  We study the possibility of occurrence of vibrational resonance in a
softening Duffing oscillator in the underdamped and overdamped cases both
theoretically as well as numerically. The oscillator is driven by two periodic
forces. Numerically we find that in the underdamped case two oscillatory
solutions are obtained in a limited range of the parameters considered (damping
coefficient and amplitude of the high frequency force) for a fixed frequency
and amplitude of the low frequency periodic force depending on the initial
conditions. These solutions have distinct response amplitude to the low
frequency force. When damping is gradually increased, only one oscillatory
solution is observed. Vibrational resonance is observed in both the regions of
oscillation. The analytical approximation yields only one oscillatory solution
for all damping values. Analytically, the peak in the area bounded by the phase
portrait as a function of the amplitude of the high frequency force is
connected to vibrational resonance. Also, the values of the frequency of the
low frequency forcing and the amplitude of the high frequency forcing at which
vibrational resonance is found to occur are obtained. In the overdamped case,
vibrational resonance is not observed for the softening Duffing oscillator thus
showing a marked contrast to the overdamped bistable oscillator
