steady finite amplitude Taylor vortices, which exist following the
first bifurcation. A complicated bifurcation structure is unravelled
dependent on the magnitude of a parameter kappa, which measures the
strength of spatial phase mixing. Only when the inner and outer
spheres almost  corotate is kappa small; otherwise kappa is large.
Two types of mode exist-- one (SP) preserves the reflectional
symmetry of the steady solutions with respect to the equatorial plane
while the other type (SB) breaks it.
For sufficiently large kappa, a supercritical SP-Hopf bifurcation
of the steady state leads to a vacillating solution
which expands into a homoclinic cycle connecting the trivial
undisturbed state to itself. SP-global bifurcations occur for all
kappa leading to limit cycles which correspond to vortices drifting
towards the equator from both sides. Further bifurcations introduce
a second frequency. For large kappa, instability generally
leads to SB-states with complicated temporal behaviours,
which we interpret as trains of pulses each dominated by distinct
frequencies.