Investigation On A Kind Of Differential Equation Models Of Energy Demand-supply

In the modern industrial society, survival of human being has been closely related to the development of energy material, such as coal, oil, gas. Furthermore, energy plays an important part in the economic growth, especially in China. Most of researches pay attention to problems of energy Using differential equation models, we study the phenomenon of energy supply and demand in this thesis.This thesis is composed of three sections. The first section analyzes stability properties of equilibrium points for the three-dimensional energy demand and supply system. We obtain the sufficient conditions of stability of the equilibrium point O, and the sufficient and necessary conditions of asymptotic stability of the equilibrium point S2, moreover, the equilibrium point S, is illustrated to be always unstable if the coefficients of the model satisfy that a1b1<a2b3N Numerical simulations illustrate effectiveness of our results. In the second section, the three-dimensional energy demand and supply system with time delay has been considered. We analyze stability properties of the equilibrium points and obtain the sufficient condition of the system where the Hopf bifurcation exists. By using the center manifold theory and normal form method, we deduce the explicit formulae which show properties of Hopf bifurcation. Numerical simulations also confirm our results. In the third section, the delayed feedback control has been imposed on the unstable system to make it stable or periodical. We analyze stability properties of the closed-loop system, and infer the explicit formulae revealing the properties of Hopf bifurcation. Numerical simulations demonstrate effectiveness of the results. At last, we consider the three-dimensional energy demand and supply system with time delay in which linear feedback control has been designed. Our analysis shows that the kind of linear feedback control may be an effective design method to make the unstable system stable.