The Investigation Of Dynamics For The Coupled Oscillators

Various complex systems in the nature and the real life can be described by the networks. The units of the complex systems can be represented by the nodes of the network, meanwhile the relationship of the units can be represented by the edges of the network. The deep research and the evolvement of the complex systems have been push forward by the development of the network science. In network science,synchronization phenomenon is one of researcher’s central issues,and Kuramoto phase oscillators’ model is the classical model for researching the synchronization. We study the synchronization behavior on the complex networks base on the Kuramoto model in this dissertation. Then we study several problems of chimera states on the systems with identical oscillators.The synchronization dynamics on the complex networks should be influenced by some factors, such as the topological structure of network,the coupling strength, and the natural frequency distribution. We introduce a Kuramoto model with a bimodal natural frequency distribution. Then we compare the dynamics of this system in homogeneous and heterogeneous complex networks. We find that the scenario of the synchronization transition depends on the network topology. For a homogeneous network (Erdos-Renyi network), the incoherent state, standing wave state and stationary synchronous state are encountered successively with the increase of the coupling strength. For a heterogeneous network (heterogeneous network), the incoherent state,standing wave state, traveling wave state and stationary synchronous state are encountered successively with the increase of the coupling strength.A spatiotemporal pattern in which identical oscillators split into two domains: one coherent and the other incoherent have been found by researcher and then Abrams give a name for this fascinating phenomenon as "chimera". In this work, we study the dependence of chimera states on initial conditions. We show that random initial conditions may lead to chimera states. Meanwhile, the probability of realizing chimera states becomes increasing when the model parameters are moving away from the boundary of their stable regime. Then, we study the relationship between the lifetimes of chimera states and the size of the systems. The observation defines that the lifetime of chimera states decrease exponentially by decreasing the size of the systems.Subsequently, we consider a system of phase oscillators with both positive controlled coupling and negative controlled coupling and discuss the dynamics of this model. We find one cluster chimera states and two clusters chimera states depend on the proportion of positive controlled coupling and negative controlled coupling. We study the scenario of the chimera states transition and the size of the coherent cluster by varying the parameters. Moreover, we verify that the coexistence of the positive controlled coupling and negative controlled coupling allows the chimera states survive in sufficiently small systems.At last, we give the summary of this dissertation and present the prospect of future research.