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

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.

Vibrational resonances in driven oscillators with position-dependent mass

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)'.

Acoustic vibrational resonance in a Rayleigh-Plesset bubble oscillator

The phenomenon of vibrational resonance (VR) has been investigated in a Rayleigh-Plesset oscillator for a gas bubble oscillating in an incompressible liquid while driven by a dual-frequency force consisting of high-frequency, amplitude-modulated, weak, acoustic waves. The complex equation of the Rayleigh-Plesset bubble oscillator model was expressed as the dynamics of a classical particle in a potential well of the Liénard type, thus allowing us to use both numerical and analytic approaches to investigate the occurrence of VR. We provide clear evidence that an acoustically-driven bubble oscillates in a time-dependent single or double-well potential whose properties are determined by the density of the liquid and its surface tension. We show both theoretically and numerically that, besides the VR effect facilitated by the variation of the parameters on which the high-frequency depends, amplitude modulation, the properties of the liquid in which the gas bubble oscillates contribute significantly to the occurrence of VR. In addition, we discuss the observation of multiple resonances and their origin for the double-well case, as well as their connection to the low frequency, weak, acoustic force field.

Spatiotemporal antiresonance in coupled reaction-diffusion systems

We present a theoretical study of the spatiotemporal antiresonance in a system of two diffusively coupled chemical reactions, one of which is driven by an external periodic forcing. Although antiresonance is well known in various physical systems, the phenomenon in coupled chemical reactions has largely been overlooked. Based on the linearized dynamics around the steady state of the two-component coupled reaction-diffusion systems we have derived the general analytical expressions for the amplitude-frequency response functions of the driven and undriven components of the system. Our theoretical analysis is well corroborated by detailed numerical simulations on coupled Gray-Scott reaction-diffusion systems exhibiting antiresonance dip in the amplitude-frequency response curve as a result of destructive interference between the coupling and the periodic external forcing imparting differential stability of the two subsystems. This leads to the emergence of spatiotemporal patterns in an undriven subsystem, while the driven one settles down to a homogeneously stable steady state.

Effect of parametric excitation on a bifractional-order damped system with a fractional-power nonlinearity

Investigation on linear/nonlinear resonance phenomena and supercritical/subcritical pitchfork bifurcation mechanism is reported in a complex bifractional-order damped system which endures a high-frequency parametric excitation and contains fractional-power nonlinearity. The approximate theoretical expression of the linear response amplitude at the primary frequency and the superharmonic response amplitude at the second and third harmonic frequencies are obtained by utilizing an analytical method and an iterative formula. A numerical approximation scheme based on the Caputo derivative for the simulation of the system is introduced, showing sufficient precision. Due to the parametric excitation, analytical approximation expressions of the stable equilibrium points are given explicitly when the exponent is not an integer so that the pitchfork bifurcation, nonlinear resonance can be studied in an analytical way, exhibiting much more operability than the external excitation case. It is found that the fractional-order derivative may bring new multibifurcation and new multiresonance phenomena, which have not yet been reported before. With the variation of different control parameters of the system, the equivalent slow-varying system can be converted from bistability to monostability and finally to bistability. Unlike the cases of the system excited by bifrequency external excitation, the optimum response amplitude of the parametric excited system is not monotonous with respect to the values of the exponent. For a range of parameters of the system, it is also found that the superharmonic resonance at the second and third harmonic frequencies is affected deeply by the parametric excitation.

Vibrational antiresonance in nonlinear coupled systems

We examine the response of a system of coupled nonlinear oscillators driven by a rapidly varying field, to a low frequency weak periodic excitation of one of the oscillators. The response amplitude of the weak field-driven oscillator at an optimal strength of the rapidly varying field exhibits a strong suppression accompanied by a large negative shift in its oscillation phase. The minimum can be identified as vibrational antiresonance in between the two maxima corresponding to vibrational resonance. This vibrational antiresonance can be observed only in nonlinear coupled systems and not in linearly coupled systems or in a single nonlinear oscillator, under similar physical condition. We discuss the underlying dynamical mechanism, the role of nonlinearity and high frequency in characterizing this counter-resonance effect. Our theoretical analysis is corroborated by detailed numerical simulations.

Subthreshold and suprathreshold vibrational resonance in the FitzHugh-Nagumo neuron model

We study the subthreshold and suprathreshold vibrational resonance in the FitzHugh-Nagumo neuron model. For the subthreshold situation, two cases where the stationary states are equilibrium point and limit cycle are considered, where different natures of vibrational resonance are observed via theoretical and numerical methods. Especially when the frequency of the high-frequency driving force is near the so-called canard-resonance frequency, the firing rate can be significantly enhanced at the presence of noise. For the suprathreshold situation, we show that the local maxima of the response amplitude are located at the transition boundaries of different phase-locking patterns. The minimal required forcing amplitudes of high-frequency signal of the firing onset are just multiples of the spiking frequency. Furthermore, phase portraits and time series show that the presence of the global maxima of the response results from not only the suprathreshold but also the subthreshold phase-locking modes. In spite of the distinct characteristics for two stationary states on subthreshold oscillation, the suprathreshold vibrational resonance showed no qualitative difference between the two cases.

Enhanced multiple vibrational resonances by Na+ and K+ dynamics in a neuron model

Some neuronal receptors perceive external input in the form of hybrid periodic signals. The signal detection may be based on the mechanism of vibrational resonance, in which a system's response to the low frequency signal can become optimal by an appropriate choice of the vibration amplitude of HFS. The vibrational resonance effect is investigated in a neuron model in which the intra- and extra-cellular potassium and sodium concentrations are allowed to evolve temporally, depending on ion currents, Na⁺-K⁺ pumps, glial buffering, and ion diffusion. Our results reveal that, compared to the vibrational resonances in the model with constant ion concentrations, the significantly enhanced vibrational multi-resonances can be observed for the single neuron system where the potassium and sodium ion concentrations vary temporally. Thus, in contradiction to a popular view that ion concentrations dynamics play little role in signal detection, we indicate that the neuron's response to an external subthreshold signal can be largely improved by sodium and potassium dynamics.

Double-maximum enhancement of signal-to-noise ratio gain via stochastic resonance and vibrational resonance

This paper studies the signal-to-noise ratio (SNR) gain of a parallel array of nonlinear elements that transmits a common input composed of a periodic signal and external noise. Aiming to further enhance the SNR gain, each element is injected with internal noise components or high-frequency sinusoidal vibrations. We report that the SNR gain exhibits two maxima at different values of the internal noise level or of the sinusoidal vibration amplitude. For the addition of internal noise to an array of threshold-based elements, the condition for occurrence of stochastic resonance is analytically investigated in the limit of weak signals. Interestingly, when the internal noise components are replaced by high-frequency sinusoidal vibrations, the SNR gain displays the vibrational multiresonance phenomenon. In both considered cases, there are certain regions of the internal noise intensity or the sinusoidal vibration amplitude wherein the achieved maximal SNR gain can be considerably beyond unity for a weak signal buried in non-Gaussian external noise. Due to the easy implementation of sinusoidal vibration modulation, this approach is potentially useful for improving the output SNR in an array of nonlinear devices.

Theoretical analysis of vibrational resonance in a neuron model near a bifurcation point

The FitzHugh-Nagumo neuron model subject to a biharmonical external force with two different frequencies is used to investigate the underlying mechanism of vibrational resonance in an excitable system in which the time scales between the fast and slow variables are separated clearly. The theoretical analysis is given based on the approximation approach and the concept of the phase-locking ratio instead of the amplification ratio widely used in the investigation of vibrational resonance in bistable oscillators. The result shows that the high-frequency subthreshold force with the frequency close to the natural frequency of the neuron model in the resting state can induce the change of potential shape of the model near the bifurcation point. This gives rise to the different phase-locking modes of the neuron responses to the same low-frequency subthreshold input. It is also shown that besides the parameters of the high-frequency force such as amplitude and frequency, the bifurcation parameter of the model can affect the vibrational resonance notably. Finally, the numerical results have verified the theoretical analysis.

Effect of multiple time-delay on vibrational resonance

We report our investigation on the effect of multiple time-delay on vibrational resonance in a single Duffing oscillator and in a system of n Duffing oscillators coupled unidirectionally and driven by both a low- and a high-frequency periodic force. For the single oscillator, we obtain analytical expressions for the response amplitude Q and the amplitude g of the high-frequency force at which resonance occurs. The regions in parameter space of enhanced Q at resonance, as compared to the case in absence of time-delay, show a bands-like structure. For the two-coupled oscillators, we explain all the features of variation of Q with the control parameter g. For the system of n-coupled oscillators with a single time-delay coupling, the response amplitudes of the oscillators are shown to be independent of the time-delay. In the case of a multi time-delayed coupling, undamped signal propagation takes place for coupling strength (δ) above a certain critical value (denoted as δu). Moreover, the response amplitude approaches a limiting value QL with the oscillator number i. We obtain analytical expressions for both δu and QL.

Effect of the heterogeneous neuron and information transmission delay on stochastic resonance of neuronal networks

We study the effect of heterogeneous neuron and information transmission delay on stochastic resonance of scale-free neuronal networks. For this purpose, we introduce the heterogeneity to the specified neuron with the highest degree. It is shown that in the absence of delay, an intermediate noise level can optimally assist spike firings of collective neurons so as to achieve stochastic resonance on scale-free neuronal networks for small and intermediate α(h), which plays a heterogeneous role. Maxima of stochastic resonance measure are enhanced as α(h) increases, which implies that the heterogeneity can improve stochastic resonance. However, as α(h) is beyond a certain large value, no obvious stochastic resonance can be observed. If the information transmission delay is introduced to neuronal networks, stochastic resonance is dramatically affected. In particular, the tuned information transmission delay can induce multiple stochastic resonance, which can be manifested as well-expressed maximum in the measure for stochastic resonance, appearing every multiple of one half of the subthreshold stimulus period. Furthermore, we can observe that stochastic resonance at odd multiple of one half of the subthreshold stimulus period is subharmonic, as opposed to the case of even multiple of one half of the subthreshold stimulus period. More interestingly, multiple stochastic resonance can also be improved by the suitable heterogeneous neuron. Presented results can provide good insights into the understanding of the heterogeneous neuron and information transmission delay on realistic neuronal networks.

Vibrational resonance induced by transition of phase-locking modes in excitable systems

We study the occurrence of vibrational resonance as well as the underlying mechanism in excitable systems. The single vibration resonance and vibration bi-resonance are observed when tuning the amplitude and frequency of high-frequency force simultaneously. Furthermore, by virtue of the phase diagram of low-frequency-signal-free FitzHugh-Nagumo model, it is found that each maxima of response measure is located exactly at the transition boundary of phase patterns. Therefore, it is the transition between different phase-locking modes that induces vibrational resonance in the excitable systems. Finally, this mechanism is verified in the Hodgkin-Huxley neural model. Our results provide insights into the transmission of weak signals in nonlinear systems, which are valuable in engineering for potential applications.

Vibrational resonance in Duffing systems with fractional-order damping

The phenomenon of vibrational resonance (VR) is investigated in over- and under-damped Duffing systems with fractional-order damping. It is found that the factional-order damping can induce change in the number of the steady stable states and then lead to single- or double-resonance behavior. Compared with vibrational resonance in the ordinary systems, the following new results are found in the fractional-order systems. (1) In the overdamped system with double-well potential and ordinary damping, there is only one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for these new resonance behaviors is the value of the fractional-order satisfies α > 1. (2) In the overdamped system with single-well potential and ordinary damping, there is no resonance, whereas there is a single-resonance for the case of fractional-order damping. The necessary condition for the new result is α > 1. (3) In the underdamped system with double-well potential and ordinary damping, there are double-resonance and one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for the new single-resonance is α < 1. (4) In the underdamped system with single-well potential, there is at most a single-resonance existing for both the cases of ordinary and fractional-order damping. In the underdamped systems, varying the value of the fractional-order is equivalent to change the damping parameter for some cases.

Novel vibrational resonance in multistable systems

We investigate the role of multistable states on the occurrence of vibrational resonance in a periodic potential system driven by both a low-frequency and a high-frequency periodic force in both underdamped and overdamped limits. In both cases, when the amplitude of the high-frequency force is varied, the response amplitude at the low-frequency exhibits a series of resonance peaks and approaches a limiting value. Using a theoretical approach, we analyse the mechanism of multiresonance in terms of the resonant frequency and the stability of the equilibrium points of the equation of motion of the slow variable. In the overdamped system, the response amplitude is always higher than in the absence of the high-frequency force. However, in the underdamped system, this happens only if the low-frequency is less than 1. In the underdamped system, the response amplitude is maximum when the equilibrium point around which slow oscillations take place is maximally stable and minimum at the transcritical bifurcation. And in the overdamped system, it is maximum at the transcritical bifurcation and minimum when the associated equilibrium point is maximally stable. When the periodicity of the potential is truncated, the system displays only a few resonance peaks.

Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback

The influence of linear time-delayed feedback on vibrational resonance is investigated in underdamped and overdamped Duffing oscillators with double-well and single-well potentials driven by both low frequency and high frequency periodic forces. This task is performed through both theoretical approach and numerical simulation. Theoretically determined values of the amplitude of the high frequency force and the delay time at which resonance occurs are in very good agreement with the numerical simulation. 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. An appropriate time delay is shown to induce a resonance in an overdamped single-well system which is otherwise not possible. For a range of values of the time-delayed parameters, the response amplitude is found to be larger than in delay-time feedback-free systems.

Controlling vibrational resonance in a multistable system by time delay

The phenomenon of vibrational resonance in a delayed multistable system that is excited by biharmonic signals is investigated in the present paper. Different from the former theory, the appearance and the disappearance of the vibrational resonance are controlled by adjusting the time delay parameter instead of modulating the amplitude of the high-frequency signal. The motion of the orbit within or between the different potential wells can also be controlled. Furthermore, based on both the methods of numerical simulation and analytical analysis, the behavior of delay-induced multiple vibrational resonance and its mechanism are investigated and discussed. The multiple vibrational resonance, which is quantified by the response amplitude at the low-frequency, is found to be periodic in the delay parameter with two periods, i.e., the periods of the two driven signals. The method used in this paper gives a new way for controlling vibrational resonance in a multistable system.