研究政府调节在能源供需关系中的作用, 考虑相对能源缺口对政府调节的影响, 建立政府调节下的能源供需系统, 分析系统的平衡点及其稳定性. 并且考虑实际政府调节过程中的时间延迟现象, 建立含时间延迟的政府调节下能源供需系统, 分析时滞系统的稳定性. 利用中心流形定理和规范性方法, 得到判定~Hopf 分支性质的计算公式. 给定参数, 模拟系统图像, 验证理论结果.

This paper considers an energy demand-supply system under the government regulation for the two regions in China. The stability properties of the equilibrium points are analyzed. Subsequently the system of energy supply and demand under government regulation with time delay has been established. Then stability of the equilibrium points is analyzed and several advantage are reflected through simulations and it is found that Hopf bifurcation appears under some conditions. By using the center manifold theorem and normal form method, we obtain the explicit formulae revealing the properties of the periodic solutions of Hopf bifurcation. Numerical simulations illustrate effectiveness of our results.