Collective behaviors of mean-field coupled Stuart–Landau limit-cycle oscillators under additional repulsive links

We study the collective behaviors of a large population of Stuart–Landau limit-cycle oscillators that coupled diffusively and equally with all of the others via the conjugate of the mean field, where the underlying interaction is shown to break the rotational symmetry of the coupled system. In the model, an ensemble of Stuart–Landau oscillators are in fact diffusively coupled via the mean field in the real parts, whereas additional repulsive links are present in the imaginary parts. All the oscillators are linked via the similar state variables, which distinctly differs from the conjugate coupling through dissimilar variables in the previous studies. We show that depending on the strength of coupling and the distribution of natural frequencies, the coupled system exhibits three qualitatively different types of collective stationary behaviors: amplitude death (AD), oscillation death (OD), and incoherent state. Our goal is to analytically characterize the onset of the above three typical macrostates by performing the rigorous linear stability analyses of the corresponding mean-field coupled system. We prove that AD is able to occur in the coupled system with identical frequencies, where the stable coupling interval of AD is found to be independent on the system’s size N. When the natural frequencies are distributed according to a general density function, we obtain the analytic equations that govern the exact stability boundaries of AD, OD, and the incoherence for a coupled system in the thermodynamic limit N → ∞. All the theoretical predictions are well confirmed via numerical simulations of the coupled system with a specific Lorentzian frequency distribution.