时滞反馈控制下五维能源供需系统的稳定性分析及Hopf分支性质  

Stabilization and Hopf Bifurcation Properties of the Five-Dimensional Energy Demand-Supply System Under Delayed Feedback Control

作　　者：杨坤一[1] 董云宁 YANG Kunyi;DONG Yunning(College of Science,North China University of Technology,Beijing 100144)

机构地区：[1]北方工业大学理学院,北京100144

出　　处：《系统科学与数学》2021年第8期2113-2136,共24页Journal of Systems Science and Mathematical Sciences

基　　金：北方工业大学重点学科研究方向(110052972027/014)项目资助课题。

摘　　要：文章考虑五维能源供需系统.首先设计时滞反馈控制,建立施加控制后的五维能源供需系统模型.对于系统的平衡点进行分析,利用特征方程性质及多项式根的分布定理,得到平衡点的各类稳定性的情形.结果发现,施加时滞反馈控制,可使得五维能源供需系统的平衡点由不稳定转化为渐进稳定或周期变化的状态.对于Hopf分支的情形,利用中心流形定理和规范型方法,得到判断Hopf分支性质的参数表达式,进而可得Hopf分支周期解的性质.对于主要结论的各类情形,给出数值图形,验证了反馈控制的稳定效果,以及Hopf分支性质判定式的有效性.In this paper,a five-dimensional energy demand-supply system has been considered.At first we design the time delayed feedback under which the mathematical model has been established.For the equilibrium points of the system,we have the different cases of the stability properties by using the characteristic equation and the distribution of the polynomial.Then we find that,the time delayed feedback control can make the five-dimensional energy demand-supply system stable or change periodically compared to the originally unstable status.For the case of the appearance of the Hopf bifurcation,we apply the center manifold theorem and normal form method to obtain the parameters’ expressions judging the properties of Hopf bifurcation.Further we have the properties of the periodic solutions for the Hopf bifurcation.For the different cases of the main results,we simulate the states of the system to show the stability of the feedback control and the effectiveness of the discriminants for the Hopf bifurcation.

关 键 词：五维能源供需系统 时滞反馈控制 平衡点 稳定性 HOPF分支 

分 类 号：O415.5[理学—理论物理] O231[理学—物理]