2-Cluster fixed-point analysis of mean-coupled Stuart-Landau oscillators in the center manifold

We reduce the dynamics of an ensemble of mean-coupled Stuart-Landau oscillators close to the synchronized solution. In particular, we map the system onto the center manifold of the Benjamin-Feir instability, the bifurcation destabilizing the synchronized oscillation. Using symmetry arguments, we describe the structure of the dynamics on this center manifold up to cubic order, and derive expressions for its parameters. This allows us to investigate phenomena described by the Stuart-Landau ensemble, such as clustering and cluster singularities, in the lower-dimensional center manifold, providing further insights into the symmetry-broken dynamics of coupled oscillators. We show that cluster singularities in the Stuart-Landau ensemble correspond to vanishing quadratic terms in the center manifold dynamics. In addition, they act as organizing centers for the saddle-node bifurcations creating unbalanced cluster states as well for the transverse bifurcations altering the cluster stability. Furthermore, we show that bistability of different solutions with the same cluster-size distribution can only occur when either cluster contains at least 1/3 of the oscillators, independent of the system parameters.

Automated summary

This study reduces the dynamics of mean-coupled Stuart-Landau oscillators and describes their structure and parameter expressions on the center manifold, investigating clustering phenomena and cluster singularities and revealing the broken symmetry dynamics of coupled oscillators. It shows that cluster singularities correspond to vanishing quadratic terms in the dynamics and serve as organizing centers for bifurcations creating unbalanced cluster states, as well as altering cluster stability. Additionally, bistability of different solutions with the same cluster-size distribution is shown to only occur when each cluster contains at least 1/3 of the oscillators, regardless of system parameters.