The Study Of Synchronization For Coupled Phase Oscillator Network

We can find synchronization from different fields in real life. We can enhance the advantageous synchronization and suppress unwanted synchronization through invesatigating synchronization. The fruitful approach was pioneered mathematically by Winfree for investigating collective synchronization. Winfree intuitively recognized that simplifications would occur if the couplings were weak, the oscillators can be characterized solely by their phases. Using the mean-field approximation, Winfree got the phase evolution equation for each oscillator. Based on Winfree’s phase model, Kuramoto presented the classical coupled phase oscillators model. Now, the Kuramoto model has become a paradigmatic model in the study of synchronization. A number of problems about phase synchronization are studied according to Kuramoto model in the dissertation.We know that the attractive (positive) and repulsive (negative) couplings widely exist in real systems, these two couplings together play an important role in different systems’evolution. The corresponding research results show that some unusual phenomena generated by the competed interaction between attractive and repulsive couplings. The evolutive law and nature of some system can be revealed by exploring and analyzing the dynamical behavior, therefore, the mixed coupling interaction has very real research significance. Accordingly, this dissertation analyzes in detail the dynamical behavior of coupled phase oscillators system with these two different couplings interaction together. We investigate the Kuramoto model with different distributions of coupling strengths and natural frequencies of oscillators by means of theoretical analysis and numerical simulation via computer. The results show that the corresponding system presents rich and complex dynamical behaviors.Firstly, a brief introduction of the historical backgrounds and the concept of phase synchronization are given in the dissertation. Then, the phase oscillator model that is Kuramoto model and synchronization on non-globally coupled networks are described in detail. Secondly, the research work of the dissertation is introduced. The research focuses mainly on three parts:1. The dynamics of Kuramoto model consisting of conformists with positive coupling strength and contrarians with negative coupling strength are investigated when the natural frequencies of oscillators follow a uniform probability density. By numerically simulating the model, we find that the model supports a modulated traveling wave state. The dependence of the observed dynamics on the parameters of the model is explored and we find that the onset of synchronization in the model displays a non-monotonic dependence on both positive and negative coupling strength.2. The dynamics of Kuramoto model consisting of conformists with positive coupling strength and contrarians with negative coupling strength are investigated when a complicated correlation between the distribution of natural frequencies and coupling strengths of oscillators is taken into consideration. For the case of correlation between the distribution of natural frequencies and coupling strengths, three correlated situations are analyzed and investigated. We find rich dynamics result from the correlation such as different types of traveling wave states and, most interestingly, another type of nonstationary state:an oscillating%state.3. The dynamics of Kuramoto model consisting of conformists with positive coupling strength and contrarians with negative coupling strength are investigated when an additional second order coupling term is added to Kuramoto model. We find that this system can take an infinite number of multi stable synchronized states in a structurally stable manner by varying the initial condition.At last, the work of this dissertation is summarized briefly, at the same time, the prospect of future research is discussed combined with the work done.