Stability and Hopf bifurcation analysis of a chaotic system using time-delayed feedback control method

In this paper, we study the effect of delayed feedback on the dynamics of a three-dimensional chaotic dynamical system and stabilize its chaotic behavior and control the respective unstable steady state. We derive an explicit formula in which a Hopf bifurcation occurs under some analytical conditions. Then the existence and stability of the Hopf bifurcation are analyzed by considering the time delay $ \tau $ as a bifurcation parameter. Furthermore, by numerical calculation and appropriate ascertaining of both the feedback strength $ K $ and time delay $ \tau $, we find certain threshold values of time delay at which an unstable equilibrium of the considered system is successfully controlled. Finally, we use numerical simulations to examine the derived analytical results and reveal more dynamical behaviors of the system.