Single and multiple vibrational resonance in a quintic oscillator with monostable potentials

We analyze the occurrence of vibrational resonance in a damped quintic oscillator with three cases of single well of the potential V(x)=1/2ω₀²x²+1/4βx⁴+1/6γx⁶ driven by both low-frequency force f cos ωt and high-frequency force g cos Ωt with Ω⪢ω. We restrict our analysis to the parametric choices (i) ω₀², β, γ>0 (single well), (ii) ω₀², γ>0, β<0, β²<4ω₀²γ (single well), and (iii) ω₀²>0, β arbitrary, γ<0 (double-hump single well). From the approximate theoretical expression of response amplitude Q at the low-frequency ω we determine the values of ω and g (denoted as ω_VR and g_VR) at which vibrational resonance occurs. We show that for fixed values of the parameters of the system when ω is varied either resonance does not occur or it occurs only once. When the amplitude g is varied for the case of the potential with the parametric choice (i) at most one resonance occur while for the other two choices (ii) and (iii) multiple resonance occur. Further, g_VR is found to be independent of the damping strength d while ω_VR depends on d. The theoretical predictions are found to be in good agreement with the numerical result. We illustrate that the vibrational resonance can be characterized in terms of width of the orbit also.