Hierarchical clustering in mean-field coupled Stuart-Landau oscillators.

Clustered solutions in oscillator networks provide an important insight into how a system might diversify from a synchronous solution into spatiotemporal complex solutions. They can, therefore, form a link between the fully synchronized and incoherent states. Despite their fundamental role in coupled oscillator dynamics, our understanding of how these clusters form and differentiate is still quite limited. Here, we study an ensemble of globally coupled Stuart-Landau oscillators and focus on how 3-cluster solutions emerge from 2-cluster solutions and how the different 3-cluster solutions are organized in parameter space. We show that the arrangement of the clusters is dictated by a codimension-two point, which we call a Type-II cluster singularity. Furthermore, our study points to a hierarchical structure of multi-cluster solutions.