Asymptotic Behavior Of Discrete Kuramoto Model On Graphs

Complex network often appears in real life and nature.With the development of network science and technology,the dynamic behavior of complex networks has gradually become one of the hot topics discussed by scholars in various fields.Because the Kuramoto model is one of the representative models for studying dynamic behaviors,based on the discrete Kuramoto model with external drive,this paper uses the theory of graph limits and the properties of complex network dynamics to study asymptotic behavior on the deterministic graphs,W-random graphs and small-world graphs.The paper first introduces the basic concepts of graphs,the basic knowledge of Kuramoto model,graph limits and random variables,and then introduces the main research contents,which are as follows:First of all,the asymptotic behavior of the discrete Kuramoto model with external drive on the deterministic graphs is analyzed.Based on the theory of graph limits,the initial value problem of the discrete Kuramoto model with external drive is studied on the simple graphs and the weighted graphs respectively.Moreover,the research results show that the solution converges to the continuous solution with respect to n.Secondly,the asymptotic behavior of the discrete Kuramoto model with external drive on the W-random graphs is analyzed.Based on the Lyapunov central limit theorem and the theory of graph limits,the initial value problem of the discrete Kuramoto model with external drive is studied on the W-random graphs.Moreover,the results show that the solution converges to the continuous solution with respect to n,and it is found that the convergence rate only depends on the Lyapunov central limit theorem on the W-random graphs.Finally,the asymptotic behavior of the discrete Kuramoto model with external drive on the small-world graphs is analyzed.Based on the Lyapunov central limit theorem and the theory of graph limits,the initial value problem of the discrete Kuramoto model with external drive is studied on the small-world graphs.Moreover,the results show that the solution converges to the continuous solution with respect to n,and it is found that the convergence rate depends on the Lyapunov central limit theorem and the regularity of graphon W_p.