Hopf Bifurcation Analysis Of A Class Of Abstract Delay Differential Equation

Differential equations have a long history of describing realistic phenomena.As an significance way for dynamic systems to express their properties,bifurcation connects nonlinear differential equations to dynamic systems,and gives detailed descriptions of various evolution systems.Therefore,it's of great actual important to study the bifurcation of dynamical systemst.Hopf bifurcation,as a class of bifurcation phenomenon,is very important in dynamical systems.As equations that can be describe the evolution systems dependent on both the present state and the past state,delay differential equations widely into the objective world we live in.In the fields of ecology,economy,medicine,etc.,delay differential equations can describe actual phenomena that occur in their own words.Therefore,delay dynamic systems can be seen everywhere in our lives.The investigation of the bifurcation problems in delay differential equations,needs not only functional,topology,but also the theory of dynamical and differential equations.Hence,a comprehensive and in-depth study of delay dynamic systems has both intense practical and theoretical background.This paper mainly studies the Hopf bifurcation problem of a class of abstract delay differential equations.The main creativities in this thesis are as follows:Firstly,by analyzing the characteristic roots,we obtain the stability of the equilibrium point,and derive the conditions for the Hopf bifurcation to occur.Secondly,we base our theory on the central manifold theory,and project the studied equatin onto the central manifold.Then we use the normal form method to gain the formula to determine the Hopf bifurcation direction and stability of bifurcation periodic solutions,amplitude,which can represent properyis of bifurcation.And then,we perform comprehensively the existence of global Hopf bifurcation to the systems.We obtain that the bifurcation periodic solution of the system equation still exists when the parameter is getting larger and larger and larger and larger distance from the bifurcation value.In addition,the number of possible bifurcation periodic solutions of the equation is discussed.