Dynamic Analysis Of Kuramoto Model With Second Order Coupling Term

In 1960s,based on Winfree's work on collective synchronization of biological systems,Kuramoto proposed a model to describe the group of weakly coupled oscillators.Kuramoto model has been widely used in physics,biology and other fields.The Kuramoto model has attracted a lot of scholars'attention,such as the stability of the equilibrium,the asymptotically complete phase synchronization or frequency synchronization behavior,and the dynamic analysis of the Kuramoto order parameter,etc.In this paper,we will discuss the dynamics of Kuramoto model with first-order and second-order coupling terms.The main work of this paper is as follows:Firstly,The phase-locked state and its stability of Kuramoto model with first-order and second-order coupling terms are discussed.Using linearized matrix and gradient flow theory,the relationship between the stability of phase locked state and the first and second-order coupling strengthk₁,k₂ is studied.We show that the complete phase synchronization equilibrium is asymptotically stable for any k₁,k₂>0.For the bipolar equilibrium,it is asymptotically stable ifk₁<2k₂.Secondly,we prove that the(?)ojasiewicz exponent of the model at the complete phase synchronization equilibrium is 1 2,which means that the convergence of a trajectory towards this equilibrium is exponentially fast.For the bipolar equilibrium,whenk₁<2k₂,the convergence rate is also exponentially fast.Furthermore,based on the monotonicity of the maximum and minimum phase,we estimate the basin of the two phase-locked states,and find that the ratio of the first and second-order coupling coefficients affects the size of basin.Lastly,we give the Kuramoto-Sakaguchi equation of the Kuramoto model and the corresponding measurement theory formula,and lift the finite dimension result for the Kuramoto model to the infinite dimension.