The Research Of Stability And Hopf Bifurcation Of Several Differential Systems

The stability and Hopf bifurcation is important in the qualitative theory of dy-namic system,they have various application in nature and the field of engineering technology.Such as biology,mechanics,artificial intelligence,pattern recognition and so on.During the nineteenth century,Lyapunov establish foundation to the theory of stability,who is a Russian mathematician.He create series of theory and method for stability.Hopf bifurcation describe the bifurcated limit cycle when the parameter cross the critical value for the variety of stability of equilibrium.In 1981,B.Hassar,N.Kazarinoff,Y.Wan concluded the Hopf bifurcation of infinite dimension dynamics.Based on the theory of operators,center manifold theorem and normal form.They give the method for computing the property of Hopf bifurcation for delayed system,such as direction of bifurcation,stability of bifurcated limit cycle.In this paper,the stability of delayed neural network system,zero-Hopf bifurcation of nuclear spin generator system,global Hopf bifurcation in a delayed phytoplankton-zooplankton system with competition and Hopf bifurcation of zooplankton-phytoplankton model with three kinds of plankton and two delays are studied.The dynamics of above system are studied extensively,they have important application in relative field.This dissertation is divided by five parts.In the first part,the definition and theorem about the stability and Hopf bifur-cation of differential system are given,besides we give introduction to the models and relative research which turn up in this dissertation.In the second part,we study the existence,uniqueness of equilibrium for cycle associative neural networks with constant delays.Then we demonstrate the global exponential stability of this system.Finding a proper Lyapunov function is important in our work.In the third part,we study the zero-Hopf bifurcation of nuclear spin generator system.There have been many works about the dynamics of this system,such as integrability and chaos.By the first or second order averaging method,we demonstrate there exist a limit cycle for the system,whose amplitude is not small.which is different from limit cycle with small amplitude by classical Hopf bifurcation.In the fourth part,we study the global Hopf bifurcation of delayed phytoplankton-zooplankton system with competition.Taking the delay as parameter,we discuss the Hopf bifurcation around the positive and singular point.Then we discuss the global existence of bifurcated limit cycle.Compare to previous work,we consider Holling?response function and the delay caused by gestation of zooplankton,furthermore we demonstrate the global existence of bifurcated limit cycle.In the fifth part,we study the bifurcation of zooplankton-phytoplankton model with three kinds of plankton and two delays,where the two delays are caused by gestation of two zooplankton.For the complex of multi-delays,we discuss the dynamics of this three dimensional system under six cases:?1??₁=0,?₂=0,?2??₁=0,?₂>0,?3??₁>0,?₂=0,?4??₁=?₂>0,?5??₁?(0,?₁₀),?₂>0,?6??₂?(0,?₂₀),?₁>0.We analysis the Hopf bifurcation of this system when the two delays coexist.Compare to previous work,we take the two gestation delays of two zooplankton as bifurcation parameter and give a new explanation to the dynamics of plankton system.