Hopf bifurcation analysis of the generalized Lorenz system with time delayed feedback control

In this work we propose a feedback approach to regulate the chaotic behavior of the whole family of the generalized Lorenz system, by designing a nonlinear delayed feedback control. We first study the effect of the delay on the dynamics of the system and we investigate the existence of Hopf bifurcations. Then, by using the center manifold reduction technique and the normal form theory, we derive the explicit formulas for the direction, stability and period of the periodic solutions bifurcating from the steady state at certain critical values of the delay.