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Large Deviations and Metastability in Dynamical Networks

动态网络中的大偏差和亚稳态

基本信息

批准号：
2009233
负责人：
Georgi Medvedev
金额：
$ 22.64万
依托单位：
Drexel University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009233&HistoricalAwards=false
关键词：
Large Deviations Metastability Dynamical Networks

项目摘要

We live in the age of networks. Our everyday lives depend on robust and predictable performance of different technological networks around us (such as power grids and communication networks). Human physiology relies on coordinated activity of hundreds of cellular networks (for example, networks of neural cells in the brain, cells in the heart and pancreas, to name a few). The patterns of connections in real world networks can be complex and exhibit nontrivial statistical properties. Understanding how the structural organization of a network affects its dynamics is the principal challenge in the theory of interacting dynamical systems that sets it apart from the theories for classical spatially extended dynamical systems, such as partial differential equations or lattice dynamical systems. The Principal Investigator (PI) seeks to develop a systematic mathematical approach to the analysis of dynamical networks. Through the development of new mathematical techniques and analyzing representative mathematical models, the PI aims to elucidate the relation between the structure and dynamics in complex networks. The PI is committed to teaching and training students. A six-month long Research Co-op for two undergraduate students will be organized in the course of this research. The PI will continue to organize minisymposia on dynamical networks at conferences on differential equations and dynamical systems.This research advances the theory for interacting dynamical systems through the development of new theoretical results and analyzing selected models. The PI and colleagues identify new dynamical phenomena and study them using the combination of tools from graph theory, probability, and analysis. The emphasis will be on the effects of random spatial organization and noise on dynamics of large networks. The following problems are addressed: the Large Deviation Principle for interacting dynamical systems on random graphs, metastability in the continuum Kuramoto model forced by noise, and synchronization and pattern formation in the Kuramoto model with inertia. The Kuramoto model of coupled phase oscillators plays a central role in the theory of synchronization with many important applications in science and technology. In particular, the analysis of synchronization and stability of clusters developed in this research elucidates the dynamics emerging in high-voltage power grids.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

我们生活在网络时代。我们的日常生活依赖于我们周围不同技术网络(如电网和通信网络)的强大和可预测的性能。人类生理依赖于数百个细胞网络的协调活动(例如，大脑中的神经细胞网络，心脏和胰腺中的细胞网络，仅举几例)。现实世界网络中的连接模式可能是复杂的，并显示出不平凡的统计特性。了解网络的结构组织如何影响其动力学是相互作用动力系统理论中的主要挑战，这使得它有别于经典空间扩展动力系统的理论，如偏微分方程组或格子动力系统。首席调查员(PI)寻求开发一种系统的数学方法来分析动态网络。PI通过发展新的数学技术和分析具有代表性的数学模型，旨在阐明复杂网络中结构和动力学之间的关系。PI致力于教学和培训学生。在本研究的过程中，将为两名本科生组织为期六个月的研究合作项目。PI将继续在微分方程组和动力系统会议上组织关于动力网络的小型会议。本研究通过发展新的理论结果和分析选定的模型来推进动力系统相互作用的理论。PI和他的同事识别新的动力学现象，并使用图论、概率和分析的工具组合来研究它们。重点放在随机空间组织和噪声对大型网络动力学的影响上。讨论了随机图上相互作用动力系统的大偏差原理，噪声强迫的连续介质Kuramoto模型的亚稳性，以及带惯性的Kuramoto模型的同步和方向图的形成。耦合相位振荡器的Kuramoto模型在同步理论中起着核心作用，在科学和技术上有许多重要的应用。特别是，这项研究中对集群的同步性和稳定性的分析阐明了高压电网中出现的动态。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。