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We reduce the dynamics of an ensemble of mean-coupled Stuart-Landauoscillators close to the synchronized solution. In particular, we map thesystem onto the center manifold of the Benjamin-Feir instability, thebifurcation destabilizing the synchronized oscillation. Using symmetryarguments, we describe the structure of the dynamics on this center manifold upto cubic order, and derive expressions for its parameters. This allows us toinvestigate phenomena described by the Stuart-Landau ensemble, such asclustering and cluster singularities, in the lower-dimensional center manifold,providing further insights into the symmetry-broken dynamics of coupledoscillators. We show that cluster singularities in the Stuart-Landau ensemblecorrespond to vanishing quadratic terms in the center manifold dynamics. Inaddition, they act as organizing centers for the saddle-node bifurcationscreating unbalanced cluster states as well for the transverse bifurcationsaltering the cluster stability. Furthermore, we show that bistability ofdifferent solutions with the same cluster-size distribution can only occur wheneither cluster contains at least $1/3$ of the oscillators, independent of thesystem parameters.

2-Cluster Fixed-Point Analysis of Mean-Coupled Stuart-Landau Oscillators in the Center Manifold