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We study two-cluster solutions of an ensemble of generic limit-cycleoscillators in the vicinity of a Hopf bifurcation, i.e. Stuart-Landauoscillators, with a nonlinear global coupling. This coupling leads to conservedmean-field oscillations acting back on the individual oscillators as a forcing.A reduction to two effective equations makes a linear stability analysis of thecluster solutions possible. These equations exhibit a $\pi$-rotationalsymmetry, leading to a complex bifurcation structure and a wide variety ofsolutions. In fact, the principal bifurcation structure resembles that of a 2:1resonance tongue, while inside the tongue we observe an 1:1 entrainment.

Two-cluster solutions in an ensemble of generic limit-cycle oscillators with periodic self-forcing via the mean-field