We analyze global bifurcations along the family of radially symmetricvortices in the Gross--Pitaevskii equation with a symmetric harmonic potentialand a chemical potential $\mu$ under the steady rotation with frequency$\Omega$. The families are constructed in the small-amplitude limit when thechemical potential $\mu$ is close to an eigenvalue of the Schr\"{o}dingeroperator 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. Whenthe parameter $\Omega$ is increased, $1+m_{0}(m_{0}-1)/2$ global bifurcationshappen. Each bifurcation results in the disappearance of a pair of negativeeigenvalues in the Hessian operator at the radially symmetric vortex. Thedistributions of vortices in the bifurcating families are analyzed by usingsymmetries of the Gross--Pitaevskii equation and the zeros of Hermite--Gausseigenfunctions. The vortex configurations that can be found in the bifurcatingfamilies are the asymmetric vortex $(m_0 = 1)$, the asymmetric vortex pair$(m_0 = 2)$, and the vortex polygons $(m_0 \geq 2)$.
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