Application Research Of Several Types Of Time-delay Dynamical Systems

In recent years,with the wide application of delay differential equation in economics,biology,engineering technology and computer science,various dynamic models with delay have been constructed from real life.The branch problem is an important problem in dynamical systems research.This paper mainly studies the application of delay differential equations in biological and Internet congestion models.The results are as follows:In the first chapter,we mainly introduces some basic definitions and theories used in the research process.In the second chapter,a FAST TCP network congestion model with feedback control is studied.Firstly,to delay the Hopf branch,a hybrid controller is added to the FAST TCP network congestion model.Communication delay is selected as a branch parameter to obtain the critical value of communication delay that keeps the original system and the controlled system stable.When the delay 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 third chapter,we mainly investigated a Predator-prey Model with two time delays and refuges effect.By analyzing the characteristic equations,we discussed the local stability of equilibrium point of the system and the sufficient condition for the existence of Hopf branch.By choosing the delay as a bifurcation parameter,we can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using the center manifold theorem and the normal form theory.At last,some numerical simulation results are confirmed that the feasibility of the theoretical analysis.In the fourth chapter,the stability and Hopf bifurcation of a partial profit model with three discrete time delays is investigated in this chapter.Based on the analysis of the root of the characteristic equation,with two combinations of three time delays ?₁,?₂,? as bifurcation parameters,we investigated the local stability of equilibrium point of the system and the sufficient condition for the existence of Hopf branch in two cases.Furthermore,we obtained the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions.At last,the validity of theoretical analysis is verified by numerical simulation.