The Property And Stability Analysis Of Equilibrium Point Of Time-delay Ordinary Differential System

This paper studies the differential equation system with time delay.On the basis of the original four-dimensional energy supply and demand system,considering the time delay in the system,a four-dimensional energy supply and demand system with time delay was established.Then,on the basis of the original two-dimensional environmental pollution model and considering the factors of economic growth,a three-dimensional environmental pollution model was established.However,there is a time delay in practical applications,so a three-dimensional environmental pollution nonlinear dynamic model with time delay is established.The article is mainly divided into three parts: the stability study of a four-dimensional energy supply and demand system with time delay;the establishment and stability analysis of a three-dimensional environmental pollution model;and the stability analysis of a three-dimensional environmental pollution model with time delay.The paper is organized as follows.In the first part,consider a four-dimensional energy supply and demand system with time delay.Considering the time delay between energy demand shortage,energy supply increase,energy import and renewable energy utilization,first establish a four-dimensional time lag system,and then analyze the stability of the system equilibrium point.After the linearization matrix is obtained,the Routh-Hurwitz criterion and the distribution of the roots of the cubic equation are used to judge the negative roots of the characteristic polynomials,and further study the stability of the system.Using the central manifold theorem and the canonical method,It has been the nature of Hopf bifurcation.Finally,a numerical simulation is performed to verify the consistency of the theoretical results.In the second part,the three-dimensional environmental pollution model and stability analysis are studied.Considering the relationship between air quality index,population and total economic growth,first establish a three-dimensional environmental pollution dynamics model to find the equilibrium point of the model,and use the Routh-Hurwitz criterion and the distribution of the roots of the cubic equation to determine the characteristic polynomial negative roots,further study the stability of the system and further analyze the stability of the equilibrium point.Finally,a numerical simulation is performed to verify the consistency of the theoretical results.In the third part,the three-dimensional environmental pollution model with time delay is studied.In practical applications,consider the time delay between the air quality index,the number of people,and the total economic growth.First,establish a three-dimensional environmental pollution model with time delay,and then analyze the stability of the equilibrium point of the three-dimensional time-delay system.Finally,the theory and results are consistent with numerical simulation.