Numerical HOPF Bifurcation Of ROSENBROCK Methods For A Class Of Delay Differential Equations

Delay differential equations play an important role in the natural sciences, social sciences and engineering fields. The theoretical research and numerical analysis are very important. It is an attractive branch in applied mathematics, especially in some realistic models of Hopf bifurcation with delays such as the Van der Pol equation, the predator - prey system and the Chemostat model. But many delay differential equations can not be solved explicitly, so numerical calculation becomes an important method to study delay differential equations. In the numerical study, it is concerned to investigate whether the corresponding numerical discrete systems can maintain the dynamical behavior of the original systems.This paper studies the stability of the delay differential system of equilibrium point and Hopf bifurcation. It focuses on the properties of discrete system, and the discrete system produced by the Rosenbrock method applied to the original. The properties include determining the bifurcation parameter value, the direction of branch and the stability of bifurcating periodic solutions.Briefly describes the stability function and characteristic equation of Rosenbrock method. A class dimensional delay differential systems with parameter is concerned. Applying the method with order p to the syetem, the corresponding numerical discrete system is obtained. It is proved that if the original system undergoes a Hopf bifurcation, then the corresponding numerical discrete system undergoes a numerical Hopf bifurcation.It is also proved that the direction of Hopf bifurcation and stability of the invariant curve of the numerical discrete system obtained by the Rosenbrock method with order p are the same as that the original system under certain conditions.The Van der Pol equation, the predator - prey system and the Chemostat model with delays are described briefly. The numerical Hopf bifurcation of the three models are obtained. Also, some numerical examples of the three models by the Rosenbrock method with stage 2 are given to support the theories above.