Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in the presence of general coupling functions. It generalizes the well-known master stability function scheme to the higher-order structures with multiple interaction layers. We verify our theoretical results by employing them on networks of paradigmatic Rössler oscillators and Sherman neuronal models, and we demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.

中文翻译：

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具有多个交互层的单纯复形中同步的稳定性

了解互连的高阶和多层结构之间的相互作用如何影响动力系统的同步行为是一个有趣的可行问题，可能应用于神经元动力学等基本主题。在这里，我们提供了一种综合方法来分析具有许多交互层的单纯复形中完整同步状态的稳定性。我们证明了同步状态作为一个不变的解决方案存在，并在存在一般耦合函数的情况下推导出稳定同步状态的必要条件。它将著名的主稳定性函数方案推广到具有多个交互层的高阶结构。