Linear stability and Hopf bifurcation analysis for exponential RED algorithm with heterogeneous delays

In this paper, the exponential RED algorithm with heterogeneous delays is considered. Local stability of the equilibrium solution of this algorithm is investigated based on analyzing the corresponding transcendental characteristic equation. Some general stability criteria involving the delays and the system parameters are derived by using generalized Nyquist criteria. In particular, using one of the delays as the bifurcation parameter, when the delays exceed a critical value, the exponential RED system undergoes a supercritical Hopf bifurcation. The explicit formulas determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al’s approaches. Finally, some numerical simulations are performed to verify the theoretical results.