  We analyze global bifurcations along the family of radially symmetric
vortices in the Gross--Pitaevskii equation with a symmetric harmonic potential
and a chemical potential $\mu$ under the steady rotation with frequency
$\Omega$. The families are constructed in the small-amplitude limit when the
chemical potential $\mu$ is close to an eigenvalue of the Schr\"{o}dinger
operator for a quantum harmonic oscillator. We show that for $\Omega$ near $0$,
the Hessian operator at the radially symmetric vortex of charge
$m_{0}\in\mathbb{N}$ has $m_{0}(m_{0}+1)/2$ pairs of negative eigenvalues. When
the parameter $\Omega$ is increased, $1+m_{0}(m_{0}-1)/2$ global bifurcations
happen. Each bifurcation results in the disappearance of a pair of negative
eigenvalues in the Hessian operator at the radially symmetric vortex. The
distributions of vortices in the bifurcating families are analyzed by using
symmetries of the Gross--Pitaevskii equation and the zeros of Hermite--Gauss
eigenfunctions. The vortex configurations that can be found in the bifurcating
families are the asymmetric vortex $(m_0 = 1)$, the asymmetric vortex pair
$(m_0 = 2)$, and the vortex polygons $(m_0 \geq 2)$.
