The Evolution And Dynamics In Complex System

Modern scientific is experiencing an epoch-making change in this transformation bred a new subject - complex system science. In recent years, the reseach of complex system has become the international forefront of scientific research and hot spots, many famous universities have set up relevant departments and research institutes, researchers from various fields, including physicists, ecologists, economists, various engineers, entomologists, computer scientists, linguists, sociologists and political scientists. A study on the complex system competition has been launched around the world. In this paper, two types of complex systems are analyzed through theoretical analysis and computer simulation. In detaided, we study the anti-synchronization in coupled chaotic system and the evolution of cooperation in population with heterogeneous structure.In chapter 1, we introduce the history and background of complex system science, specially, things related to this paper, such as chaotic system, complex network and society system.In chapter 2, we investigate the phenomenon of anti-synchronization (AS) in coupled chaotic systems. Firstly we prove that the anti-synchronization can not be regarded as a special form of synchronization. Then according to the eigenvalue analysis, the necessary conditions and the parameters of the anti-synchronization are presented. The dynamics behaviors of coupled Lorenz system and chua system are investigated intensively. Various phenomena are explored such as hysteresis, coexistence of AS and un-AS state. Druing the inverstigation of the routine to AS, we find in the coupled Lorenz system, the coupled chaotic oscillators undergo on-off intermittency to AS. And in the coupled Chua system, the anti-synchronization state is achieved by competition between anti-synchronization and other states.From chapter 3 to chapter 6, we study the evolution of cooperation in structured populations. Firstly, in chapter 3, we introduce the basic knowledge of game theory and some related conclusions. Then, in chapter 4, chapter 5 and chapter 6, we respectively study the evolution of cooperation under the effect of different update rules, heterogeneous structure and heterogeneous payoff matrix.In chapter 4, by introducing Helbing-Schlag proportional imitation update rule (rule-A) and its modified version (rule-B), we study the evolution of cooperation on star-like networks. During the investigation of single hub structure, we find the state of hub nodes leads the evolution of the whole system. Especially under rule-A, the strategy of hub node will be the strategy taken by the whole population. Under rule-B, the dependence on the state of hub node is weakened. But the evolutionary stable state is still initially dependent. However, system shows a rich dynamic. The phase space develops to three different regions:never-jump-area, one-jump-area and circulatory-jump-area. The investigation on coupled two single hub structure shows the dynamic under rule-B is less affected than that under rule-A. However, there is still a strong relationship in dynamics between the whole system and the subsystem.In chapter 5, we try to build a thoeretical model to give an explanation to the widely proved phenomenon that cooperation can be improved when the population of selfish players is located on a heterogenous network, such as scale-free network in the prisoner's dilemma. Thus, we build three simple networks with increasing complexity on their network structures. Then according to a master equation, we develop the replication equations for the evolution of cooperation on these networks. The explicit formulations for the cooperator frequency on these networks are deduced and the relationship between the cooperator frequency and the network heterogeneity is discussed.In chapter 6, from the point of the heterogeneity of payoff matrix, we investigate the evolution of cooperation on regular lattice. In detailed, the heterogeneity of payoff matrix is introduced to the population in two different ways:the heterogenous character of payoff matrix assigned to every player (model-A) or the heterogenous character of payoff matrix assigned to every link between any two players (model-B). An enhancement of cooperation in model-A is observed while an inhibition of cooperation in model-B is prominent. The explanations on the enhancement or inhibition of cooperation are presented for these two cases.