Statistical Physics Studies On Explosive Synchronization Phase Transition Of Kuramoto Model On Complex Networks

Synchronization is one of the most common dynamical behaviors in nature. Recently, it has attracted much attention of researchers from science and engineering technology fields. In this dissertation, we study the process and property of synchronization phase transition of Kuramoto model on complex networks. Previous studies demonstrated that there is a continuous phase transition from an incoherent state to a synchronized state in the original Kuramoto model. But one recent study indicated that a discontinuous phase transition can show up in the process of synchronization transition of the scale-free network by incorporating a linear correlation between the dynamical property and the topological structure, which is commonly known as the explosive synchronization. The excitation mechanism and transition process of explosive synchronization are studied systematically from theoretical analysis and numerical simulation by using a series of analysis methods related to statistical physics and nonlinear dynamics. The main contents of this dissertation are summarized as follows:(1) At present the studies on synchronization of complex networks are mainly focused on the uniform coupling, which is actually a kind of ideal scenario. In the real world, the coupling between the nodes may not be uniform because of the difference of the node itself. In order to reflect the differences between nodes, the effects of three different types of correlations, i.e. sub-linear, linear, and super-linear correlations, on synchronization transition process are investigated via introducing a microscopic correlation between coupling and frequency in globally coupled Kuramoto model. The results show that the linear correlation between coupling and frequency can excite an explosive synchronization, which also can save the average node cost.(2) It is widely believed that the large frequency mismatch of a pair of nodes, also known as disassortativity in frequency, is a direct cause of an explosive synchronization. Our study suggests that this is not a comprehensive understanding of the physical mechanism. We perform a large number of numerical simulations for this question by using the control variable method. The criterion for the emergence of explosive synchronization transitions is given from two aspects of the topological structure and the dynamical property. The results show that only when the degrees and natural frequencies of the network’s nodes are both disassortative can an explosive synchronization happen.(3) The weighted network not only describes the topology of the network but also reflects the details of interaction between nodes, which highlights the complexity of network. We extend the study of explosive synchronization into weighted networks. The frequency-gap-weighted coupling scheme is introduced into Kuramoto model on a family of static scale-free networks, in which the degree distribution exponent of network is adjustable. Thus, the influence of the structure control parameter and the weighted exponent on process of synchronization transition is studied. The results show that explosive synchronization cannot occur in the heterogeneous network, and the synchronization transition is continuous. We also find that in homogeneous networks the explosive synchronization is replaced by a continuous phase transition when the microscopic correlation between edge strength and frequency gap is changed from positive to negative.(4) The nodes and edges are the most basic elements of complex networks. We indicate two nodes and their connection as a pair of nodes. Taking a cue from the processing method of two-body problem in Newton mechanics, the notion of reduced frequency, similar to the reduced mass of two-body problem, is introduced to characterize and quantify the dynamical state of each pair of oscillators. It is found that the network undergoes an explosive synchronization transition when the linking probability is positively correlated with reduced frequency. However, the explosive synchronization transition is replaced by a continuous phase transition when their correlation is changed from positive to negative. Besides, the results show that the spontaneous degree-frequency correlation in the networks primarily contributes to an explosive synchronization.