chaos control and hopf bifurcation analysis of a three-dimensional chaotic system

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 bi-furcation are analyzed by considering the time delay τ as a bifurcation parameter. furthermore, by numerical calculation and appropriate as-certaining of both the feedback strength k and time delay τ , we find certain threshold values of time delay at which an unstable equilibrium of the considered system is successfully controlled. finally, we use nu-merical simulations to examine the derived analytical results and reveal more dynamical behaviors of the system.