Synchronization Of High-dimensional Kuramoto Model Based On Digraphs

High-dimensional Kuramoto model, also called Lohe model, is a class of non-linear multi-agent systems. A multi-agent system is a complex dynamical system composed of some differential or difference equations coupled each other according to the structure of a graph. High-dimensional Kuramoto model has a good physical background, which can be used to explain the synchronization phenomenon of quan-tum oscillators. In the existing contributions of high-dimensional Kuramoto model,most of them only consider the case of undirected graphs. But for the case of general digraphs, few results are obtained on high-dimensional Kuramoto model.This thesis investigates the criteria of equilibria and synchronization for high-dimensional Kuramoto model based on digraphs. First, for judging equilibria, the condition of strong connectedness is weakened to the existence of a directed span-ning tree and a necessary and sufficient condition of equilibria is obtained. Second,some results on synchronization are generalized from the case of undirected graphs to that of digraphs, which are listed in details as follows: (1) If the digraph is strongly connected and the initial states of all the oscillators lie in a half sphere, then synchro-nization can be realized; (2) If the digraph has a directed spanning tree and the first strongly connected component has only one vertex, then the states of all the agents will synchronize as the initial states are on a half sphere; (3) If the digraph has a di-rected spanning tree and every inner product of any two initial states is positive, then the states of all the agents synchronize. In this thesis, the main tools are the general-ized LaSalle invariant principle and mathematical induction. Moreover, a new method is used to deal with the synchronization problem of the high-dimensional Kuramoto model, which is obviously different from the existing methods.