Synchronization Optimization And Explosive Synchronization Of Coupled Oscillators On Complex Networks

Synchronization is the most typical collective behavior and one of the important fields of complex systems and statistical physics.Since Barabasi and Albert found that many actual network connectivity distributions have power-law forms in 1999,people began to carry out extensive research on complex networks.The synchronization of coupled oscillators on complex networks has become a research hotspot.Studies in the past few decades have shown that most synchronous phase transitions in the system are continuous phase transitions(second-order phase transitions),that is,the order parameters increase continuously after passing through the phase transition point.However,it has recently been found that synchronization can be explosive(first-order phase transition)in some coupled oscillator systems.It is characterized by a sudden jump of the order parameter after passing through the phase transition point,and changes with the coupling strength.The forward and backward phase transition process is irreversible and there is a hysteresis region.Because explosive synchronization is essentially different from continuous phase transition synchronization,this discovery opens up a new research direction and attracts a large number of scholars.However,the conditions for explosive synchronization are different for different networks.Explosive synchronization can occur in heterogeneous Scale-Free networks with positive frequency correlation,but will not occur in homogeneous ER networks with positive frequency correlation.The internal mechanism of explosive synchronization is not clear,and whether there are general conditions?As a research direction of synchronization,synchronization optimization has many research results.After synchronous optimization changes the network structure or node frequency,the order parameter curve of the system will become steep.Then,with the deepening of synchronous optimization,whether the order parameter curve of the system may become vertical,that is,become a firstorder discontinuous phase transition.Starting from synchronous optimization,this paper explores the relationship between synchronous optimization and explosive synchronization,in order to better reveal the internal mechanism of explosive synchronization.In the first part of this paper,the synchronization optimization of coupled phase oscillators on two representative simple networks(heterogeneous star network and homogeneous ring network)is studied.In the star network,the synchronization is optimized by changing the node frequency and introducing the control node,and the synchronization optimization problem on multiple coupled star networks is further considered.After synchronous optimization on a homogeneous ring network,it is found that the frequency satisfies a special allocation form(the sum of frequency differences between connected nodes is the largest).With the increase of coupling strength,the system first appears partial synchronization,and then the order parameters appear discontinuous jump.This synchronization process is named mixed explosive synchronization.According to the boundary conditions of global synchronization and partial synchronization,the nodes on the ring are divided into synchronization zone,buffer zone and asynchronous zone.Under different conditions,the performance of the oscillator in the buffer zone is different,and there can be two states:synchronous and asynchronous,which corresponds to the bistable state of the hysteresis zone.Referring to the optimal frequency allocation method on the ring network,we give a better frequency allocation method on the two-dimensional grid network with periodic boundary.It is found that when the frequency meets the special allocation method,the synchronization process of the system is mixed explosive synchronization.The second part of this paper discusses the optimal synchronization problem on complex networks.First,consider adding edge optimization network structure.The network structure is optimized based on the cluster condition when the system is desynchronized.The specific edge adding position is obtained by using the eigenvector corresponding to the maximum eigenvalue of Jacobian matrix near the critical coupling strength of the system.The local optimal networks with adding edge optimization under different frequency distributions basically meet the following conditions:(Ⅰ)the deviation between node frequency and average value is linear with node degree;(Ⅱ)the network is a binary network,in which the oscillators with lower or higher frequency are divided into two groups;(Ⅲ)the region aggregation with adjacency matrix 1 exists and is far away from the diagonal.Based on these three conditions,a global optimal network is constructed,and its critical coupling strength can approach the theoretical minimum.In order to solve the problem that the adding edge optimization network structure cannot optimize the frequency,we propose a set of constructive optimization method based on structural index.According to the change rate of Jacobian matrix eigenvalue with coupling strength,two structural indexes S1 and S2 affecting the critical coupling strength of the system are obtained.The optimization scenarios under three limited conditions are solved by using the structural index(increasing S1 and decreasing S2):optimize frequency allocation for given network structure and frequency distribution,construct frequency distribution for given network structure,and construct network structure for given frequency distribution.When optimizing the frequency allocation,it is found that the areas with small critical coupling strength and explosive synchronization(S1 is larger and S2 is smaller)are highly coincident,that is,explosive synchronization will occur when the critical coupling strength is small.When constructing the frequency distribution,the synchronization process obtained by the same construction method on different types of networks(homogeneous ER network,NW small world network and heterogeneous Scale-Free network)is explosive synchronization.When the optimal network is constructed with a given frequency distribution,the corresponding synchronization process of the system is also explosive synchr-onization.It is found that after optimization,the synchronization process of the system changes from two-stage continuous phase transition to one-stage explosive synchronization,that is,optimized synchronization will lead to explosive synchronization.We analyze the explosive synchronization and get the conditions that need to be met when explosive synchronization appears.Comparing the conditions of optimal synchronization,it is found that when the conditions of optimal synchronization are met,the conditions of explosive synchronization are naturally met.The theory explains the internal reason of optimal synchronization.In this paper,an efficient optimization synchronization method is used to generate explosive synchronization.By connecting the optimization synchronization with explosive synchronization,explosive synchronization can be easily generated under different limited conditions.The research of this paper deepens the understanding of explosive synchronization and provides guidance for controlling the emergence of explosive synchronization.