Study On Synchronization Of Kuramoto Model And Chimera State Of Complex Networks

Complex networks have some or all features of self-organization,self-similarity,attractor,small-world,and scale-free,and can be used to describe related models in many fields such as natural sciences,social sciences,and engineering sciences.Complex network is a cross-disciplinary subject that has attracted much attention in the 21st century.It uses statistical physics,computer software,mathematics and other disciplines as research tools and complex systems as research objects.It has become one of the important research tools for describing and understanding complex systems.People have obtained many theoretical results and practical applications concerning complex networks.The synchronization has always been an important issue in the study of complex networks.In the last decade,people have been studying synchronization and related issues.The research scope covers many disciplines such as physics,chemistry,biology,and sociology.The chimera state was discovered when studying the synchronous dynamics of the Kuramoto phase oscillator.Chimera state is closely related to the nervous system.The study of chimera state can be of great help in solving medical problems.Therefore,research on the synchronization and chimera state related problems of complex networks is of great significance to interpreting real-world phenomena or solving practical problems.The main research work of this paper is as follows:(1)based on the Kuramoto local coupled oscillator’s mean-field model,a dynamic model of a nonlinear oscillator on a unidirectional near-neighbor ring is proposed.Firstly,the theoretical analysis of the dynamic stability of the few-body system is performed,and then the fourth order Runge-Kutta method is used to simulate the dynamic equation of the system.The results show that when the number of oscillators in the system is N≤6,the average frequency of all the oscillators converges to the same frequency with the increase of the coupling strength,resulting in a single synchronous state.When the number of system oscillators is N≥7,as the coupling strength increases,there are multiple stationary branches in the system synchronization area.The order parameter R decreases with the increase of the number of oscillators in the system and tends toward zero.(2)based on the generalized Kuramoto model with bi-harmonic coupling function,we consider the positive and negative second harmonic coupling strengths in the system,the dynamic behavior of the system is simulated by the fourth-order Runge-Kutta method.We find that the phase distribution of oscillators has abundant dynamic characteristics.The system has synchronous and traveling wave states.The second harmonic order parameter and the pattern of the system can better reflect the degree of synchronization of the system.When the probability p increases or decreases,there is a sharp discontinuity transition between p=0.45 and p=0.40,reflecting the existence of hysteresis in the system.By the pattern of the system,it is found that the system has the existence of synchronous and traveling wave states.(3)based on the time-discrete Rulkov map neuron system in two-dimensional system,the dynamic model of the neurons with the nearest neighbor and nearest diagonal in the coupled system is proposed.The system is numerically simulated by iterative method.The results show that the snapshots of the membranes of the neurons located at(ij)at the time step n at different chemical synaptic coupling strengths can well reflect the system’s incoherent state,chimera state and incoherent state.Through calculating the order parameter.standard deviation and the strength of incoherence,we find that when the chemical synaptic coupling strength is 0.6<ε<0.8,the system appears chimera state.Finally,we summarize and prospect the research on synchronization of Kuramoto model and chimera state on complex networks.