Dynamics of Large Networks

科学技术中的许多非常重要的问题导致了对相互作用动力系统的大网络的分析。我们预测癫痫发作、有效控制电网或协调一组机器人的能力，取决于我们对耦合动力系统集体行为基础原理的理解。我们周围的许多自然和人造网络都具有极其丰富和复杂的互连。这类网络的数学建模对非线性科学提出了新的挑战，需要将组合和概率方法结合到复杂系统的动态分析中。图论具有非凡的潜力，可以激发扩展动力系统的新的强大技术，并将其应用于技术、社会、经济和生物网络。在这项研究中，首席研究员(PI)将最先进的图论技术与动力系统的分析方法相结合，开发了一套有效的工具来研究耦合动力系统及其在神经科学中的应用。这项研究的结果将被整合到动力系统和数学神经科学的研究生课程中。在这个项目中，PI开发了一种统一的方法来研究动态网络作为三种类型空间域上的演化方程：Caley图、准随机图和图极限。对于每一类中的问题，确定了与基本空间结构相适应的分析和代数技术。这些技术被用来研究Caley图上耦合相位振荡器系统的空间模式的稳定性，耦合混沌映射系统中的同步，不同随机图(包括小世界图和表现出幂规律行为的图)上的微分方程，以及在神经网络学习模型中表现出时间结构稳定模式的那些。PI寻求系统的方法来描述网络连通性在塑造耦合系统中的动力学中的作用。这项工作旨在发展一种关于正则图和随机图上的相互作用的动力系统的理论。

A number of very important problems in science and technology lead to the analysis of large networks of interacting dynamical systems. Our ability to predict epileptic seizures, to effectively control a power grid, or to coordinate a group of robots rely on our understanding of the principles underlying collective behavior in coupled dynamical systems. Many natural and man-made networks around us feature extraordinary richness and complexity of interconnections. Mathematical modeling of such networks poses new challenges for nonlinear science and requires new approaches incorporating combinatorial and probabilistic methods into dynamical analysis of complex systems. Graph Theory holds an extraordinary potential to inspire new powerful techniques for extended dynamical systems and applications to technological, social, economic, and biological networks. In this research, the Principal Investigator (PI) combines state-of-the-art techniques of Graph Theory with analytical methods for Dynamical Systems to develop an effective set of tools for studying coupled dynamical systems and their applications in neuroscience. The results of this research will be integrated into graduate courses in Dynamical Systems and Mathematical Neuroscience. In this project, the PI develops a unified approach for studying dynamical networks as evolution equations on three types of spatial domains: Caley graphs, quasirandom graphs, and graph limits. For problems in each class, analytical and algebraic techniques, which mesh well with the underlying spatial structures, are identified. These techniques are used to study stability of spatial patterns in systems of coupled phase oscillators on Caley graphs, synchronization in systems of coupled chaotic maps, and differential equations on different random graphs including small-world and graphs that exhibit power law behavior, as well as those that exhibit temporally structured stable patterns in a neural network model of learning. The PI seeks systematic ways for describing the role of network connectivity in shaping the dynamics in coupled systems. This work is aimed toward development of a theory for interacting dynamical systems on regular and random graphs.