John Abidemi Laoye

Vibrational Resonance (VR), which is characterised by the enhancement of the maximum response of a weakly driven system’s output signal induced by a high-frequency (HF) periodic signal, was numerically examined in a bi-harmonically driven dimensionless model of an enzymesubstrate reaction with fractional-order damping, which models coherent oscillations in brain waves. The model incorporates a damping force that depends on a non-integer (fractional) order derivative rather than the typical first-order derivative in classical damping models. The output response was obtained by solving the model numerically using Grunwald-Letnikov’s fractional derivatives definition. The response amplitude,¨ computed from the Fourier spectrum of the output signal, was used to characterise VR. The effect of the fractional-order damping coefficient on the observed VR was considered for different damping strength coefficients. Single-peak resonances were observed. The fractional-order damping modulated the observed VR in a manner similar to the damping strength in an integer-order system, by reducing the high-frequency signal amplitude at which VR occurs. Increased brain wave activity from enzyme-substrate reaction may be due to inherent energy transfers from changes in the rate of decay, hence significant behavioural changes in brain wave activity could be linked to inherent changes in the decay rate of the excited enzymes, even when there is no change in the number of enzyme-substrate carriers. This study reveals the potential of fractional-order damping for enhancing biophysical system modelling, with implications for understanding brain wave activities.

Living systems are composed of molecules and atoms that can exhibit specific quantum effects, including coherent electric vibration, and vibrating systems that undergo resonance. Due to their resonance, biological materials can transfer energy when they vibrate at a specific frequency. In this paper, we establish the occurrence of vibrational resonance in a biophysical system modeling activated enzyme molecules in the brain waves. The VR was characterised using the response amplitude, Q which was numerically calculated using the Fourier coefficient of the output signal. The external high-frequency signal affected the enzyme-substrate reaction and amplified the combination process observed at resonance. Importantly, double resonances were induced by the strength of the decay rate. Resonance, induced by the strength of the decay rate and the external high-frequency signal would lead to significant brain wave activity related to inherent energy transfers from changes to the decay rate, and enzyme-substrate combination, and could be observed from the Electroencephalogram.

We examine the impacts of time-delay and phase shift between two acoustic driving forces on vibrational resonance (VR) phenomena in the oscillations of a spherical gas bubble. Using the approximate method of direct separation of the motions, we obtain the equation of slow motion and the response amplitude, and we validate the theoretical predictions with numerical simulations. We find that the response amplitude of the system at the lower frequency varies periodically with respect to the phase shift. When the phase shift consists of an even number of periods, it can be optimized to enhance the system’s response in the relevant parameter space of the high-frequency driving force. In addition to the enhancement of the VR peak by variation of the phase shift, our results show that the time-delay also plays a significant role in the bubble’s response to dual-frequency acoustic driving fields. It and can be exploited either to suppress drastically, or to modulate, the resonance peaks, thereby controlling the resonances. Our analysis shows further that cooperation between the time-delay and the amplitude of the high-frequency component of the acoustic waves can induce multiple resonances. These results could potentially be exploited to control and enhance ultrasonic cleaning processes by varying the time-delay parameter in the presence of phase shifted dual-frequency acoustic waves. Moreover, it could be employed to achieve improved accuracy in ultrasonic biomedical diagnosis and tumour therapy, as well as for targeted delivery of reagents transported within bubbles.

The vibrational resonance (VR) phenomenon has received a great deal of research attention over the two decades since its introduction. The wide range of theoretical and experimental results obtained has, however, been confined to VR in systems with constant mass. We now extend the VR formalism to encompass systems with position-dependent mass (PDM). We consider a generalized classical counterpart of the quantum mechanical nonlinear oscillator with PDM. By developing a theoretical framework for determining the response amplitude of PDM systems, we examine and analyse their VR phenomenona, obtain conditions for the occurrence of resonances, show that the role played by PDM can be both inductive and contributory, and suggest that PDM effects could usefully be explored to maximize the efficiency of devices being operated in VR modes. Our analysis suggests new directions for the investigation of VR in a general class of PDM systems. This article is part of the theme issue ‘Vibrational and stochastic resonance in driven nonlinear systems (part 1)’.

The role of nonlinear dissipation in vibrational resonance (VR) is investigated in an inhomogeneous system characterized by a symmetric and spatially periodic potential and subjected to nonuniform state-dependent damping and a biharmonic driving force. The contributions of the parameters of the high-frequency signal to the system's effective dissipation are examined theoretically in comparison to linearly damped systems, for which the parameter of interest is the effective stiffness in the equation of slow vibration. We show that the VR effect can be enhanced by varying the nonlinear dissipation parameters and that it can be induced by a parameter that is shared by the damping inhomogeneity and the system potential. Furthermore, we have apparently identified the origin of the nonlinear-dissipation-enhanced response: We provide evidence of its connection to a Hopf bifurcation, accompanied by monotonic attractor enlargement in the VR regime.