Bifurcation analysis for Chenʹs system with delayed feedback and its application to control of chaos

Time-delayed feedback has been introduced as a powerful tool for control of unstable periodic orbits or control of unstable steady states. In the present paper, regarding the delay as parameter, we investigate the effect of delay on the dynamics of Chenʹs system with delayed feedback. We first consider the effect of delay on the steady states, and then investigate the existence of local Hopf bifurcations. By using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.