Mathematical investigations of dynamical systems on complex networks

拟议研究的目标是开发新的数学理论和分析工具，用于调查科学和工程许多领域中产生的大规模动力系统。PI提出了一个数学框架的互联系统的非线性微分方程定义在一个网络表示的有向图。图的每个顶点或网络的节点可以是生态系统中的空间斑块、传染病的同质宿主群体、生物调节系统中的DNA或蛋白质的种类或细胞的类型、神经网络中的单个神经元、化学反应中的化学复合物、耦合振荡器网络中的机械、电或生物化学振荡器，无人机或飞行编队网络中的纳米卫星。每个顶点的动力学可以用称为顶点系统的低维微分方程系统来描述。互联系统的复杂性在于它的规模大和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.