Mathematical investigations of dynamical systems on complex networks

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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.