Bifurcation Analysis Of Rossler Chaotic System With Delayed Feedback

Time-delayed feedback has been introduced as a powerful tool for control of unstable periodic orbits or unstable steady states. In the present paper, we discuss the effect of Rossler system with delayed feedback.When parameters satisfy some assumptions, Rossler system is a chaotic dynamical system. In order to discuss the dynamical character of Rossler system and use delayed feedback control chaos, in the first, we discuss the stability of the steady states of Rossler system with delayed feedback. When the engenvalues of the linear part are pure imaginary numbers, we obtain the corresponding delay value. The stability of the steady state is lost when the delay passes through the critical value, and simultaneity chaos may vanish, as well as there will be a family periodic solutions bifurcate from the steady states. Then, we derive the explicit formulae for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady states, by using the normal form method and center manifold theorem.At last, we apply the analytical results to an example for the purpose of control of chaos. In numerical simulation, we obtain some figures which support our theoretical results about Rossler system with delayed feedback. Those figures indicate that when delay is very small, the stability of steady states doesn't change and chaos still exist. As delay increases, the stability of steady states is changed. When delay is near some critical value, a family of periodic solutions bifurcate from the steady state.