超图上的动力系统
Dynamical systems on Hypergraphs
摘要：
网络是一种广泛使用的有效范例，用于模拟基本单元成对交互的真实世界系统。许多身体的相互作用经常在起作用，不能通过诉诸二元交换来建模。在这项工作中，我们考虑一类锚定在超图上的动力系统。理想情况下，任意大小的超边包围着单个单元，以便考虑多个同时发生的相互作用。后者介导的组合拉普拉斯算子，这是在这里介绍和特点。主稳定性函数的形式适应于当前的设置。研究了一类在超图上演化的系统的图灵斑图和非线性（正则和混沌）振子的同步。对外部施加的扰动的响应带有相互作用的高阶性质的印记。
英文摘要：
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.
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