Numerical Hopf Bifurcation For Delay Differential Equations Of Fractional Order

Fractional delay differential equations are widely used in cybernetics, biology,computer science, economics and other fields. In these areas, we can optimize the performance of the system and improve the system stability through the study of the fractional delay differential equation parameters. Therefore, the study of fractional order delay differential system is very necessary. In this thesis, we study the numerical Hopf bifurcation of the fractional order delay differential equations, mainly including the following aspects.Firstly, we analysize the Hopf bifurcation of the fractional delay differential equation theoretically, and obtain the condition for the parameters of the Hopf bifurcation of the fractional delay differential equation.Secondly, assumed the parameter of the fractional order delay differential equation is the critical value of Hopf bifurcation. By using linear multistep method for discrete fractional order delay differential equations, the discrete scheme is processed. And the ability to maintain the Hopf bifurcation of fractional order delay differential equations is analyzed by using the fractional order linear multistep method. When the parameters of the fractional order delay differential equation passes through the critical value, the solution of the equilibrium point of the equation will be changed, and the phenomenon of Hopf bifurcation occurs. We obtain that the numerical Hopf bifurcation phenomenon occurs when the discrete schemes are experiencing numerical critical points. We also give the approximation result between the critical value of Hopf bifurcation which generated by discrete scheme and the analytic ones. Finally, by applying the fractional backward differentiation formula, the fractional Euler method, and the fractional trapezoidal rule to the fractional order delay differential equation, we give the numerical experiments to confirm the derived conclusion.