The Dynamics In The Coupled Phase Oscillator System

There are many kinds of collective behaviors in nature, and synchronization behavior is one of the most attractive form, which occurs between interacting individuals. Coupled oscillator system is often used to study the synchronization behavior in mathematics. Furthermore, the oscillators can be characterized by their phase when the couplings are weak. Thus, the coupled oscillator can be approximated by the coupled phase oscillator.In the study of the dynamical behavior of the coupled phase oscillators, the Kuramoto model has become a typical model. In the classical Kuramoto model, there is an assumption that is the coupling strength between the oscillators is positive and equal. That is to say, the interaction between the oscillators is attracted to each other. However, in the actual system, the interaction between the oscillators can be either attractive or repulsive, and the interaction strength between the oscillators can also be related to the oscillators. Improvement has been made to the Kuramoto model, such as add on repulsive coupling. A study found that there are some interesting dynamic phenomena in the deformation of different Kuramoto models.In this paper, we assume that the coupling strength between the oscillators is different, and there is also a phase lag in the interaction between the oscillators. We not only make numerical simulation, but also use the Ott-Antonsen ansatz to analyze the synchronous behavior of the oscillators. A brief introduction of synchronization and Kuramoto model is given in the dissertation. And my main research work is as following :1, First of all, we study a model which consists of two subpopulations, each with a different phase lag and interaction strength.We find that there are two kinds of synchronization states in this system:one is the traveling wave states, and the other is the stationary synchronous states. We find that the stationary and traveling wave states can be smoothly connected through the properly chosen parameter paths without bifurcation, and two of traveling wave states can also be smoothly connected without bifurcation.2, Secondly, we study the synchronization dynamics in a system of two interacting populations of phase oscillators. In this system the inter-population coupling and the intra-population interaction are different. We explore different types of synchronization dynamics when the incoherent state becomes unstable, and we find that the inter-population coupling is crucial to the synchronization. When the intra-population interaction is repulsive, the local synchronization can still be maintained through controlling the inter-population coupling.3, Finally, we study the synchronization dynamics in a system of multiple interacting populations of phase oscillators based on the second work. We find that, for attractive inter-population coupling, the local order parameters in different populations are of in-phase while the local synchronization are of anti-phase for repulsive inter-population coupling.At the end of the paper, the work of this dissertation is summarized briefly.