| Physical problems in complex systems have attracted the attention from physical community. In the thesis, we investigate five such problems. The related systems is scientific map, traffic flow and simple pendulum. About the scientific map, A scientific map different from the traditional one is retrieved using the method to construct production space in economics, the the origin of this difference is also discussed. About the traffic flow, we study the problems:the effect of human rhythm on packet delivery and the flux fluctuation phenomena in this system, and each problems is organised as a chapter. About the simple pendulum, several approximate expressions of the pendulum period is retrieved, and these may shade light on the method to construct the approximate expression and a more simple and accurate expression may be obtained in the near future.The method to construct scientific map is presented in chapter2. Scientific map is a useful tool to travel in scientific knowledge. After it has been proposed in1973, there is only one method to construct such a map. The method is based on citation analysis. Although it is recognized that the scientific maps constructed by means of this method is not accurate, a detailed discuss on this method is not presented at presented. This is due to the lack in the researching method. Here, we using the method proposed by Hidalgo in2007to construct a different type of scientific map, and discuss the problem in the method to construct such maps. The original method is used to construct produce space. The Higalgo’s method is consisted of two parts:the method to retrieve correlation matrix and the method to visualize this matrix using complex network. Both method we will use in mapping knowledge domains. Then, we obtain a correlation matrix about knowledge domains which is called discipline and its visulization using complex network. By comparing the matrix and the visulization of the Higalgo’s method and those of the traditional method, we discuss the difference between the two menthods.Special, after the matrix is visuallized as a complex network, we layout the network according to its community property. We investigate the these two networks corresponding to the Higalgo’s method and the traditional method respectively. And it is found that there are significant difference between those two networks. Furthermore, we discuss only two parts of the difference. The results suggest that the network based on citation analysis represents the correlation of the problems focused on by two disciplines, and the network based on Hidalgo’s represents the correlation that how much knowledge a discipline use in another. This difference is very important and may shade light on the the further investigation in this field.The effect of human rhythm on packed delivery is presented in chaper3. Communication networks such as the Internet, the relationship between packet generation rate and time is similar to a rectangle wave function due to the rhythm of human. Thus, we investigate the traffic dynamics on network with rectangle wave packet generation rate. It is found that the critical delivering capacity parameter βc (which separates the congested phase and the free phase) decreases significantly with the duty circle r of rectangle wave for package generation. And, in the congested phase, more collective generation of packets (smaller r) is helpful to decrease packet aggregation rate. Moreover, it is found that the congested phase can be divided into two regions, i.e., regionl and region2, where the distribution of queue lengthes are nonlinear and linear, respectively. Also, the linear expression of the distribution of queue lengthes in region2is achieved analytically. Our work reveals the obvious effect of rectangle wave on traffic dynamics and the queue length distribution in the system, which is of essential interest and may provide insights into the designing of work-rest schedule and routing strategies.The flux fluctuation phenomena in network traffic flow is presented in chaper4. Recent work has revealed a law governing flux fluctuation and the average flux in complex dynamical systems. We establish the universality of this flux-fluctuation law through the following steps:(1) we derive the law in a more general setting, showing that it depends on a single parameter characterizing the external driving,(2) we conduct extensive numerical computations using distinct external driving, different network topologies, and multiple traffic routing strategies, and (3) we analyze data from an actual vehicle traffic system in a major city in China to lend more credence for the universality of the flux-fluctuation law. Besides its fundamental importance in complex systems, the flux-fluctuation law can be used to infer certain intrinsic property of the system for potential applications such as control of complex systems for improved performance.The method to constructed an approximate expression of the period of a simple pendulum in the case of a large amplification is discussed in chapter5. Comparing with the series expansion of the exact period, a new approximate formula and its modulated form for the period of the simple pendulum are pre-sented. The constructed formula and its modulated form can be applied in the amplitudes of114°and136°, respectively, its relative error is less than1‰Some analyses on constructing of the period of a simple pendulum are discussed.Two approximate expressions of the period of a simple pendulum in the case of a large amplification and its modification is presented in chapter6. Accu-rate approximate analytical formulae of the pendulum period composed of a few elementary functions for any amplitude are constructed. Based on an approxi-mation of the elliptic integral, two new logarithmic formulae for large amplitude close to180°are obtained. Considering the trigonometric function modulation results from the dependence of relative error on the amplitude, we realize accu- rate approximation period expressions for any amplitude between0°and180°. A relative error less than0.02%is achieved for any amplitude. This kind of modulation is also effective for other large-amplitude logarithmic approximation expressions.At last we conclude the whole thesis in chapter7. |
