We investigate the principal parametric resonance of a Rayleigh–Duffing oscillator with
time-delayed feedback position and linear velocity terms. Using the asymptotic perturbation
method, we obtain two slow flow equations on the amplitude and phase of the oscillator.
We study the effects of the frequency detuning, the deterministic amplitude, and the
time-delay on the dynamical behaviors, such as stability and bifurcation associated with
the principal parametric resonance. Moreover, the appropriate choice of the feedback gain
and the time-delay is discussed from the viewpoint of vibration control. It is found that the
appropriate choice of the time-delay can broaden the stable region of the non-trivial
steady-state solutions and enhance the control performance. Theoretical stability analysis
is verified through a numerical simulation.