Research On Numerical Hopf Bifurcations Of Nonlinear Delay Differential Equations

This paper deals with stability of numerical solution of Hopf bifurcation and the stability of equilibrium of nonlinear delay differential equation.By using Euler method and nonstandard finite difference scheme,we investigate the numerical solution of Hopf bifurcation of delay differential equation.Finally,our conclusions are tested by examples.The paper has four chapters together.The first chapter is the introduction,which mainly introduces the background and significance of the research.In chapter two,the stability of positive equilibrium and Hopf bifurcation of the Mackey-Glass system are studied by Euler method,and the conclusion is verified by numerical examples.In the third chapter,the stability of positive equilibrium and the Hopf bifurcation phenomenon of x'(t)=-px(t-t)+q?(t)/r+?~?(t) are studied by the nonstandard finite difference scheme.The correctness of the conclusion is verified by numerical examples.The fourth chapter studies the stability of the numerical of Hopf bifurcation and numerical solution of x'(t)=-px(t-t)+q?(t)/r+?~?(t) by Euler method,and gives the theoretical proof and numerical experiments.At last,this paper summarizes the content of this research and gives a prospect for future research.