On Dynamic Behavior Of High-dimensional Kuramoto Model

Synchronization is a common phenomenon in nature.The behavior of many an-imal groups is usually disordered at the beginning.After a period of time,the groups will display some synchronization behavior.For example,the fireflies glowing regu-larly,the frogs singing together in summer night and birds flying in the same direc-tion.The collective behavior of these animals has aroused researchers' strong interest in exploring synchronous behavior.In order to better explain these synchronous phe-nomena,researchers have established many mathematical models.Kuramoto model is one of the classical mathematical models that can describe some collective behav-ior and solve the synchronization problem.Many scholars have done a lot of research work on Kuramoto model,which is widely used in physics,bioengineering,mechan-ical engineering and other fields.High dimensional Kuramoto model is a complex dynamic system composed of multiple differential equations.The dynamic equations of individuals are coupled according to the structure of a graph.In the study of the ex-isting high-dimensional Kuramoto model,there are many research results in the case of complete graphs and a large number of results have not been extended when the connection graph is not a complete graph.When the connection graph is an m-regular graph,general undirected graph or directed graph,the existing research results on the high-dimensional Kuramoto model are still relatively few.In this thesis,some dynamic behaviors are investigated for the high-dimensional Kuramoto model in the case that the connection graph not being complete,some exist-ing research results are generalized,new problems are proposed and new theoretical results are obtained.Firstly,for the first-order high-dimensional Kuramoto model,some results in the case of complete graphs are generalized to the case of m-regular graphs.Secondly,for the second-order high-dimensional Kuramoto model under con-nected undirected graphs,sufficient conditions for the convergence to the equilibrium points are obtained.In addition,for the case of directed graphs,the first-order high-dimensional Kuramoto model composed of three individuals is studied.Not only the exponential synchronization is proved,but also the exponential decay rate of the syn-chronization errors is accurately provided.Finally,the obtained theoretical results are verified by simulations.