  This study numerically investigates the bifurcation aspect of the wide-gap
spherical Couette flow (SCF), with an emphasis on the competition among
polygonal coherence with different wave numbers observed over transitional
Reynolds numbers. Focusing on a representative case, the half-radius ratio, we
confirm that the axisymmetric state becomes unstable over the first
transitional Reynolds number at which the 4-fold spiral state bifurcates, using
the continuation method based on the Newton-Raphson algorithm. The
Galerkin-spectral method was employed to numerically solve the governing
equations. It is found that the 3-fold spiral state bifurcates from the
axisymmetric state at a slightly higher Reynolds number than the first
transitional Reynolds number. The attraction of the 3-fold spiral state expands
rapidly with an increase in the Reynolds number, which is determined by
verifying the distance of the unstable periodic-like state to both spiral
states in the state space. This aspect of the state space explains the
experimentally bistable realization of different equilibrium states over the
first transitional Reynolds number. This study also found that the
periodic-like state is composed of the 3- and 4-fold spiral states, similar to
a beat with two different frequencies.
