Vibrational resonance in a higher-order nonlinear damped oscillator with rough potential

Acoustic vibrational resonance in a Rayleigh-Plesset bubble oscillator

“…It is conjectured that the system can also admit additional equilibrium solutions depending on the choice of truncation of the binomial expansion of in connection with higher-order nonlinear terms of the potential functions. Research shows that up-to-triple well solutions are possible in higher-order nonlinear systems in contrast with the familiar single and double equilibrium solutions reported earlier for lower-order potential functions [53] , [115] , [123] , [124] , [125] . Similarly, a flip-flop between hard-spring and soft-spring bistabilities due to higher-order truncation of the Toda oscillator was observed and analyzed by Goswami [126] while, earlier, a third-order approximation reduced the Toda oscillator model to the Heńon–Heiles (HH) Hamiltonian system which is non-integrable in contrast to the original integrable Toda Hamiltonian [127] , [128] , [129] .…”

“…Note that the effective resonance frequency can play a direct and significant role analogous to the effective nonlinear dissipation reported in Refs [7] , [10] , [54] , [53] in the enhancement of signals by modulating the parameters of the fast signal. The effective resonance frequency parameters, which can be computed from the second term of Eq.…”

“…The net flow of heat energy is characterised by a thermal viscosity that has been reported to influence the free oscillations of the bubble strongly, particularly during the transition between the adiabatic and the isothermal states [103] , [104] , [105] . Recently, the impact of non-linear damping/dissipation on the occurrence of VR in nonlinear systems has also gained considerable research attention [7] , [10] , [53] , [54] . In view of this, and to account for the roles of all the individual factors contributing to damping of the bubble dynamics, and their influence on the VR phenomenon, it is essential to treat the liquid and thermal viscosities, as well as the acoustic radiation, independently in the derivation of the equation of motion.…”

“…Previous reports had shown that the system potential may consist of a smooth background potential plus a rapidly and randomly varying potential that may alter the dynamics of the system when the particles are insufficiently energetic to overcome the net potential barriers. When a dynamical system with energetic particles encounters such potential, the system is capable of annulling the effects on its trajectories [53] . This case of (double-well-double-hump potential) is therefore consistent with the stable and unstable equilibria depicted as case II provided that the potential barrier at the hump ( ) located at is significantly higher than the barrier ( ) near .…”

“…The phenomenon has gained enormous research attention in the last two decades and has been extensively investigated due its several potential industrial and biomedical applications in a wide range of contexts including bistable systems [45] , [46] , [47] , [48] , multistable systems [19] , [49] , systems with various forms of potential structures [13] , [15] , linearly damped systems [1] , [50] , electrical circuits and time-delayed systems [5] , [6] , [51] , [52] , as well as plasma models [10] , [53] . Although earlier VR studies focused on the parameters of the high-frequency component of the bi-harmonic force, some recent VR research has explored the effects of parameters such as nonlinear damping/dissipation [7] , [10] , [53] , [54] , the complementary roles played by system’s innate bifurcation parameters [13] , [48] , [55] , the effect of time-delay [52] , [56] and fractional derivatives [57] , [58] , [59] , antiresonance [60] , the depth of a potential and the location of its minimum [61] , the effects of potential deformation [62] and potential roughness [53] , and mass-dependent VR [63] , amongst others. More importantly, VR has been realized experimentally in vertical cavity surface emitting lasers and optical systems [45] , [46] , [49] , [64] , [65] .…”