Hopf bifurcation of a five-dimensional delay differential system

Abstract

A five-dimensional delay differential system is proposed to describe oscillatory behaviour in microbial continuous culture. Stability and Hopf bifurcation are discussed, including the changing regularity of the bifurcation value. An explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and centre manifold. Finally, some numerical simulations are performed to illustrate the analytical results found.

Keywords:

• delay
• Hopf bifurcation
• stability
• periodic solution
• oscillation

2000 AMS Subject Classifications :

• 92C40
• 34K13
• 34K17

Acknowledgements

This work was supported by HIT(WH)2008009, HIT(WH)ZBQD201006, Weihai Municipal Science and Technology Development Program, the National Natural Science Foundation of China (No. 10471014, No. 10671126), and National Basic Research Program of China (No. 2007CB714304).