Study On Collective Dynamic Behavior Of Coupled Oscillator Systems In Complex Network

With the development of science and technology,complex network comes into our life more and more.As an emerging interdisciplinary discipline in the21 st century,its research content involves various fields: mathematics,physics,sociology,economics and so on.Among them,collective behavior on complex network has always been a very important research topic in the field.In recent years,a variety of research achievements have emerged: the initial study of synchronization in nonlinear systems,the discovery of chimera state and explosive synchronization and so on.These findings have led to a better understanding of complex phenomena in the real world.Although there have been a lot of research results,there are still a variety of problems that have not been solved,therefore,the collective dynamic behavior and characteristics of coupled oscillator system in complex network are studied in this paper.The main research work of this paper is as follows:1、Based on the Kuramoto model,a two-layer one-way coupled phase oscillator model is proposed.In this model,the coupling mode of the first layer is nearest neighbor coupling,the coupling mode of the second layer is global coupling,and there is a one-way point-to-point couple from the second layer to the first layer.Through numerical simulation and analysis,it is found that when the system parameters are changed,the oscillators in the first layer will change from unsynchronized state to the synchronized state,and the system will have chimera state within a certain range of parameter.The phase diagram of the system about these three states are obtained,it is found that the critical value of state transition increases with the increase of coupling strength between the same layers.Maximum Lyapunov exponent of the system is calculated,when the system is in the unsynchronized state and chimera state,the Maximum Lyapunov exponent is greater than zero,it means that the system is sensitive to initial values and is in the chaotic state.When the system is in synchronized state,the Maximum Lyapunov exponent is less than zero,the system loses its initial sensitivity and is in an ordered state.2 、 Based on the phase oscillator model of two-body interaction,a unidirectional driven coupled phase oscillator model with three-body interaction is presented.The response layer is the Kuramoto model with three-body interaction coupling,and the driven layer is the Kuramoto model with global coupling.For this system,in a certain range of the parameter,the system will be in the traveling wave state.The velocity and the direction of the motion of the traveling wave is obtained by numerical simulation.When the response layer can synchronize,the phase of the response layer oscillator is hysteretic with that of the driver layer oscillator.Increasing the driving strength between layers or decreasing the coupling strength within layers will reduce the lag time.At the same time,there is a power-law relationship between the lag time and the driving strength,and the power-law exponent between the two will decrease with the increase of the coupling strength within the layer.3、An adaptive network model is proposed based on Kuramoto model.The instantaneous order parameter of the system is used as adaptive control factor,and noise is introduced to study the effect of noise on dynamic behavior of the system.The explosive synchronization is found in this system and adding noise does not change the characteristics of explosive synchronization of the system.The width of the hysteresis curve of the order parameter fluctuates slightly when the noise strength is small,and increases rapidly when the noise strength is strong.At the same time,the system bifurcates due to the addition of noise,and some frequency synchronization clusters appear.For this model,similar dynamic behaviors are shown when the network structure is changed.