Mathematical investigations of dynamical systems on complex networks

这项研究的目的是开发新的数学理论和分析工具，用于研究来自许多科学和工程领域的大规模动力系统。PI提出了一个由有向图表示的网络上定义的相互关联的非线性微分方程组的数学框架。图的每个顶点或网络的一个节点可以是生态系统中的空间斑块、传染病的同质宿主种群、生物调控系统中的一种DNA或蛋白质或一种细胞、神经网络中的单个神经元、化学反应中的化学复合体、耦合振荡器网络中的机械、电气或生化振荡器、飞行编队网络中的无人飞行器或纳米卫星。每个顶点处的动力学可以用一个称为顶点系统的低维微分方程组来描述。互联系统的复杂性在于其规模之大，在于顶点系统之间的相互作用。

The objective of the proposed research is to develop new mathematical theories and analytic tools for the investigation of large-scale dynamical systems arising from many fields of science and engineering. The PI proposes a mathematical framework of interconnected systems of nonlinear differential equations defined over a network represented by a directed graph. Each vertex of the graph or a node of the network can be a spatial patch in an ecological system, a homogenous host population for infectious diseases, a species of DNA or protein or a type of cells in biological regulatory systems, a single neuron in neural networks, a chemical complex in chemical reactions, a mechanical, electrical or biochemical oscillator in a network of coupled oscillators, an unmanned aerial machine or a nano-satellite in flight formation networks. The dynamics at each vertex can be described by a low-dimensional system of differential equations called vertex system. The complexity of the interconnected systems lies in its large scale and in the interactions among vertex systems.