Delay-induced bifurcations in a nonautonomous system with delayed velocity feedbacks

This paper investigates the bifurcations due to time delay in the feedback control system with excitation. Based on an self-sustained oscillator, the delayed velocity feedback control system is proposed. For the case without excitation, the stability of the trivial equilibrium is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or periodic solutions may occur. For the case with excitation, the main attention is focused on the effect of time delay on the obtained periodic solution when primary resonance occurs in the system under consideration. To this end, the control system is changed to be a functional differential equation. Functional analysis is carried out to obtain the center manifold and then a perturbation approach is used to find periodic solutions in a closed form. Moreover, the unstable regions for the limit cycles are also obtained, predicting the occurrence of some complex behaviours. Numerical simulations are employed to find the routes leading to quasi-periodic motions as the time delay is varied. It has been found that: (i) Time delay can be used to control bifurcations; and (ii) time delay can be applied to generate bifurcations. This indicates that time delay may be used as a "switch" to control or create complexity for different applications.