Mean Field Analysis of Dynamical Networks

自然界、社会和技术中的大量重要结构和现象可以通过相互作用的动力系统网络来建模。例子包括技术中的电力和通信网络，生物学中的神经元和遗传网络，以及社会和经济网络。理解这些系统的行为需要先进的数学技术来研究复杂网络中的动力学。该项目将使用数学建模，分析和数值模拟来阐明复杂网络中结构和动力学之间的联系。特别是，研究者的目的是找到新的方法来描述网络组织的动力学模型和研究关键现象，如同步和出现和分叉的空间模式在网络中的耦合振荡器的发病。这项研究的结果和将在其过程中开发的工具，将提高我们理解，预测和控制真实的世界网络行为的能力。平均场近似是研究相互作用动力系统大系综最有效的分析工具之一。这种方法最初是为了解决统计物理学中的问题而开发的，在研究各种物理、化学、生物和技术系统的耦合振荡器模型中的集体动力学方面非常成功。耦合相位振子的仓本模型中的同步分析和嵌合态的分叉分析依赖于在极限中正式导出的平均场方程，因为振子的数量趋于无穷大。尽管它在应用中取得了巨大的成功，但网络上耦合动力系统的平均场近似的数学基础还没有得到很好的理解。 在这项研究中，目的是推导和严格证明的平均场描述的动力学耦合系统的收敛族图。这在两个方面推进了耦合系统的平均场理论：第一，通过将该理论的主要结果扩展到一大类随机网络，包括那些具有小世界和无标度连通性的网络;第二，通过放松关键的正则性假设，阻止该理论应用于真实的世界网络。新的理论结果被用来研究在系统的耦合相位振荡器与空间结构的相互作用，稳定的动力学制度的仓本模型的幂律图，和图案的形成在随机介质中的神经领域的同步。

A large number of important structures and phenomena in nature, society, and technology can be modeled by networks of interacting dynamical systems. Examples include power and communication networks in technology, neuronal and genetic networks in biology, as well as social and economic networks, to name a few. Understanding behavior of these systems requires advanced mathematical techniques for studying dynamics in complex networks. This project will use mathematical modeling, analysis, and numerical simulations to elucidate the link between the structure and dynamics in complex networks. In particular, the investigator aims to find new ways for describing network organization in dynamical models and study critical phenomena, such as the onset of synchronization and emergence and bifurcations of spatial patterns in networks of coupled oscillators. The results of this research and the tools that will be developed in its course, will enhance our ability to understand, predict, and control the behavior of real world networks. The mean field approximation is one of the most effective analytical tools available for studying large ensembles of interacting dynamical systems. Originally developed for problems in statistical physics, this method has been extremely successful for studying collective dynamics in coupled oscillator models of various physical, chemical, biological, and technological systems. The analysis of synchronization in the Kuramoto model of coupled phase oscillators and the bifurcation analysis of chimera states rely on the mean field equation formally derived in the limit, as the number of oscillators goes to infinity. Despite its spectacular success in applications, the mathematical basis of the mean field approximation of coupled dynamical systems on networks is not well understood.  In this research, the aim is to derive and rigorously justify the mean field description of dynamics of coupled systems on convergent families of graphs. This advances the mean field theory for coupled systems in two ways: first, by extending the main results of this theory to a large class of random networks, including those with small-world and scale free connectivity, and, second, by relaxing the key regularity assumptions, blocking the application of this theory to real world networks. The new theoretical results are used to study the onset of synchronization in systems of coupled phase oscillators with spatially structured interactions, stable dynamical regimes in the Kuramoto model on power law graphs, and pattern formation in neural fields in random media.