Higher-order interactions and heteroclinic network dynamics

耦合动态节点网络在科学技术中无处不在，影响着我们日常生活的方方面面。事实上，由相互作用的物种、大脑中的神经元和耦合转子组成的生态网络作为电网的模型，都是耦合振荡节点网络的例子。这样的网络可以产生广泛的集体动态--耦合节点的联合动态--例如同步。至关重要的是，网络的功能或功能障碍往往取决于集体的动态。例如，神经疾病，如帕金森氏病，长期以来一直与过度的神经同步有关。网络的集体动力学，即节点是否同步或表现出其他动态行为，取决于网络结构和相互作用。网络结构确定一个节点是否影响其他节点。网络交互决定了一个节点如何影响其他节点。在许多现实世界的网络中，网络相互作用包括高阶耦合，例如，一个节点对另一个节点的影响可能取决于第三个节点的状态。然而，在通常研究的网络中，这种相互作用常常被忽略。该项目将阐明耦合振子网络的集体动力学。我们在这里讨论的主要问题是网络结构和互动如何塑造集体动力--尤其关注更高级别的互动。我们将研究被称为异宿结构的对象，并阐明它们如何组织与真实世界网络相关的相互作用的动力学。该项目将在动力系统理论及其应用方面产生新的结果。此外，我们还将研究这些结果如何导致控制动力学的新方法。对网络结构和相互作用如何塑造动态的洞察可以用来理解必须调整网络的哪一部分才能使振荡器同步(或不同步)。由于帕金森氏症与过度同步有关，这最终可能导致新的方法来调整网络参数，以恢复健康的大脑功能。

Networks of coupled dynamical nodes are ubiquitous in science and technology and influence many parts of our everyday lives. Indeed, ecological networks of interacting species, neurons in the brain and coupled rotors as a model for a power grid are examples of networks of coupled oscillatory nodes. Such networks can give rise to a wide range of collective dynamics - the joint dynamics of the coupled nodes - such as synchrony. Crucially, the network function or dysfunction often depends on the collective dynamics. For example, neurological diseases, such as Parkinson's disease, have long been associated with excessive neural synchronization.The collective dynamics of a network, that is, whether the nodes synchronize or show other dynamical behaviour, depends on the network structure and interactions. The network structure determines whether a node influences other nodes. The network interactions determine how a node influences other nodes. In many real-world networks, the network interactions include "higher-order" coupling, for example, the influence of one node onto another may depend on the state of a third node. However, such interactions are often omitted in commonly studied networks.The proposed project will elucidate the collective dynamics of coupled oscillator networks. The main question we address here is how network structure and interactions - with a particular focus on higher-order interactions - shape the collective dynamics. We will investigate objects called heteroclinic structures and elucidate how they organize the dynamics for interactions that are relevant for real-world networks.The project will yield new results in dynamical systems theory and their application. Moreover, we will investigate how the results can lead to new ways to control dynamics. Insights into how network structure and interactions shape the dynamics can be employed to understand what part of the network one has to tune to get oscillators to synchronize (or not). Since Parkinson's disease has been associated with excessive synchrony, this could eventually lead to new ways to tune network parameter to restore healthy brain functionality.