Research On The Application Of Time-delay Dynamic Systems In Economic And Network Congestion Control Models

In recent years,with the rapid development of economy and society,time-delayed dynamical systems have been widely used in many fields such as economics,population ecology,finance,engineering technology and social science.Branch is usually used to study structurally unstable systems and is an important topic in the study of dynamical systems and nonlinear differential equations.This paper mainly studies the application of delay differential equations in economic and Internet congestion models.The results are as follows:The first chapter is mainly about the general introduction and understanding of research topics.The second chapter mainly introduces some basic definitions and theories used in the research process.In the third chapter,we study a business cycle model with time-delayed feedback.Firstly,a time-delay feedback controller is added to the business cycle model,and a new model is proposed.Secondly,the linear stability of the model and the local Hopf bifurcation are studied.Using the center manifold theorem and the the normal form theory,the stability of the Hopf bifurcation direction and the bifurcation periodic solution are studied.At last,some numerical simulation results are presented to confirm that the controller can effectively increase the stability region of the business cycle model.In the fourth chapter,the fluid flow model with time delay is studied.First,the communication delay is chosen as the bifurcation parameter.When the dealy value passes through the critical value,the equilibrium loses its stability and a Hopf bifurcation emerges.In addition,the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are studied by using the center manifold theorem and the normal form theory.Finally,some numerical simulation results with mathematical software are confirmed that the feasibility of the theoretical analysis.In the fifth chapter,the stability and Hopf bifurcation of a price Rayleigh model with time-delayed feedback is investigated in this chapter.We discussed the effect of the delay on the system.Some numerical simulation results are confirmed that the feasibility of the theoretical analysis.