M. Anisha Nashrin, S. M. Abdul Kader, M.V. Sethu Meenakshi, V. Chinnathambi, S. Rajasekar
Analysis of Vibrational Resonance in a Parametric Quintic Oscillator with Double-Well Potentials
Dynamic Systems and Applications 32 (2023) 294-313
https://doi.org/10.46719/dsa2023.32.16
ABSTRAT:
The phenomenon of vibrational resonance (VR) has been investigated in a parametric quintic oscillator with five cases of double-well potential V(x) = – (1/2) (ωo)2 (1 + q cos ωpt) x2 + (1/4) βx4 + (1/6) ϒ x6 while 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) (ωo)2, β, ϒ > 0 (double-well), (ii) (ωo)2, ϒ > 0, β < 0 (double-well) (iii) (ωo)2, β > 0, ϒ < 0, 4 (ωo)2 ϒ < β2 < (16/3)(ωo)2 ϒ (double-hump double-well). (iv) (ωo)2, β > 0, ϒ < 0, β2 = 4(ωo)2 ϒ (double-hump double-well), and (v) (ωo)2, β > 0, ϒ < 0, β2 > (16/3)(ωo)2 ϒ (double-hump double-well). For Ω >> ω, the solution of the system consists of slow motion with frequency ω and fast motion with frequency Ω. The flow equation approach is used to drive the response amplitude ao(ω) analytically from the equation for slow motion of the system, in terms of the parameters of the high-frequency signal and the parametric excitation. From the analytical expression of ao(ω), we determine the values of g (denoted as gVR ) at which VR occurs. Numerical simulations are carried out to validate the theoretical results. We show that for fixed values of the parameters of the system, as g is varied, single or multiple vibrational resonances occur in the double-well cases of the system. gVR is found to be independent of the damping strength d. Moreover, the effect of damping strength d is found to decrease the response amplitude ao(ω).
AMS (MOS) Subject Classification. 34K18, 337C29, 65P20, 65P30, 74H65.
Key Words and Phrases. Parametric quintic oscillator, Double-well potential, Vibrational resonance, Bifurcation, Chaos.