Death Of Coupled Oscillators And The Two Clustered Chimera State

Rich dynamics of the chaotic systems has been explored and revealed by researchers since great efforts have been stressed on nonlinear dynamics research. The method of chaos control and the synchronization of chaotic systems have been developed rapidly. In the last decades, various synchronization phenomena and the amplitude death caused by parameter mismatches or time delay in coupled systems have been found among coupled chaotic oscillators. In this article, we have studied the phase death of the coupled systems consisted of two Lorenz oscillators with parameter mismatch, and explained the mechanism behind the phase death. Then we introduce the chimera state in one-dimensional nonlocally coupled system, in which exist several synchronizing domains, while between them the oscillators are asynchronous. We have described the picture of chimera state and explained it by means of OA ansatz. At last, we have investigated the cooperation in an evolutionary prisoner’s dilemma on complex networks and have pointed that the heterogeneity caused by the degree-degree correlations of networks suppressed cooperation.In chapter one, we introduce the knowledge of nonlinear dynamics and chaos, describe the chaotic synchronization and the cooperation in the evolutionary games in the complex networks.In chapter two, we introduce several chimera states in different systems including clustered chimera, spiral chimera and’breathing’ chimera. We find that two clustered chimera state in the simplest system (one-dimensional nonlocal system) without time-delay coupling. We describe the picture of two clustered chimera in detail and use the OA ansatz to explain the phenomena in theory.In chapter three, we introduce the phase death of the coupled Lorenz oscillators with multistable states. The mechanism behind it is that: increasing the coupling strength, the basin of the stationary equilibrium expands and finally causes the phase death. (We classify these phase deaths into two types:complete phase death and incomplete phase death, and show the different parameter spaces they exist. Finally, we give the basin of the stable equilibrium in the incomplete phase death.In chapter four, we introduce the degree-degree correlations of networks, and describe the heterogeneity caused by the networks with degree-degree correlations by means of the generating functions. Finally, we point out that the heterogeneity suppresses the cooperation level. Chapter five is the summary of the whole article.