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.

我们降低了均值耦合的Stuart-Landau振荡器的整体动力学，使其接近同步解决方案。特别是，我们将系统映射到本杰明-费尔不稳定性的中心流形上，分叉使同步振荡不稳定。使用对称参数，我们描述了中心流形上高达三次阶的动力学结构，并推导了其参数的表达式。这使我们能够研究Stuart-Landau系综描述的现象，例如低维中心流形中的聚类和聚类奇点，从而为耦合振荡器的对称破坏动力学提供了进一步的见解。我们表明，Stuart-Landau系集中的簇奇点对应于中心流形动力学中消失的二次项。此外，它们充当鞍形节点分叉的组织中心，从而创建不平衡的簇状态以及横向分叉改变簇的稳定性。此外，我们表明，只有当任一群集包含至少1/3的振荡器时，才会出现具有相同群集大小分布的不同解决方案的双稳态，而与系统参数无关。