Application Of The Differential Equations Theory Under The Actual Background

In this paper,the stability of the differential equations systems under practical backgrounds have been considered.On the one hand,the conditions for the stability properties of the equilibrium points have been given in certain range of the coefficients for the system,considering the time delay phenomenon in the energy transportation process,a five-dimensional energy demand-supply system with time delay have been established,and the dynamic behavior of the system is analyzed.Given the system parameters,the state diagram of the system change is simulated.On the other hand,we established three-dimensional environmental pollution system,a delayed feedback controller is proposed to stabilize the system,and the dynamic analysis of the system have been analyzed.This paper is mainly composed of the following three parts.In the first part,we have obtained the conditions for stability of the equilibrium pointsS₀ and S by using the Hurwitz Criterion.Simulation results show availability of the conditions for stability properties.In the second part,we have been devoted to the five-dimensional energy demand-supply system which suffers from a given time delay.Firstly,the system with time delay has been modeled.Then the linear approximation method is used to calculate the corresponding characteristic equation of the linear system,and analyzing the negative roots of the characteristic equation by the Hurwitz Criterion and the quartic equation.Stability of the equilibrium points are analyzed by the switching theorem.Explicit formulae of Hopf bifurcation has been deduced by the central manifold theorem and poincare normalization method.At last,numerical simulations illustrate effectiveness of the main theorems.The third part,firstly we establish the three-dimensional environment pollution dynamical system which describes the relationship among the air quality index,population and economic growth.This mathematical model has been obtained after introducing the economical growth factors in the two-dimensional environment pollution dynamical system.Secondly we compute the the equilibrium points of the three-dimensional environment pollution dynamical system.And then we analyze stability of the equilibrium points by using the theory of differential equations,which can be shown to be available in the following simulation numerical results.Finally,a delayed feedback controller has been imposed on the three-dimensional environment pollution dynamical system,which may make the originally unstable system stable for some appropriate feedback coefficients.Moreover,numerical simulations illustrate the effectiveness of the feedback control.