Three Essays On Exploring Classical Nonautonomous Systems And Econophyisical Systems

In this dissertation, we study two typical examples of nonautonomous systems. One is the one-dimentional classical harmonic oscillator, the other is one-dimentional generalized Fokker-Planck equation. The former is a simple but useful system. The latter describes the phenomena of fat tail distribution of log-return in stock market, which is a complex system. Moreover, another complex system studied is the one in national economy which describes the gap between increasing rate of aggregate demand and aggregate supply of China from 1992 to 2001. Finally, we also report some results of the research on mesoscopic systems in collaboration with other graduates.In the process of studying the one-dimentional classical harmonic oscillator, we address the question that how to get the exact solution of this system by means of algebraic dynamics. The one-dimentional classical harmonic oscillator is an su(1, 1) nonautonomous classical system which is an important model in physics. However, to obtain its exact solution is too hard to achieve. In this paper, we use algebraic dynamical method to get its exact solution for the first time. As the time-dependent coefficients of the harmonic oscillator assumes some elementary functions, such as power functions, trigonal functions, exponential11111c? l ions etc., the exact solutions become analytic. The recent I y proposed "analytic approximate solution" can be deduced from our corresponding analytic solution obtained. At last, we compare this two solutions by means of numerical compul ;i t i on.In 1 ho study of the one-dimentional generalized l-'okkor Planck equation, we also use algebraic dynamical iikmIkxI to get its exact solution for the first time. Then Based on the exact solution, an important issue in modern econophysics— the fat-tail distribution in stock market is analyzed. We propose a generalized Fokker-Planck equation with time dependent drift-diffusion coefficients and aquadratic potential. Because this system has SU (1,1) ?UQ)dynamical symmetry, the exact solution can be obtained by means of algebraic dynamics. Based on the exact solution, we analyze an important issue in econophysics— the fat-tail (list ri hut ion and get some important conclusions. For example, for the generalized FPE system there can exist only the lime1 dependent correlation potential and the static eoiT'elat ion potential is not allowed in consistence with the equilibrium steady Gaussian distribution.I'or another complex system—the national economic system, we const met a nonlinear model and explore the gap between increasing rate of aggregate demand and aggregate supply of China from 1992 to 2001. In economics, the study of the gap...