Bifurcation analysis in the control of chaos by extended delay feedback

Bifurcation scenario in control of chaos with extended delay feedback is considered. Lorenz system is investigated as a neutral delay differential equation. Stability and Hopf bifurcation properties are obtained which show that chaos vanishes at a backward Hopf bifurcation and again appears after a forward Hopf bifurcation with the increasing of time delay, and that chaos is hardly detected under feedback with large delay. We find the branch of Hopf bifurcation being interrupted by chaotic attractor. Extended delay feedback is compared with the traditional delay feedback. These results are obtained in either theoretical or numerical way which provides a clear interpretation for control of chaos with extended delay feedback. Finally some simulations are carried out and applications are given with respect to Makey–Glass equation and Rössler system.