  We consider standing waves in the focusing nonlinear Schr\"odinger (NLS)
equation on a dumbbell graph (two rings attached to a central line segment
subject to the Kirchhoff boundary conditions at the junctions). In the limit of
small $L^2$ norm, the ground state (the orbitally stable standing wave of the
smallest energy at a fixed $L^2$ norm) is represented by a constant solution.
However, when the $L^2$ norm is increased, this constant solution undertakes
two bifurcations, where the first is the pitchfork (symmetry breaking)
bifurcation and the second one is the symmetry preserving bifurcation. As a
result of the first symmetry breaking bifurcation, the standing wave becomes
more localized in one of the two rings. As a result of the second symmetry
preserving bifurcation, the standing wave becomes localized in the central line
segment. In the limit of large norm solutions, both standing waves are
represented by a truncated solitary wave localized in either the ring or the
central line segment. Although the asymmetric wave supported in the ring is a
ground state near the symmetry breaking bifurcation of the constant solution,
it is the symmetric wave supported in the central line segment which becomes
the ground state in the limit of large $L^2$ norm. The analytical results are
confirmed by numerical approximations of the ground state on the dumbbell
graph.
