Hopf bifurcations in time-delay systems with band-limited feedback

To discuss the effects of parameter change of time-delay and feedback control on the dynamics of a nonlinear device with feedback, the stability and Hopf bifurcation analyses on the role played by delay and control parameter in a time-delay system with band-limited feedback were investigated. By analyzing the associated characteristic equations, it is found that the control parameter and delay can qualitatively change the dynamics, and the Hopf bifurcation occurs when the parameters pass through a sequence of critical values. The stability and direction of Hopf bifurcation are determined by applying the Hassard normal form theory and the center manifold theorem. Results of some numerical simulations on Hopf bifurcation agree well with the theoretical results, which mean that the system control can be realized by adjusting the magnitude of time delay and feedback gain.