Existence and convergence of Neimark–Sacker bifurcation for delay differential equations using Runge–Kutta methods

Abstract

This paper considers the Runge–Kutta methods used on one-dimensional delay differential equations (DDEs), which undergo Hopf bifurcation from the origin. It not only shows the numerical approximations undergo Neimark–Sacker bifurcation, but also these Neimark–Sacker bifurcation points close the Hopf bifurcation point of the DDEs with the order of methods. Meanwhile, compared with another interpolation method, we conclude that the given interpolation method is better in the convergence of Neimark–Sacker bifurcation points. Finally, by using 2-stage Gauss method to delay Logistic equation, the theoretical analysis is supported.

Keywords:

• Neimark–Sacker bifurcation
• Hopf bifurcation
• delay differential equation
• Runge–Kutta methods
• interpolation method

2000 AMS Subject Classifications :

• 34K18
• 65P30
• 74G15

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (No. 10671047) and by the Science Research Foundation in Harbin Institute of Technology (200713).