Vibrational Resonance Phenomenon Research Articles

Pitchfork bifurcation and vibrational resonance in a neutral delayed system

Abstract The pitchfork bifurcation and vibrational resonance phenomenon are analytically and numerically investigated in the neutral delayed system. The results show that the neutral delayed time induces the pitchfork bifurcation transiting between supercritical and subcritical cases while the neutral delayed strength may induce supercritical pitchfork bifurcation. The neutral delayed time can also induce the periodic or quasi-periodic vibrational resonance. This period simultaneously contains the period of the low-frequency harmonic excitation and period of the high-frequency harmonic excitation. The results extend the investigations of the pitchfork bifurcation and vibrational resonance.

Ultrasensitive vibrational resonance induced by small disturbances.

We have found two kinds of ultrasensitive vibrational resonance in coupled nonlinear systems. It is particularly worth pointing out that this ultrasensitive vibrational resonance is transient behavior caused by transient chaos. Considering a long-term response, the system will transform from transient chaos to a periodic response. The pattern of vibrational resonance will also transform from ultrasensitive vibrational resonance to conventional vibrational resonance. This article focuses on the transient ultrasensitive vibrational resonance phenomenon. It is induced by a small disturbance of the high-frequency excitation and the initial simulation conditions, respectively. The damping coefficient and the coupling strength are the key factors to induce the ultrasensitive vibrational resonance. By increasing these two parameters, the vibrational resonance pattern can be transformed from ultrasensitive vibrational resonance to conventional vibrational resonance. The reason for different vibrational resonance patterns to occur lies in the state of the system response. The response usually presents transient chaotic behavior when the ultrasensitive vibrational resonance appears and the plot of the response amplitude vs the controlled parameters shows a highly fractalized pattern. When the response is periodic or doubly periodic, it usually corresponds to the conventional vibrational resonance. The ultrasensitive vibrational resonance not only occurs at the excitation frequency, but it also occurs at some more nonlinear frequency components. The ultrasensitive vibrational resonance as transient behavior and the transformation of vibrational resonance patterns are new phenomena in coupled nonlinear systems.

Theory and numerics of vibrational resonance in a three-level atomic optical bistable system

The vibrational resonance phenomenon in a three-level atomic optical bistable system driven by two kinds of different frequency signals is investigated. By virtue of the method of the direct separation of fast and slow motions, the approximate analytical expression for the response amplitude Q at the low-frequency is obtained. It is found that the response amplitude presents a peak with the amplitude and frequency of the high-frequency signal varying, and it is identified as the characteristic signature of the vibrational resonance. Meanwhile, the position, at which the vibrational resonance occurs, is determined by the analytical expression of the response amplitude. The effects of the other parameters on vibrational resonance in the three-level atomic optical bistable system are also investigated. The mechanism of vibrational resonance phenomenon is given by the phase synchronization between the input and output signals. The numerical simulation and theoretic results are in basic agreement.

Dynamical response and vibrational resonance of a tri-stable energy harvester interfaced with a standard rectifier circuit.

This paper investigates the dynamical response and vibrational resonance (VR) of a piecewise electromechanically coupled tri-stable energy harvester (TEH), which is driven by dual-frequency harmonic excitations. To achieve a stable DC output, the TEH is interfaced with a standard rectifier circuit. Using the harmonic balance method combined with the separation of fast and slow variables, a steady-state response together with the analytical expressions of displacement and harvested power is derived. The multi-solution feature in the amplitude-frequency response is observed and can improve the harvesting performance of the TEH under a low-frequency environment. There is an optimal time constant ratio and electromechanical coupled coefficient to maximize the harvested DC power. Meanwhile, the VR phenomenon of the TEH is explored through the response amplitude of the low-frequency input signal, which implies that an appropriate combination can induce the occurrence of VR and improve the rectified voltage. Similarly, the nonlinear stiffness coefficients can be adjusted by changing the magnet distance to induce the appearance of VR. The theoretical solutions are well supported by numerical simulation and experimental verification. Specifically, the theoretical analysis and experimental evidence illustrate that the harvested power under the VR effect is much higher than that without VR.

Reentrance-like vibrational resonance in a fractional-order birhythmic biological system

The occurrence of vibrational resonance in an enzymatic substrate reaction described by a fractional-order birhythmic biological model driven by a low-frequency and a high-frequency signal is investigated. A novel reentrance-like vibrational resonance phenomenon is found, i.e., a switch from “single peak→ double peaks → single peak” with the order of the fractional time-derivatives q, the parameters α, and β varying. However, the nonlinearity parameter μ induces only a switch from “single peak→ double peaks”. The mechanics is demonstrated based on the best phase synchronization between the input and the output signal. Meanwhile, the theoretical analysis is carried out by means of the method of the separation of fast and slow variables. The theoretical results show the same phenomenon as the numerical results.

Entropic vibrational resonance.

We demonstrate the existence of vibrational resonance associated with the presence of an uneven boundary. When the motion of a Brownian particle is confined in a region with an uneven boundary, constrained to a double cavity, a high-frequency signal may produce a peak in the spectral power amplification of the other low-frequency signal and therefore to the appearance of the vibrational resonance phenomenon. The mechanism of vibrational resonance in constrained boundaries is different from that in energetic potentials and is termed entropic vibrational resonance (EVR). The EVR can be observed even if the bias force is absent in any direction. Through careful analysis, we clarify two types of mechanisms of the EVR. The one mechanism is ascribed to the transition from a bistable system to a monostable system, and the other corresponds to the match between the escape rate and the natural frequency of the low-frequency signal. Our work merges the vibrational resonance with an uneven boundary, thus extending the scope of the vibrational resonance and shedding new light on the concept of resonance.

Numerical analyses of the vibrational resonance occurrence in a nonlinear dissipative system

We carry out numerical analyses on a system which describes the motion of a particle experiencing periodic potential and submitted to both bichromatic input excitation and space-dependent nonlinear dissipative damping. The bichromatic driving models a low frequency signal corrupted by an additive high frequency perturbation which can enhance the detection of the low frequency stimulation via the Vibrational Resonance (VR) phenomenon. We focus on the combined effect of nonlinear dissipation and the mass of the particle on the occurrence of vibrational resonance. Depending on the mass of the particle, we demonstrate that the system exhibits two different behaviors, for one of which VR ceasing to exist. Furthermore, we propose an indicator which enable us to identify a critical value of mass below which VR phenomenon can be observed and beyond which no enhancement can be expected by perturbing the system with the high frequency component.

Perception enhancement of subthreshold noisy image with vibrational resonance

The authors reveal the first evidence of subthreshold noisy image perception, which takes benefit of the vibrational resonance phenomenon. Indeed, an appropriate amount of a high-frequency perturbation allows revealing a grey level noisy image through a threshold detector.

On the LFM signal improvement by piecewise vibrational resonance using a new spectral amplification factor

The authors construct the piecewise vibrational resonance (VR) method by introducing the piecewise idea and combine the re-scaled and twice sampling methods for the linear frequency modulated (LFM) signal. Importantly, they put forward a new spectral amplification factor to quantify the VR. The new spectral amplification factor has some advantages as compared with the cross-correlated coefficient, which can also be used to quantify the aperiodic VR. The top value of the spectrum amplification factor is the necessary and sufficient condition for the occurrence of the optimal VR phenomenon. Here, they discuss the effects of different parameters on VR by using some numerical examples. The method described in this study provides an effective way to improve the LFM signal, and even other kinds of frequency modulated signals.

Experimental application of vibrational resonance on bearing fault diagnosis

The vibrational resonance (VR) method is used to analyze the simulated and experimental vibration signals with different bearing faults. At first, we get the curve of the response amplitude at the characteristic frequency versus the amplitude of the auxiliary signal to obtain the VR phenomenon. Based on the curve, we obtain the optimal amplitude of the auxiliary signal which can induce the strongest VR. Then, we process the simulated and experimental bearing fault signals accordingly. In addition, we made a comparison of the proposed methodology with the envelope spectrum of the time domain signal. The results show that the response amplitude at the characteristic frequency is amplified excellently compared with the corresponding characteristic frequency in the raw signal. The VR method is better than the envelope spectrum in characteristic frequency extraction. Furthermore, the amplitude at the characteristic frequency is much larger than those at other frequency components after the VR processing. Therefore, the VR can be used for bearing fault diagnosis. We think it also has a certain values of reference in other weak feature extraction issues.

Vibrational Resonance in an Overdamped System with a Fractional Order Potential Nonlinearity

We investigate the vibrational resonance by the numerical simulation and theoretical analysis in an overdamped system with fractional order potential nonlinearities. The nonlinearity is a fractional power function with deflection, in which the response amplitude presents vibrational resonance phenomenon for any value of the fractional exponent. The response amplitude of vibrational resonance at low-frequency is deduced by the method of direct separation of slow and fast motions. The results derived from the theoretical analysis are in good agreement with those of numerical simulation. The response amplitude decreases with the increase of the fractional exponent for weak excitations. The amplitude of the high-frequency excitation can induce the vibrational resonance to achieve the optimal response amplitude. For the overdamped systems, the nonlinearity is the crucial and necessary condition to induce vibrational resonance. The response amplitude in the nonlinear system is usually not larger than that in the corresponding linear system. Hence, the nonlinearity is not a sufficient factor to amplify the response to the low-frequency excitation. Furthermore, the resonance may be also induced by only a single excitation acting on the nonlinear system. The theoretical analysis further proves the correctness of the numerical simulation. The results might be valuable in weak signal processing.

Amplification of the LFM signal by using piecewise vibrational methods

We propose the piecewise re-scaled vibrational resonance (VR) method and the piecewise twice sampling VR method to amplify the weak linear frequency-modulated (LFM) signal. The system used to amplify the weak LFM signal is a typical bistable system with fractional-order deflection nonlinearity. The concrete procedures of both the piecewise re-scaled VR method and the piecewise twice sampling VR method are explained in detail. Through studying the effect of the factional-order exponent on VR, we find that the traditional bistable system is not the optimal model to improve the weak LFM signal. By investigating different parameters on the VR phenomenon, we verify the effectiveness of the two proposed methods.

On the correlation between phase-locking modes and Vibrational Resonance in a neuronal model

We numerically and experimentally investigate the underlying mechanism leading to multiple resonances in the FitzHugh-Nagumo model driven by a bichromatic excitation. Using a FitzHugh-Nagumo circuit, we first analyze the number of spikes triggered by the system in response to a single sinusoidal wave forcing. We build an encoding diagram where different phase-locking modes are identified according to the amplitude and frequency of the sinusoidal excitation. Next, we consider the bichromatic driving which consists in a low frequency sinusoidal wave perturbed by an additive high frequency signal. Beside the classical Vibrational Resonance phenomenon, we show in real experiments that multiple resonances can be reached by an appropriate setting of the perturbation parameters. We clearly establish a correlation between these resonances and the encoding diagram of the low frequency signal free FitzHugh-Nagumo model. We show with realistic parameters that sharp transitions of the encoding diagram allow to predict the main resonances. Our experiments are confirmed by numerical simulations of the system response.

Controllable parametric excitation effect on linear and nonlinear vibrational resonances in the dynamics of a buckled beam

In this study, the effect of a controllable parametric excitation on both linear and nonlinear vibrational resonances on the dynamic of a buckled beam excited by a combination of uncontrollable low- and high-frequency periodic forces are investigated. First of all, the beam dynamic is assumed to be constrained by two periodic and independent ambient solicitations, such as wind and earthquake. An axial load of the beam represented by a periodic and parametric excitation is used to control the vibrational resonance phenomenon, induced by the presence of the two external excitations. Approximate analytical expressions for the linear response and the high-frequency force amplitude at which linear vibrational resonance occurs are obtained. An analytical expression of the amplitude of the nonlinear response at the superharmonic equal to the double of the low-frequency, is obtained. For all these expressions, we show the effect of the parametric excitation. We compare all the obtained results with the ones of the case where, the parametric force is absent. It is shown that, the presence of the parametric excitation permit the suppression of both linear and nonlinear vibrational resonances. Moreover, the vibration amplitudes of the buckled beam are significantly reduced, around certain threshold values for the amplitude and the frequency of the parametric excitation.

Vibrational resonance in a periodic potential system with stable noise

A periodic potential system excited by multi-low frequency weak signals, the high frequency signal and additive stable noise is constructed. Based on this model, the vibrational resonance phenomenon under stable noise is investigated by taking the mean signal-noise-ratio gain (MSNRI) of output as a performance index. Then the influences of stability index (0 2), the skewness parameter (-1 1) of stable noise, the amplification factor D and the high frequency signal amplitude B, and frequency on the resonant output effect are explored. The results show that under the different distributions of stable noise, the multi-low frequency weak signals detection can be realized by adjusting the high frequency signal parameter B or to induce vibrational resonance within a certain range. When (or ) is given different values, the curve of MSNRI-B has multiple peaks with the increase of B for a certain frequency , and the values of MSNRI corresponding to peaks of the curve of MSNRI-B are equal. So the intervals of B which can induce vibrational resonances are multiple, and the multiple resonance phenomenon turns periodic with the increase of B. Similarly, the curve of MSNRI- also has multiple peaks with the increase of for a certain amplitude B, so the intervals of which can induce vibrational resonances are also multiple. The difference is that the multiple resonance phenomenon becomes irregular with the increase of . Besides, the resonance intervals of B and do not change with nor . Under the different values of amplitude factor D, the resonance intervals of B (or ) do not change with the increase of D, indicating that only the energy of the high frequency signal transfers toward the signals to be measured, and the energy of stable noise does not transfer toward the signals to be measured. Besides, when B and are fixed, it can still be realized to detect the weak signal with the increase of D, which shows that the weak signal detection method based on vibrational resonance can overcome the shortcoming that noise intensity in industrial sites cannot be regulated and controlled. The results provide a new method of detecting the weak signal, and have potential application value in signal processing.

Vibrational resonance in feed-forward-loop neuronal network motifs

A neuron carries out its functions in networks receiving contacts from roughly 104 presynaptic neurons. Such a dense connectivity profile for a single neuron may give rise to enormous complex neuronal topologies, which might be very difficult to understand the underlying mechanisms of neurological functions and diseases. Extensive experimental data from neuroanatomical studies have uncovered that neural networks include some recurring topologies of microcircuits, known as network motifs which serve as characteristic building blocks of complex networks [1,2]. Therefore, it is widely assumed that a clear explanation on dynamical and functional features of these network motifs can be considered as the first step to understand large networks. Following this motivation, in this study, we investigate the Vibrational Resonance (VR) phenomenon in a triple-neuron feed-forward-loop (FFL) which is one of the most significant brain network motifs, shown in Figure ​Figure1.1. VR is a physical phenomenon found in nonlinear systems where a weak signal can be detected and processed by the system with the assistance of another high frequency signal. It is very similar to the well-known stochastic resonance phenomenon, where the role of noise is replaced in VR by a high frequency signal. In recent years, there exists a growing interest in applications of VR to neuroscience because two frequency signals are pervasive in neural systems, i.e. bursting neurons exhibit two widely different time scales, simultaneous arrival of vocal signals having distinct frequencies to auditory neurons. Due to the significance of its potential in neural signal processing, the VR phenomenon has already been studied with computational models of neurons and their networks [3,4]. However, a population of excitatory and inhibitory neurons has not been considered, yet. Here, we study the VR dynamics depending on whether the neurons in the FFL motifs are excitatory or inhibitory considering eight possible structural configurations of the considered microcircuit (Table ​(Table1).1). Such an approach provides us to test the functional influences of structural configuration on VR dynamics in neural populations. Figure 1 Schematic illustration of the considered feed-forward-loop motif. LF and HF refers to low and high frequency signals, respectively. Neuron 1 and 3 is considered as input and output neurons, respectively. Table 1 Eight possible FFL types. We found that the weak signal transmission in the network via the VR mechanism is possible depending on both the coupling strength between neurons and the network structure. For instance, when a weak coupling strength is present between connected neurons, weak signal injected to the input neuron can be transmitted to the output with only T8-FFL motif. In this case, signal transmission is impossible for other types of network motifs. For intermediate coupling strengths, signal transmission performance is high for all motifs and motif type does not change very much VR dynamics. Finally, for strong coupling strengths, the best VR performance is obtained with T1-FFL motif where all the neurons in the network are excitatory. We also clarify the mechanisms that underlies the differences in performance of network motifs.

Optimizing the Electrical Power in an Energy Harvesting System

In this paper, we study the vibrational resonance (VR) phenomenon as a useful mechanism for energy harvesting purposes. A system, driven by a low frequency and a high frequency forcing, can give birth to the vibrational resonance phenomenon, when the two forcing amplitudes resonate and a maximum in amplitude is reached. We apply this idea to a bistable oscillator that can convert environmental kinetic energy into electrical energy, that is, an energy harvester. Normally, the VR phenomenon is studied in terms of the forcing amplitudes or of the frequencies, that are not always easy to adjust and change. Here, we study the VR generated by tuning another parameter that is possible to manipulate when the forcing values depend on the environmental conditions. We have investigated the dependence of the maximum response due to the VR for small and large variations in the forcing amplitudes and frequencies. Besides, we have plotted color coded figures in the space of the two forcing amplitudes, in which it is possible to appreciate different patterns in the electrical power generated by the system. These patterns provide useful information on the forcing amplitudes in order to produce the optimal electrical power.

Vibrational resonance in an asymmetric bistable system with time-delay feedback

Vibrational resonance is a resonant dynamics induced by a high-frequency periodic force at the low-frequency of the input periodic signal, and the input periodic signal is enhanced by a high-frequency signal. In this paper, a linear time-delayed feedback bistable system with an asymmetric double-well potential driven by both low-frequency and high-frequency periodic forces is constructed. Based on this model, the vibrational resonance phenomenon is investigated. Making use of the method of separating slow motion from fast motion under the conditions of Ω>>ω (Ω is the frequency of the high-frequency signal and ω is the one of the low-frequency signal), equivalent equations to the slow motion and the fast motion are obtained. Neglecting the nonlinear factors, the analytical expression of the response amplitude Q can be obtained, and the effects of the time-delay parameter α and the asymmetric parameter r on the vibrational resonance are discussed in detail. Moreover, the locations at which the vibrational resonance occurs, are obtained by means of solving the condition for a resonance to occur. A major consequence of time-delayed feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the time-delayed parameter, i.e. in Q-α plot, α can induce the Q which is periodic with the periods of the high-frequency signal and the low-frequency signal. The locations at which the vibrational resonance occurs are not changed by the asymmetric parameter r. However, the resonance amplitude is enhanced with increasing r. Specifically, the resonance amplitude is greatly enhanced when r>0.15. On the other hand, in the symmetric case (r=0), BVR at which the vibrational resonance occurs is periodic with the periods of high-frequency signal and low-frequency signal as α increases, which is shown in BVR-α (B is the amplitude of the high-frequency signal) plot. In Q-Ω plot, Q is presented by multi-resonance at the small values of B and Ω, but Q tends to a fixed value at the small values of B and the large values of Ω. We believe that the above theoretical observations will stimulate the experimental study of vibrational resonance in nonlinear oscillators and electronic circuits with time-delayed feedback.

Is the High-Frequency Signal Necessary for the Resonance in the Delayed System?

In a delayed system excited by low-frequency and high-frequency signals, the necessity of the high-frequency signal on the resonance is discussed. By adjusting the delay time, the resonance occurs in a wide scope of frequencies, including the primary, subharmonic and superharmonic frequencies. Only for very few cases does the high-frequency signal have a positive effect on the resonance. It is the traditional vibrational resonance phenomenon. In most situations, the high-frequency excitation is unnecessary for the resonance. An appropriate delay, rather than the high-frequency signal, is the key factor in improving the weak low-frequency signal.

High-frequency effects in the FitzHugh-Nagumo neuron model

The effect of a high-frequency signal on the FitzHugh-Nagumo excitable model is analyzed. We show that the firing rate is diminished as the ratio of the high-frequency amplitude to its frequency is increased. Moreover, it is demonstrated that the excitable character of the system, and consequently the firing activity, is suppressed for ratios above a given threshold value. In addition, we show that the vibrational resonance phenomenon turns up for sufficiently large noise strength values.