论文标题
2群集的固定点分析中心歧管中的均值耦合Stuart-Landau振荡器
2-Cluster Fixed-Point Analysis of Mean-Coupled Stuart-Landau Oscillators in the Center Manifold
论文作者
论文摘要
我们减少了接近同步解决方案的均值耦合Stuart-Landau振荡器的集合的动力学。特别是，我们将系统映射到本杰明 - 弗脉不稳定性的中心歧管上，分叉破坏了同步振荡。使用对称参数，我们描述了该中心歧管上的动力学的结构，直至立方顺序，并为其参数得出表达式。这使我们能够研究由斯图尔特 - 兰道集团描述的现象，例如聚类和聚类奇异性，在较低维的中心歧管中，提供了对耦合振荡器的对称性折叠动力学的进一步见解。我们表明，Stuart-Landau合奏中的集群奇异性对应于中心歧管动态中消失的二次术语。此外，它们充当鞍形节点分叉的组织中心，从而产生了不平衡的聚类状态，以及横向分叉改变了群集稳定性。此外，我们表明，只有在任何一个群集包含至少$ 1/3 $的振荡器，与系统参数无关的振荡器的$ 1/3 $时，才能发生不同的解决方案的双稳定性。
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.
