Delay Induced Hopf Bifurcation in a Small-world Network Model

This paper is committed to study the stability of delay induced Hopf bifurcation in a nonlinear small-world network model with delay, which can be demonstrated by a one-order delay differential equation. With emphasis on the relationship between the Hopf bifurcation and the time-delay, we investigate time-delay as a bifurcation parameter by using tools from control and bifurcation theory. It is proved that there exists a critical value of time-delay for the stability of the model. When the time-delay passes through the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the bifurcating periodic solution of the system is determined according to the perturbation method. Finally, numerical simulations are given to verify theoretic analysis.