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许多方面。事实上，相互作用物种的生态网络、大脑中的神经元和作为电网模型的耦合转子都是耦合振荡节点网络的例子。这样的网络可以产生大范围的集体动力学——耦合节点的联合动力学——比如同步。至关重要的是，网络功能或功能障碍往往取决于集体动力。例如，神经系统疾病，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。网络结构决定了一个节点对其他节点的影响。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，网络交互包括“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，这种相互作用往往被忽略。提出的项目将阐明耦合振荡器网络的集体动力学。我们在这里讨论的主要问题是网络结构和相互作用——特别关注高阶相互作用——如何塑造集体动力。我们将研究被称为异斜结构的对象，并阐明它们如何组织与现实世界网络相关的相互作用的动态。该项目将在动力系统理论及其应用方面产生新的成果。此外，我们将研究结果如何导致控制动力学的新方法。深入了解网络结构和相互作用如何塑造动态，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，这可能最终导致调整网络参数以恢复健康大脑功能的新方法。

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.