Dynamics In The Kuramoto Model With A Discontinuous Bimodal Distributions Of Natural Frequencies

With the rapid development of science, people begin to use network thinking to study nature and social life. Along with the advancement of research, scholars have proposed many different types of network models, such as regular networks, random network, deterministic network, small-world networks and scale-free networks.For the research of complex networks, people are interested in synchronization in dynamic phenomenon. In real life, we can also find synchronization phenomenon in different fields. Through understanding of it, we can enhance the synchronizations which are beneficial to the people, and suppress those which are harmful for us. Winfree found a mathematical method, which can effectively deal with the collective synchronization phenomenon. When the coupling between the oscillators is relatively small, we can use phase to make the description for the Oscillator motion state simpler. Later, he gave the evolution equations of oscillator phase by the mean-field approximation method. Based on the research of Winfree, Kuramoto proposed a network model of coupled phase oscillators which is also called Kuramoto model.In this paper, we consider a Kuramoto model in which the natural frequencies of oscillators follow a discontinuous bimodal distribution constructed from a Lorentzian one. Different synchronous dynamics (such as different types of travelling wave states, standing wave states, and stationary synchronous states) are identified and the transitions between them are investigated. On the basis of the Kuramoto model, we change the distribution of the natural frequency of oscillators, making it into a discrete distribution. We also research how the changes of parameters influence the synchronous state, observe the transition between each synchronization state, and analyze its theoretical significance. We change the symmetry of the distribution of by ωd, and find not only the critical value between each synchronous states changed, but also the synchronous state changed. Increasing the asymmetry in frequency distribution brings the critical coupling strength to a low value and that strong asymmetry is unfavorable to standing wave states. In this paper, we elaborate on the reasons and the detailed process.