Networks of limit cycle oscillators can show intricate patterns of synchronization such as splay states and cluster synchronization. Here we analyze dynamical states that display a continuum of seemingly independent splay clusters. Each splay cluster is a block splay state consisting of subclusters of fully synchronized nodes with uniform amplitudes. Phases of nodes within a splay cluster are equally spaced, but nodes in different splay clusters have an arbitrary phase difference that can be fixed or evolve linearly in time. Such coexisting splay clusters form a decoupled state in that the dynamical equations become effectively decoupled between oscillators that can be physically coupled. We provide the conditions that allow the existence of particular decoupled states by using the eigendecomposition of the coupling matrix. We also provide an alternate approach using the external equitable partition and orbital partition considerations combined with symmetry groupoid formalism to develop an algorithm to search for admissible decoupled states. Unlike previous studies, our approach is applicable when existence does not follow from symmetries alone and also illustrates the differences between adjacency and Laplacian coupling. We show that the decoupled state can be linearly stable for a substantial range of parameters using a simple eight-node cube network and its modifications as an example. We also demonstrate how the linear stability analysis of decoupled states can be simplified by taking into account the symmetries of the Jacobian matrix. Some network structures can support multiple decoupled patterns. To illustrate that, we show the variety of qualitatively different decoupled states that can arise on two-dimensional square and hexagonal lattices.

中文翻译：

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线性耦合极限周期振荡器网络中的解耦同步状态

极限周期振荡器的网络可以显示复杂的同步模式，例如展开状态和群集同步。在这里，我们分析动态状态，这些状态显示了看似独立的八字形簇的连续体。每个扩展集群是一个块扩展状态，由状态完全一致的节点的子集群组成，且振幅均匀。八字形群集中的节点的相位等距分布，但不同八字形群集中的节点具有任意相位差，该相位差可以固定或随时间线性变化。这样的共存八字簇形成解耦状态，因为动力学方程在可以物理耦合的振荡器之间有效地解耦。通过使用耦合矩阵的本征分解，我们提供了允许存在特定解耦状态的条件。我们还提供了一种替代方法，该方法使用外部公平分区和轨道分区考虑因素以及对称的类群形式主义来开发一种算法，以搜索可容许的解耦状态。与以前的研究不同，当存在不仅仅来自对称性时，我们的方法是适用的，并且可以说明邻接和拉普拉斯耦合之间的差异。我们显示，使用简单的八节点立方网络及其修改作为示例，对于大量参数，解耦状态可以线性稳定。我们还演示了如何通过考虑雅可比矩阵的对称性来简化解耦态的线性稳定性分析。某些网络结构可以支持多种解耦模式。为了说明这一点，