  In this paper, an attempt has been made to understand the parametric
excitation of a periodic orbit of nonlinear oscillator which can be a limit
cycle, center or a slowly decaying center-type oscillation. For this a delay
model is considered with nonlinear feedback oscillator defined in terms of
Li\'enard oscillator description which can give rise to any one of the periodic
orbits stated above. We have characterized the resonance and antiresonance
behaviour for arbitrary nonlinear system from their stability and bifurcation
analyses in reference to the standard delayed van der Pol system. An
approximate analytical solution using Krylov--Bogoliubov (K-B) averaging method
is utilised to recognize the sub-harmonic resonance and antiresonance, and
average energy consumption per cycle. Direction of Hopf bifurcation and
stability of the periodic solution bifurcating from the trivial fixed point are
carried out using normal form and center manifold theory. The parametric
excitation is also thoroughly investigated via bifurcation analysis to find the
role of the control parameters like time delay, damping and nonlinear terms.
