Hopf Bifurcation Analysis In A Lorenz Equation With Extended Delay Feedback

Delay feedback control, an important method in the control theory, is very effective, because both current state and the past states are considered in the design of the controller. It is becoming an important issue in the nonlinear dynamical science to investigate the role that delay feedback plays in controlling chaos.This paper focuses on the bifurcation analysis of differential equations with extended time delay feedback control. Further more, we choose a chaotic system, Lorenz system, as the basic model in order to reveal how this control acts on the chaos.Firstly, we conclude that a system of differential equations with extended delay feedback can be transformed into functional differential equations with infinite time delay or equations of neutral type.Secondly, local dynamical analysis of the model is obtained. By analyzing the roots distribution of the characteristic equation of the linearized equation, stability results of the three equilibria and the existence of Hopf bifurcation are obtained. Stability switches with respect to parameter delay are detected. Numerical simulations indicate that certain'chaos switches'appears simultaneously. To explore more dynamics near the bifurcation points, we give formula to calculate the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions, by center manifold theorem and the normal form method.Finally, some simulations are given to support the theoretical results.