The Synchronization Process Of The Generalized Kuramoto Model And The Synchronization Ability Of Graphene Network

In the study of synchronization theory,Kuramoto model is one of the most classic and successful example.Although the model is simple,it can grasp the nature of the physical.Previously,many studies have shown that,the Kuramoto model’s phase transition process is continuous.Biut in the year of 2011,Spanish team found that the Kuramoto model produced a discontinuous phase transition,known as explosive synchronization,when the frequency was related to the coupling strength.Later,Zhang et al.proposed a frequency-weight model according to the characteristics of some systems.Then,Zhou et al.found the phase transition process changed from the first-order phase transition to the second-order phase transition with the expectation of the probability distribution increasing in the frequency-weight Kuramoto model which does not satisfy the rotation invariance in the fully connected network.On the basis of the predecessors,this article gives the critical value of the forward orbit under the asymmetric Gaussian frequency distribution based on the mean field method.At the same time,the analytical solution is verified by simulationly.And we have explained the the phenomenon that the critical value of the forward orbit decreases with the increasing of the expectation of the Gaussian distribution.Then,based on the analytical results and the simulation results,the critical value μ*in which frequency-weight Kuramoto model changes from the first order phase transition to the second transition is predicted,and we have found the variance of the Gaussian distribution and μ*are related linearly which is explained mathematically.Finally,we find the phenomenon that the critical value of the forward orbit decreases and the critical value of the backward orbit is almost invariant with the expected increasing in the random and scale-free networks,thus the frequency-weight Kuramoto model changes from the first order phase transition to the second order phase transition.In reality,most of the complex networks do not exist in the form of isolated mono-layer networks,and form a multi-layers network or the network of networks with other networks through the structure and function.In this paper,through the research on the structure of graphene and graphite,we observe the influences of the network struc-ture on the network dynamics.In this paper,we first construct the adjacency matrix of graphene,and solve the eigenvalues of the Laplacian matrix corresponding to the mono-lithic graphene network,having found that the synchronization ability of the graphene monolayer network decreases with the increasing of the number of nodes.Then,the monolayer graphene is compared with the monolayer 2D Lattice in the eigenvalues of the Laplacian matrix,and it is found that in the type B network,the synchronization ability of graphene is stronger than that of 2D Lattice,which may be one of the reasons why the graphene is better that other substances(such as copper)in conductivity and thermal conductivity.By comparing the characteristic values of monolayer and mul-tilayer networks,it is found that the synchronization ability of multi-layer network is weaker than that of single-layer network,which may be one of the reasons of that the conductivity of graphene is better than that of graphite.