Networks are a widely used and efficient paradigm to model real-world systems where basic units interact pairwise. Many body interactions are often at play, and cannot be modelled by resorting to binary exchanges. In this work, we consider a general class of dynamical systems anchored on hypergraphs. Hyperedges of arbitrary size ideally encircle individual units so as to account for multiple, simultaneous interactions. These latter are mediated by a combinatorial Laplacian, that is here introduced and characterised. The formalism of the master stability function is adapted to the present setting. Turing patterns and the synchronisation of non linear (regular and chaotic) oscillators are studied, for a general class of systems evolving on hypergraphs. The response to externally imposed perturbations bears the imprint of the higher order nature of the interactions.

中文翻译：

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超图上的动力学系统

网络是一种广泛使用的有效范例，可用于对基本单元成对交互的现实系统进行建模。许多身体的相互作用经常在起作用，并且不能通过采用二进制交换来建模。在这项工作中，我们考虑锚定在超图上的一类动力学系统。理想情况下，任意大小的超边都围绕单个单元，以解决多个同时发生的相互作用。后者是由组合式拉普拉斯算子介导的，在此进行介绍和表征。主稳定性功能的形式适用于当前设置。对于超图上发展的一类通用系统，研究了图灵模式和非线性（规则和混沌）振荡器的同步性。