Department of Mathematics, University of California, Davis, Davis, California 95616, USA.
Department of Computer Science and Department of Mechanical and Aerospace Engineering, University of California, Davis, Davis, California 95616, USA.
Chaos. 2022 May;32(5):053121. doi: 10.1063/5.0095094.
Cascading failures abound in complex systems and the Bak-Tang-Weisenfeld (BTW) sandpile model provides a theoretical underpinning for their analysis. Yet, it does not account for the possibility of nodes having oscillatory dynamics, such as in power grids and brain networks. Here, we consider a network of Kuramoto oscillators upon which the BTW model is unfolding, enabling us to study how the feedback between the oscillatory and cascading dynamics can lead to new emergent behaviors. We assume that the more out-of-sync a node is with its neighbors, the more vulnerable it is and lower its load-carrying capacity accordingly. Also, when a node topples and sheds load, its oscillatory phase is reset at random. This leads to novel cyclic behavior at an emergent, long timescale. The system spends the bulk of its time in a synchronized state where load builds up with minimal cascades. Yet, eventually, the system reaches a tipping point where a large cascade triggers a "cascade of larger cascades," which can be classified as a dragon king event. The system then undergoes a short transient back to the synchronous, buildup phase. The coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to the standard exogenous ones, and we show their respective roles. We establish the phenomena from numerical studies and develop the accompanying mean-field theory to locate the tipping point, calculate the load in the system, determine the frequency of the long-time oscillations, and find the distribution of cascade sizes during the buildup phase.
复杂系统中存在级联故障，Bak-Tang-Weisenfeld（BTW）沙堆模型为分析它们提供了理论基础。然而，它并没有考虑到节点可能具有振荡动力学的可能性，例如在电网和大脑网络中。在这里，我们考虑了一个 Kuramoto 振荡器网络，BTW 模型在该网络上展开，使我们能够研究振荡和级联动力学之间的反馈如何导致新的涌现行为。我们假设，节点与其邻居的同步程度越低，它就越脆弱，相应地，其承载能力就越低。此外，当一个节点倒塌并卸下负载时，其振荡相位会随机重置。这导致在新兴的长时标下出现新的循环行为。系统大部分时间都处于同步状态，在这种状态下，负载会在最小级联的情况下累积。然而，最终，系统达到一个临界点，一个大级联触发了一个“更大级联的级联”，这可以被归类为龙王事件。然后，系统会经历一个短暂的瞬态，回到同步的累积阶段。容量和同步之间的耦合除了标准的外生耦合之外，还产生了内源性级联种子，我们展示了它们各自的作用。我们从数值研究中建立了这些现象，并开发了伴随的平均场理论来定位临界点，计算系统中的负载，确定长时间振荡的频率，并找到在累积阶段的级联大小的分布。
