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

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