# Vibrational resonance: A review

Jianhua Yang  Jiangsu Key Laboratory of Mine Mechanical and Electrical
Equipment, School of Mechatronic Engineering, China University of Mining and
Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China  S. Rajasekar
School of Physics, Bharathidasan University, Tiruchirapalli, 620024,
Tamilnadu, India  Miguel A. F. Sanjuán<EMAIL_ADDRESS>Nonlinear
Dynamics, Chaos and Complex Systems Group, Departamento de Fisica, Universidad
Rey Juan Carlos, Tulipan s/n, Mostoles, 28933, Madrid, Spain

###### Abstract

Over the past two decades, vibrational resonance has garnered significant
interest and evolved into a prominent research field. Classical vibrational
resonance examines the response of a nonlinear system excited by two signals:
a weak, slowly varying characteristic signal, and a fast-varying auxiliary
signal. The characteristic signal operates on a much longer time scale than
the auxiliary signal. Through the cooperation of the nonlinear system and
these two excitations, the faint input can be substantially amplified,
showcasing the constructive role of the fast-varying signal. Since its
inception, vibrational resonance has been extensively studied across various
disciplines, including physics, mathematics, biology, neuroscience, laser
science, chemistry, and engineering. Here, we delve into a detailed discussion
of vibrational resonance and the most recent advances, beginning with an
introduction to characteristic signals commonly used in its study.
Furthermore, we compile numerous nonlinear models where vibrational resonance
has been observed to enhance readers’ understanding and provide a basis for
comparison. Subsequently, we present the metrics used to quantify vibrational
resonance, as well as offer a theoretical formulation. This encompasses the
method of direct separation of motions, linear and nonlinear vibrational
resonance, re-scaled vibrational resonance, ultrasensitive vibrational
resonance, and the role of noise in vibrational resonance. Later, we showcase
two practical applications of vibrational resonance: one in image processing
and the other in fault diagnosis. This presentation offers a comprehensive and
versatile overview of vibrational resonance, exploring various facets and
highlighting promising avenues for future research in both theory and
engineering applications.

###### keywords:

vibrational resonance , nonlinear systems , nonlinear response , aperiodic
signal , logical computation , noise
PACS: 05.45.-a , 33.20.Tp , 05.40.-a , 05.40.Ca

††journal: Physics Reports

##  1 Introduction

Multiple excitations with different time scales are frequently utilized across
various disciplines. These excitations may manifest as forces, electric
currents, or other signal types. When a nonlinear system is subjected to
simultaneous multiple excitations, the response to the slow varying excitation
closely depends on the fast varying excitation. This phenomenon is known as
vibrational resonance discovered by Landa and McClintock in their seminal
paper [1]. Inspired by another well known nonlinear phenomenon, stochastic
resonance [2], [3], Landa and McClintock termed this new phenomenon as
vibrational resonance. In their work, the role of noise in stochastic
resonance is replaced by a high-frequency harmonic signal in vibrational
resonance. Although vibrational resonance shares some similarities with
stochastic resonance, particularly regarding dynamical behaviors, they possess
distinct essential properties, thus constituting a new subfield of nonlinear
dynamics [4]. Initially, vibrational resonance primarily focused on the
response amplitude of a nonlinear system to a weak low-frequency harmonic
excitation. Landa and McClintock conducted their initial studies of
vibrational resonance in a typical bistable Duffing oscillator, examining both
overdamped and underdamped versions. They found that the response amplitude of
the nonlinear system at low frequencies depends nontrivially on the amplitude
of the high-frequency signal. Specifically, the curve depicting the response
amplitude at low frequencies versus the amplitude of the high-frequency signal
exhibits a resonance-like shape. This shape resembles the well-known
frequency-response curve. At the resonance peak, the amplitude of the weak,
low-frequency excitation component in the response is significantly amplified
by the high-frequency signal. The classical vibrational resonance comprises
three basic elements: (i) nonlinear systems, (ii) slowly varying excitation
(characteristic signal), and (iii) fast varying excitation (auxiliary signal).
It is important to note that the terms ”fast” and ”slow” are primarily used
for comparing the two excitation signals, with the slowly varying excitation
typically being weak. Through the interaction of these three basic elements,
vibrational resonance emerges. In many cases, valuable information is conveyed
through a weak, low-frequency, or slowly varying signal. Therefore, the
development of vibrational resonance theory is crucial for detecting and
amplifying weak signals.

In their pioneering work, Landa and McClintock introduced vibrational
resonance based on numerical simulations. Subsequently, Gitterman analyzed a
second-order bistable system and provided an analytical explanation for the
occurrence of vibrational resonance [5]. Later, Blekhman and Landa conducted a
more specific analytical study on vibrational resonance in overdamped and
underdamped Duffing oscillators [6], deriving conditions for single and double
resonance patterns. Furthermore, they highlighted errors between analytical
and numerical simulations. The analytical technique commonly used to analyze
vibrational resonance is known as the method of direct separation of motions
[7]-[9]. This method is typically employed to obtain responses at the
excitation frequency of the slowly varying harmonic signal. For more accurate
results or results at nonlinear frequency components, such as the response
amplitude at subharmonic, superharmonic, or combination frequencies, other
analytical methods like the average method, perturbation method, multiscale
method, and harmonic balance method can be used [7], [10]]. Compared to other
methods, the derivation process of the method of direct separation of motions
is simpler and more straightforward, making it the preferred choice when
responses primarily involve fast and slow motions. Blekhman and Landa
extensively explored vibrational resonance using the method of direct
separation of motions. They analyzed vibrational resonance with single or
multiple resonance patterns in various systems, including quintic oscillators
with potential function in different configurations [11]-[14], coupled
anharmonic oscillators [15], [16], one-way coupled bistable systems [17],
rough potential systems [18], Toda potential systems [19], periodic potential
systems [20]-[23], quantum systems [24]-[26], nano-electromechanical systems
[27], and chemical reaction models [28], among others.

Effects of time delay have been incorporated into the study of vibrational
resonance. The delay parameter has been observed to induce periodic
vibrational resonance in a simple delayed bistable system, with the period of
the resonance peak relative to the time delay parameter being equal to the
period of the high-frequency excitation signal [29]. In coupled systems
[30]-[32], delay-induced vibrational resonance exhibits two periods when the
delay time is sufficiently long, coinciding with the periods of the two
excitations [30], [32], [33]. Analysis of vibrational resonance considering
time delay has been conducted in various systems, including two coupled
overdamped anharmonic oscillators [15], the Duffing oscillator [34], [35], an
asymmetric bistable system [36], the FitzHugh-Nagumo system [37]-[39], the
discrete Rulkov neuronal model [40], a genetic toggle switch [41], a gene
transcriptional regulatory system [42], and a harmonically trapped potential
system [43], among others. In time-delayed systems, it has been observed that
high-frequency excitation alone is not always sufficient to induce resonance
in some cases [44], [45]. The high-frequency excitation does influence the
waveform of the response. When considering time delay in vibrational
resonance, it is necessary to account for the Hopf bifurcation induced by the
time delay [46]. Additionally, due to the coexistence of time delay and
multiple excitations, it is essential to consider the combined effect of high-
frequency excitation and the delay term on the response simultaneously.

In recent years, fractional nonlinear systems have become an important type of
system for addressing various science and engineering problems. Fractional
systems incorporate fractional-order derivatives, originating from the long-
standing issue of fractional calculus. Specifically, while the traditional
derivative order is an integer, in some special cases, it can be non-integer.
Fractional nonlinear systems find widespread use in fields such as dynamic
modeling, automatic control, electromagnetism, among others, and are even
referred to as the the $21^{st}$ century system [47]. There are many
interesting findings regarding vibrational resonance in fractional systems. A
double resonance pattern caused by a fractional-order derivative term has been
observed in an overdamped fractional Duffing oscillator [48]. Vibrational
resonance has also been observed in the Duffing system with fractional-order
external and intrinsic damping [49], [50], fractional multistable systems
[51], fractional anharmonic coupled systems [52], fractional quintic systems
[53]-[55], fractional birhythmic biological systems [56], and fractional Toda
oscillators [57], among others. Furthermore, the effects of fractional-order
damping terms on vibrational resonance are directly related to the static
bifurcation of fixed points of the equivalent system [58]. Specifically, at
critical points of the pitchfork bifurcation [48], [59]-[63], the saddle-node
bifurcation [64], and the transcritical bifurcation [65], the response
amplitude curve exhibits an inflection point, thus influencing the resonance
pattern of the curve.

Vibrational resonance in neuronal models and complex networks is currently a
topic of significant interest. Ullner et al. [66] investigated vibrational
resonance in the FitzHugh-Nagumo neuron model. Additionally, vibrational
resonance in the FitzHugh-Nagumo model has been studied from different
perspectives [67]-[76], along with analysis in a neuron-astrocyte coupled
model [77]. Vibrational resonance is also observed in single or coupled
Hindmarsh-Rose neuronal systems [78]. These networks may consist of various
clusters of neuronal oscillators or other types of oscillators. The influences
of different network factors on vibrational resonance have been analyzed,
leading to many interesting findings [79]-[93].

In a nonlinear system subjected to a harmonic characteristic excitation,
vibrational resonance is not limited to linear response but also extends to
nonlinear response frameworks. Specifically, vibrational resonance manifests
at nonlinear frequencies, including the second harmonic frequency [94] and
even higher-order frequencies [95]-[97] of the low-frequency signal. It occurs
at subharmonic [98]-[100], superharmonic, and combination frequencies [98], as
well as at the sum and difference frequencies of the low-frequency and the
natural frequency of a system [101]. This phenomenon has been investigated in
a two-level quantum system for both linear and nonlinear vibrational resonance
responses [26].

One type of vibrational resonance is ghost vibrational resonance, which occurs
as a high-frequency excitation-induced resonance at a fundamental frequency
absent in the input low-frequency excitation [102]-[105]. Similar to
stochastic resonance, ghost vibrational resonance exhibits features akin to
ghost stochastic resonance [106]. Another variant, entropic vibrational
resonance [107], [108], shares similarities with entropic stochastic resonance
[109].

Vibrational resonance induced by various characteristic signals has garnered
significant attention. Apart from harmonic signals, characteristic signals can
take the form of anharmonic periodic signals, aperiodic binary signals,
frequency modulated signals, and other complex forms in engineering
applications. Notably, binary aperiodic signals can induce aperiodic
vibrational resonance. Chizhevsky and Giacomelli [110] analyzed aperiodic
vibrational resonance induced by aperiodic binary excitation, introducing the
cross-correlation coefficient index for analysis.

Furthermore, traditional vibrational resonance typically involves dealing with
slowly varying characteristic signals. However, in many scenarios, the
characteristic signal manifests as fast-varying. Vibrational resonance has
been investigated in the presence of a fast characteristic signal [111]-[113],
amplitude-modulated signals [114], aperiodic binary signals [115]-[117], and
frequency-modulated signals [118], [119]. Exploring resonance with a complex
frequency-modulated signal amidst strong noise background is intriguing [120].
To quantify frequency-modulated signal-induced vibrational resonance, spectral
amplification factors [119] and fusion indices [121] have been introduced. The
fusion index serves to measure the necessary and sufficient conditions for
vibrational resonance and describe amplification and the “similarity” of the
output waveform compared to the input waveform simultaneously. Additionally, a
modified cross-correlation coefficient has been utilized to eliminate errors
between the input signal and system output induced by phase delay [122],
thereby enhancing performance measures for aperiodic vibrational resonance.

Noise is nearly ubiquitous across various scientific and engineering fields.
Understanding how nonlinear systems respond to simultaneous noise, fast, and
slow excitations is a crucial concern [123]. Particularly, much work has been
done to explore the beneficial effects of different types of noise on
vibrational resonance. Zaikin et al. [124] have investigated vibrational
resonance in coupled oscillators excited by parametric noise. They utilized a
piecewise linear system for each oscillator, approximating the classical
bistable system. Their findings revealed that multiplicative noise applied to
each oscillator induces a phase transition, resulting in bistability.
Consequently, vibrational resonance occurs, enhancing the weak signal.
Essentially, a certain amount of noise fosters vibrational resonance.
Subsequent analyses have delved into the interplay between stochastic
resonance and vibrational resonance in various nonlinear models [125]-[147].
Some of these studies have identified instances where vibrational resonance
enhances and controls stochastic resonance.

Very recently, there has been a growing interest in exploring vibrational
resonance across various engineering applications. One notable area of
exploration is vibrational resonance within energy harvesting systems
[148]-[151]. Additionally, research has been done to investigate its potential
utility in equipment fault diagnosis [152]-[157]. Moreover, vibrational
resonance in image processing has also been investigated [158], [159].
Recognizing the significance of vision research, there is a growing interest
in further exploring vibrational resonance in the realm of image perception.
We believe that vibrational resonance will continue to play a vital role in
various engineering applications, prompting increased attention and research
in this domain.

In 2020, to commemorate the 20th anniversary since the inception of the
concept of vibrational resonance, as well as McClintock P V E’s 80th birthday,
Vincent et al. organized a thematic/special issue on nonlinear resonances in
driven systems, encompassing vibrational and stochastic resonance [160],
[161]. This issue was published by Philos. T. R. Soc. A. The first part of the
issue was released online in January 2021, followed by the second part in
April 2021. Within this compilation, there are 11 papers dedicated to
vibrational resonance [26], [77], [93], [159], [162]-[168]. These papers delve
into vibrational resonance within quantum systems, laser systems, signal
processors, logic gates, position-dependent mass systems, and energy
harvesters.

Although there are monographs on vibrational resonance [58], [169], a
comprehensive review article is still lacking to aid researchers in gaining a
better understanding of vibrational resonance and its related topics. This
serves as the primary motivation behind this work. The slow-varying excitation
may manifest in periodic or aperiodic forms. On one hand, different forms of
excitations can induce various resonance patterns. On the other hand,
vibrational resonance occurs across a spectrum of nonlinear systems.
Therefore, it is essential to provide summaries of these excitations and
nonlinear models. In Section 2, we introduce some typical characteristic
signals in periodic or aperiodic forms. Next, we summarize the nonlinear
models for vibrational resonance in Section 3. Standard performance measures
of different kinds of vibrational resonance are presented in Section 4. In
Section 5, we delve into the theoretical formulation of the method of direct
separation of motions, linear and nonlinear vibrational resonance, and
ultrasensitive vibrational resonance. Additionally, the re-scaled vibrational
resonance is introduced to address the fast-varying characteristic signal,
with a brief analysis of the role of noise in vibrational resonance. In
Section 6, two applications of vibrational resonance are presented. Finally,
in Section 7, we offer the main conclusions of the review paper and provide
some future outlooks.

##  2 The excitations

Typically, vibrational resonance excitation consists of two signals: the
characteristic signal and the auxiliary signal, which can take various forms.
The characteristic signal in the following common signal types is often
considered.

###  2.1 Periodic signal

A periodic signal is the most frequently used in various fields. Among them,
the commonly used characteristic or auxiliary signal is a harmonic function.
The harmonic signal with amplitude $A$ can be in a sine waveform
$s(t)=A\sin(\omega t+\phi)$ or $s(t)=A\sin(2\pi ft+\phi)$, and in a cosine
waveform $s(t)=A\cos(\omega t+\phi)$ or $s(t)=A\cos(2\pi ft+\phi)$. For
$\omega$ and $f$, their units are $rad/s$ and $Hz$, respectively. The initial
phase is $\phi$. Additionally, there are anharmonic periodic signals, each
with different effects on vibrational resonance and related phenomena. Below,
we provide examples of typical anharmonic periodic signals, including:

(i) Square signal

$s(t)=\left\\{\begin{array}[]{rl}A,&(2n-2)\pi/\omega<t<(2n-1)\pi/\omega,\\\
-A,&{\rm{(}}2n-1)\pi/\omega<t<2n\pi/\omega,\\\
\end{array}\right.\;\;n=1,2,\cdots.$ (1)

(ii) Asymmetric sawtooth signal

$s(t)=\left\\{\begin{array}[]{rl}\frac{2At}{T},&(2n-2)\pi/\omega<t<(2n-1)\pi/\omega,\\\
\frac{2At}{T}-2A,&(2n-1)\pi/\omega<t<2n\pi/\omega,\\\
\end{array}\right.\;\;n=1,2,\cdots.$ (2)

(iii) Symmetric sawtooth signal

$s(t)=\left\\{\begin{array}[]{rl}\frac{4At}{T},&(2n-2)\pi/\omega<t<(\frac{4n-3}{2})\pi/\omega,\\\
-\frac{4At}{T}+2A,&(\frac{4n-3}{2})\pi/\omega<t<(\frac{4n-1}{2})\pi/\omega,\\\
\frac{4At}{T}-4A,&(\frac{4n-1}{2}{)}\pi/\omega<t<2n\pi/\omega,\\\
\end{array}\right.\;\;n=1,2,\cdots.$ (3)

(iv) Modulus of sine signal

$s(t)=A\left|\sin\left(\frac{\omega}{2}t\right)\right|.$ (4)

(v) Rectified sine signal

$s(t)=\left\\{\begin{array}[]{rl}A\sin\omega
t,&(2n-2)\pi/\omega<t<(2n-1)\pi/\omega,\\\
0,&(2n-1)\pi/\omega<t<2n\pi/\omega,\\\ \end{array}\right.\;\;n=1,2,\cdots.$
(5)

In the above formulas, $T$ is the period of the excitation that is equal to
$2\pi/\omega$. The slow excitation in anharmonic periodic form has been used
in some works of stochastic resonance in nonlinear systems [170], [171],
however not much work on vibrational resonance has been done. The form of the
periodic signal is found to influence directly the static bifurcation and the
appearance of the vibrational resonance [172]. By Fourier series expansion, an
anharmonic excitation can be viewed as the superposition of constant and
infinite harmonic terms. Moreover, an anharmonic periodic excitation may
induce much more complex dynamical behaviors [142], [173]-[176]. This suggests
that it is crucial to conduct more in-depth analysis of vibrational resonance
induced by various anharmonic periodic excitations, encompassing both the
characteristic and auxiliary signals in an anharmonic periodic form. The
waveforms of a harmonic and anharmonic periodic signals mentioned above and
certain other signals are given in Fig. 1. Among them, in Fig. 1 (a)-(e), the
periodic signals corresponding to (i)-(v) are sequentially presented. The
parameters of these signals, such as the period, the duty ratio, etc., have
important effects on vibrational resonance and related dynamic phenomena.

Figure 1: Waveforms of different types of characteristic signals. (a) square
signal, (b) asymmetric sawtooth signal, (c) symmetric sawtooth signal, (d)
modulus of sine signal, (e) rectified cosine signal, (f) aperiodic binary
signal, (g) frequency modulated signal, (h) echo chirp signal, (i) amplitude
modulated harmonic signal, (j) vibration signal of a bearing with fault in out
race (constant speed condition), (k) vibration signal of a bearing with fault
in inner race (constant speed condition), (l) vibration signal of a bearing
with fault in out race (time-varying speed condition).

###  2.2 Aperiodic binary signal

The aperiodic vibrational resonance is reported in [110], [115], [116]. The
aperiodic binary signal is governed by

$s(t)=A\sum\limits_{j=-\infty}^{+\infty}{R_{j}}\Gamma(t-jT),\quad\Gamma(\tau)=\left\\{\begin{array}[]{ll}1,&\tau\in[0,\>T]\\\
0,&\tau\notin[0,\>T]\\\ \end{array}\right..$ (6)

and $R_{j}$ is a random number sequence of $+1$ or $-1$ with an independent
stochastic distribution. $A$ is the amplitude of the aperiodic signal.
$\Gamma(t)$ is a random pulse sequence with minimal pulse width $T$. The time
series of an aperiodic binary signal is shown in Fig. 1(f).

The aperiodic vibrational resonance has been used in noisy image enhancement
[158], [159]. For monochrome image processing, the image is usually
transformed to an aperiodic binary signal. Hence, a further study on aperiodic
vibrational resonance is important. As an extension, stochastic resonance
induced by an M-ary signal has been investigated in certain systems
[177]-[179]. As an example of an application, if we deal with a gray image, we
must first transform the image into an M-ary signal. Therefore, the
vibrational resonance induced by an M-ary signal is worth exploring.

###  2.3 Frequency-modulated signal

The linear frequency modulated signal is a simple and typical signal in
science and engineering, whose form is

$s(t)=A\cos(\pi\gamma{t^{2}}+2\pi ft+\phi),$ (7)

where $A$ and $\phi$ are the amplitude and the initial phase, respectively.
$\gamma$ (the chirp rate) and $f$ (the starting frequency) characterize the
modulation characteristics of the frequency. The instantaneous frequency
$f_{\mathrm{in}}(t)$ is the derivative of the phase, i.e.,
$f_{\mathrm{in}}(t)=2\pi\gamma t+2\pi f$. The vibrational resonance caused by
the linear frequency modulated signal has been studied in [118] and [119].

Another typical frequency modulated signal is the echo chirp signal, which is
widely applied in compressed sensing, range measurement, velocity estimation,
and so on [121]. The expression of each echo chirp component is

$e(t)=\left\\{\begin{array}[]{ll}{\mathrm{e}}^{j2\pi\left[(B/(2T))t^{2}+(f_{\mathrm{c}}-B/2)t\right]},&0\leq
t\leq T\\\ 0,&\mathrm{otherwise}\end{array}\right..$ (8)

Herein, $B$, $T$ and $f_{\mathrm{c}}>0$ are the bandwidth, the duration and
the central frequency, respectively. The instantaneous frequency of the echo
chirp component is $f_{\mathrm{in}}=(B/T)t+f_{\mathrm{c}}-B/2$. The echo chirp
signal which is composed by $n$ chirp components is defined as [180]

$s(t)=\sum\limits_{i=1}^{n}{{\alpha^{i-1}}w(t-id-c)}\Re(e(t-id-c)),$ (9)

where $\alpha$ is the attenuation rate, $d$ is the delay between adjacent
chirp components, and $c$ is the initial delayed time. In addition, the symbol
$\Re(\bullet)$ represents the real part of the chirp signal, and $w(t)$ is
given by

$w(t)=\left\\{\begin{array}[]{ll}0.5[1-\cos(2\pi t/T)],&0\leq t\leq T\\\
0,&\mathrm{otherwise}\end{array}\right\\}.$ (10)

The plots of the linear frequency modulated signal and the echo chirp signal
are shown in Figs. 1(g) and 1(h), respectively. Usually, the frequency
modulated signals have a very clear physical or engineering background. For
example, the characteristic frequency of a vibration signal of a faulty
rolling bearing is in the frequency-modulated form when the equipment
operating in a variable speed condition [120]. Further, it is interesting to
analyze the features of vibrational resonance induced by different forms of
the nonlinear frequency modulated signal [181]-[185] in different engineering
fields.

###  2.4 Amplitude modulated signal

The amplitude modulated harmonic signal induced vibrational resonance have
been studied in [31], [114], [186]-[190]. The corresponding signal model is

$s(t)=\left[{A+B\cos\Omega t}\right]\cos\omega t.$ (11)

Its plot is given in Fig. 1(i). In addition, the amplitude modulated aperiodic
binary signal is [114]

$s(t)=\left[{A+B\cos\Omega
t}\right]\sum\limits_{j=-\infty}^{+\infty}{{R_{j}}\Gamma(t-jT)}.$ (12)

In fact, the vibration signal of a faulty bearing can be considered as an
amplitude modulated signal. For the vibration signal of a bearing with outer
race fault, its simulated signal form is [191]-[193]

$\left\\{{\begin{array}[]{*{20}{c}}{s\left(t\right)=A\sin\left({2\pi{f_{\text{n}}}t}\right)\exp\left\\{{-B{{\left[{t-i\left(t\right)/{f_{0}}}\right]}^{2}}}\right\\}}\\\
{i\left(t\right)={\text{floor}}\left[{{f_{0}}t}\right]}\\\
\end{array}}\right.,$ (13)

where $A$ is the amplitude of the pulse signal, $B$ is the attenuation
coefficient, $f_{\mathrm{n}}$ and $f_{0}$ are the natural frequency and the
characteristic fault frequency, respectively. The function $i(t)$ represents
the repetitions, and the function $floor[\bullet]$ is the floor function.

With an inner race or a ball element fault, the bearing vibration simulated
signal is [194], [195]

$\left\\{{\begin{array}[]{*{20}{c}}{s\left(t\right)=\sum\limits_{i}{H\left(t\right)}h\left({t-iT}\right)}\\\
{H\left(t\right)={A_{\text{0}}}\cos\left({2\pi{f_{\text{r}}}t}\right){\text{ +
}}C}\\\
{h\left(t\right)=A\cos\left({2\pi{f_{\text{n}}}t}\right)\exp\left({-B{t^{2}}}\right)}\\\
\end{array}}\right.,$ (14)

where the characteristic frequency is $1/T$, and $T$ is the period of the
pulse. The function $H(t)$ is the modulated signal with amplitude $A_{0}$, $C$
is the modulation bias, and $f_{\mathrm{r}}$ is the rotating frequency.

As shown in Figs. 1(j) and 1(k), for the signals in Eq. (13) and Eq. (14), the
periodic property and the amplitude modulation are apparently manifested.
Vibrational resonance is investigated with the signals of Eq. (13) and Eq.
(14) on bearing fault diagnosis [152]-[157]. Different from other signal
amplifier, not only the characteristic frequency is enhanced but other
disturbed frequencies are suppressed.

Sometimes, the frequency and the amplitude of the characteristic signal are
modulated simultaneously. We still use the bearing fault diagnosis as the
engineering background. While the rotating machinery is operating in variable
work conditions, the vibration signal of the bearing has both the frequency
and the amplitude modulated property. For example, under the variable speed
condition, the vibration signal of a bearing with outer race fault is
simulated by the following signal [196]

$\left\\{\begin{gathered}s(t)=\sum\limits_{m=1}^{M}A\exp\left\\{{-B[t-v(t)]}\right\\}\sin\left\\{{2\pi{f_{\text{n}}}[t-v(t)]+\phi}\right\\}\hfill\\\
v(t)=\sum\limits_{i=1}^{m-1}{{T_{i}}}\hfill\\\ \end{gathered}\right.,$ (15)

where $m$ is the impulse index, $T_{i}$ is the repeating period of the $i$th
impulse. The function $v(t)$ is the total occurrence time of $(m-1)$ impulses
and is a function of time $t$. The plot of this signal is presented in Fig.
1(l). The effect of the above signal has been investigated by the stochastic
resonance method [196]. We still need to analyze the vibrational resonance
caused by this signal in the bearing fault diagnosis under different variable
working conditions. With the rapid development of mechanical industry, we need
to pay enough attention to the research of vibrational resonance in the field
of mechanics.

###  2.5 Logical signal

Logical vibrational resonance has been investigated and interesting results
have been reported [143], [197]-[201]. As a result, the logical inputs can be
as the characteristic signals, which are random permutations consisting of
four logic states (0, 0), (0, 1), (1, 0), (1, 1). Since logical operations are
widely used in the field of electronics and computer technology, the
theoretical analysis, as well as the analysis oriented towards applications of
logical vibrational resonance are of great importance for the future.

##  3 Nonlinear models of vibrational resonance

Depending on whether the system incorporates fractional damping, time delay,
or noise, nonlinear models for vibrational resonance are typically classified
into five main categories: ordinary differential systems, mapping systems,
fractional differential systems, delayed differential systems, and stochastic
differential systems. In the literature, authors may adopt different
expressions or symbols to label the excitations and the equations. To simplify
the system model without the need for a specific notation, we employ
$s_{\mathrm{L}}(t)$, $s_{\mathrm{H}}(t)$, and $\xi(t)$ to denote the slowly-
varying characteristic signal, the fast-varying auxiliary signal, and the
noise, respectively.

###  3.1 Models with ordinary differential systems

We present several common nonlinear systems characterized by ordinary
differential equation models in the investigation of vibrational resonance.
These models are categorized based on the shape of the potential function, the
type of damping (linear or nonlinear), the number of subsystems, and the
application context of the systems.

####  3.1.1 Symmetric bistable oscillators

The overdamped and underdamped symmetric bistable Duffing oscillator systems
are governed by

$\dot{x}+\omega_{0}^{2}x+\beta x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)$ (16)

and

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta
x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (17)

respectively.

In Eqs. (16) and (17), $\omega_{0}^{2}<0$, $\beta>0$ and $\delta>0$. The
potential function is

$V(x)=\frac{1}{2}\omega_{0}^{2}x^{2}+\frac{1}{4}\beta x^{4}\,.$ (18)

In absence of $s_{\mathrm{L}}(t)$ and $s_{\mathrm{H}}(t)$, these two systems
have three equilibria $x=0$ and $x_{\pm}=\pm\sqrt{\omega_{0}^{2}/\beta}$. Many
vibrational resonance works have considered this kind of system [1], [5], [6],
[94], [96], [98], [101], [102], [111], [119], [121], [127], [132], [141],
[152]-[155], [202]-[205]]. Vibrational resonance in a bistable system is shown
in an experimental setup of Eq. (17) [206]. The symmetric bistable system is a
typical model for many physical and engineering systems. In addition, if
$\omega_{0}^{2}>0$ and $\beta>0$, the potential function is in a monostable
form with only a stable equilibrium $x=0$; if $\omega_{0}^{2}>0$ and
$\beta<0$, then the system is a softening Duffing system and the potential
function has a stable equilibrium $x=0$ and two unstable equilibria
$x_{\pm}=\pm\sqrt{\omega_{0}^{2}/\beta}$ [207]. In current researches on
vibrational resonance, the bistable systems of Eqs. (16) and (17) are the most
typical and studied models. There is a need for further research on the
connection between vibrational resonance and more complex dynamical phenomena,
including various bifurcation patterns, safe basin erosion [208], and others,
within the framework of this typical bistable system model.

Another overdamped symmetric bistable system wherein vibrational resonance is
investigated [114], [115], [118], [209]] is

$\dot{x}+\omega_{0}^{2}x+\beta
x\left|x\right|^{\alpha-1}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (19)

The potential function is

$V(x)=\frac{1}{2}\omega_{0}^{2}x^{2}+\frac{\beta}{{\alpha+1}}\left|x\right|^{\alpha+1},\quad\omega_{0}^{2}<0,\;\beta>0,\;\alpha>0.$
(20)

When $\alpha>1$, the potential function has two stable equilibria
$x_{\pm}=\pm{\left({-\frac{{\omega_{0}^{2}}}{\beta}}\right)^{\frac{1}{{\alpha-1}}}}$
and an unstable equilibrium $x=0$. The parameter $\alpha$ is a real irrational
or rational number. In engineering applications, the value of $\alpha$ is
usually related to the material property. The detailed physical background has
been explained in previous works [210]-[212]. For the potential function, the
value of $\alpha$ mainly influences its deepness. In the study of vibrational
resonance, this kind of system can be viewed as a signal processor. For the
case $\alpha=3$, the shape of the potential function presents the typical
bistable configuration which is described by Eq. (18). In addition, a suitable
value of $\alpha$ can cause the system response to be in a stronger resonance
state. It is important to investigate this kind of potential further by using
analytical, experimental and numerical ways. The plots of the symmetric
bistable potential function corresponding to Eq. (18) and Eq. (20) are given
in Figs. 2(a) and 2(b), respectively. Herein, we choose four different values
of $\alpha$ to obtain the curves in Fig. 2(b).

Figure 2: Plots of the potential function of some nonlinear oscillators, (a)
symmetric bistable potential in Eq (18), (b) symmetric bistable potential in
Eq. (20), (c) asymmetric bistable potential in Eq. (22), (d) asymmetric
bistable potential in Eq. (24), (e) monostable potential in Eq. (18), (f)
Morse potential function in Eq. (29), (g) Tietz-Hua potential in Eq. (31), (h)
function in Eq. (33) with monostable potential, (i) function in Eq. (33) with
bistable and tristable potentials, (j) periodic potential function, (k)
harmonically trapped potential function, (l) asymmetric deformable potential
function.

####  3.1.2 Asymmetric bistable oscillators

There are many interesting asymmetric bistable systems, such as

$\ddot{x}+\delta\dot{x}+\frac{\mathrm{d}V}{\mathrm{d}x}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(21)

where the potential function is

$V(x)=\frac{1}{2}\omega_{0}^{2}x^{2}+\frac{1}{3}\alpha x^{3}+\frac{1}{4}\beta
x^{4}.$ (22)

The potential function is in asymmetric double-well form for
$\omega_{0}^{2}<0$, $\beta>0$ and $\alpha\neq 0$ (see Fig. 2(c)). Vibrational
resonance also takes place in the asymmetric bistable system of Eq. (21) for a
wide range of input signal frequencies for which the resonance is impossible
in the symmetric bistable system [213].

Another asymmetric bistable potential is

$V(x)=\frac{1}{2}\omega_{0}^{2}x^{2}+\frac{1}{4}\beta x^{4}+\gamma
x,\quad\omega_{0}^{2}<0,\;\beta>0,$ (23)

where $\gamma$ is a parameter indicating the asymmetry. Vibrational resonance
is investigated also in overdamped systems with the above asymmetric potential
function theoretically, numerically and experimentally [95], [110], [129],
[214]. Interestingly, the asymmetric property is necessary for the occurrence
of certain types of resonances. As an illustration, logical vibrational or
stochastic resonance typically manifests only in the presence of a bias
parameter, and the distinct logical resonances achieved through $AND$, $NAND$,
$OR$, and $NOR$ calculations are contingent upon the tilting direction of the
potential function. In addition, the asymmetry of the potential also
influences the response amplitude curve of vibrational resonance, especially
when the initial conditions are located in different potential wells in
numerical calculations [64].

Another asymmetric bistable potential exists in the three-level atomic optical
bistable system, e.g.,

$\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{d}V}{\mathrm{d}x}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(24a) $\displaystyle V(x)$ $\displaystyle=$ $\displaystyle-
yx+{x^{2}}\left({\frac{{{c_{1}}}}{2}+\frac{{{c_{2}}}}{3}x+\frac{{{c_{3}}}}{4}{x^{2}}}\right)$
(24b)

where $y$, $c_{1}$, $c_{2}$, $c_{3}$ are all constants [215]. Here, the
potential $V(x)$ is an asymmetric bistable form. The curve of the potential
function is given in Fig. 2(d).

The groundwater-dependent plant ecosystem also has an asymmetric potential
with three equilibria (two stable and one unstable) [216]

$\displaystyle\dot{x}=x(x_{\mathrm{cc}}-x),$ (25a) where $\displaystyle
x_{\mathrm{cc}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}a\left[d(t)-d_{\mathrm{inf}}\right]\left[d_{\mathrm{sup}}-d(t)\right],&\mathrm{if}\;\;d_{\mathrm{inf}}<d<d_{\mathrm{sup}}\\\
0,&\mathrm{otherwise}\\\ \end{array}\right.$ (25d) $\displaystyle d(t)$
$\displaystyle=$ $\displaystyle d_{0}+\beta
x+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (25e)

where $x$ is the phreatophyte biomass of the considered species,
$x_{\mathrm{cc}}$ is the carrying capacity of the studied ecosystem, and
$d(t)$ is the depth of groundwater level. The two excitations describe the
variation of the groundwater table that caused by some factors. The potential
function of the system presents an asymmetric bistable form with $x>0$.

####  3.1.3 Monostable oscillators

The nano-electromechanical resonator is a monostable system [27]

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta
x^{3}=F[1+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)]\cos\omega_{\mathrm{f}}t,$ (26)

where $\omega_{0}^{2}>0$, $\beta>0$, and $\omega_{\mathrm{f}}$ is in the
vicinity of the fundamental mode frequency. Its potential function is given by
Eq. (18) with $\omega_{0}^{2}>0$ and $\beta>0$. The plot of the monostable
potential function is shown in Fig. 2(e).

The monostable system excited by biharmonic signals is [217]

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\alpha x^{2}+\beta
x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (27)

The potential function is given by Eq. (22). When $\alpha=0$,
$\omega_{0}^{2}$, $\beta>0$, the potential function is monostable and
symmetric about $x=0$. When $0<\alpha^{2}<4\omega_{0}^{2}\beta$, the potential
function is still in monostable form but asymmetric. When
$\alpha^{2}>4\omega_{0}^{2}\beta$, the potential turns to a bistable form. An
experimental setup for the system Eq. (27) was built and the underlying
vibrational resonance was also investigated in this work.

The Morse oscillator can describe the photodissociation and the interatomic
potential of molecules. It is a monostable system and the dynamical equation
of motion is [218]

$\ddot{x}+\delta\dot{x}+\beta\mathrm{e}^{-x}\left(1-\mathrm{e}^{-x}\right)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(28)

The corresponding potential function is expressed as

$V(x)=\frac{1}{2}\beta\mathrm{e}^{-x}\left(1-\mathrm{e}^{-x}\right).$ (29)

The plot of the potential function of the Morse oscillator is given in Fig.
2(f), and $V(x)$ has only one local minimum.

The system with a two-frequency-excited Tietz-Hua quantum well is also a
monostable system exhibiting quantum resonance [25]. The Hamiltonian of the
system is

$H=\frac{1}{2m}p_{x}^{2}+V_{\mathrm{TH}}+\lambda
x\left(s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)\right).$ (30)

The Tietz-Hua potential function is

$V_{\mathrm{TH}}(x)=V_{0}\left[\frac{1-\mathrm{e}^{-b_{h}(x-x_{\mathrm{e}})}}{1-c_{\mathrm{h}}\mathrm{e}^{-b_{\mathrm{h}}(x-x_{\mathrm{e}})}}\right]^{2}.$
(31)

Its graph of $V_{\mathrm{TH}}(x)$ is presented in Fig. 2(g). The Tietz-Hua
potential reduces to the classical Morse potential when its potential constant
$c_{\mathrm{h}}$ is zero.

In the above four kinds of monostable systems, all the excitations are in
harmonic form, and yet the monostable system especially has advantages in
processing aperiodic pulse signal under noisy background, which has been
verified in extracting steel wire rope flaw detection signal based on
stochastic resonance [219], [220]. At present, there is very little work on
vibrational resonance of monostable system under the action of different
aperiodic pulse signals. The related theoretical and applied researches are
worthwhile to do in the near future.

####  3.1.4 Quintic oscillators

Single or multiple vibrational resonance can occur in a quintic oscillator
[11]-[14], [221]

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta x^{3}+\gamma
x^{5}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (32)

The potential function is

$V(x)=\frac{1}{2}\omega_{0}^{2}x^{2}+\frac{1}{4}\beta x^{4}+\frac{1}{6}\gamma
x^{6}.$ (33)

By choosing different system parameters, the potential may be in a monostable,
bistable, or tristable configuration. When $\omega_{0}^{2}$, $\beta$,
$\gamma>0$, the potential has a single-well. When $\omega_{0}^{2}$,
$\gamma>0$, $\beta<0$, and $\beta^{2}<4\omega_{0}^{2}\gamma$, the potential
also has a single well. When $\omega_{0}^{2}$, $\beta>0$, and $\gamma<0$, the
potential has a double-hump single-well. The potential curves with different
single-well shapes are presented in Fig. 2(h). When $\omega_{0}^{2}<0$,
$\beta$, $\gamma>0$, the potential is bistable. When $\omega_{0}^{2}$,
$\gamma>0$, $\beta<0$, and $\beta^{2}>16\omega_{0}^{2}\gamma/3$, the potential
has a triple-well shape. The plots of the potential with bistable and
tristable __cases are plotted in Fig. 2(i). The potential $V(x)$ can be used
in different occasions, e.g., modelling dynamics of optical bistability,
magnetoelastic beam, folded intermediates of proteins, among some other
problems [11].

####  3.1.5 Periodic potential oscillators

There is a series of vibrational resonances appearing in the pendulum system
[20], [140], with overdamped version

$\dot{x}+\sin x=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)$ (34)

or with underdamped version

$\ddot{x}+\delta\dot{x}+\sin x=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (35)

The potential function is $V(x)=-\cos x$ and its form is shown in Fig. 2(j).
There are also some other modified periodic potential functions, such as
$V(x)=-(1/k)\left[\sin kx+(1/4)\sin 2kx\right]$ which represents a rocking
ratchet [21]. Vibrational resonance has been explored in a system featuring
periodic potential, and it has been investigated in the context of cold atoms
confined in an optical lattice [222].

Furthermore, vibrational resonance has been studied in a nonlinear dissipative
system which has a symmetric periodic potential and a space-dependent
nonlinear damping coefficient [22], [223]. The system is

$\ddot{x}+\gamma_{0}\left[1-\lambda\sin(kx+\phi)\right]\dot{x}+V_{0}\sin
kx=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (36)

which is a more general case of Eq. (35).

####  3.1.6 Harmonically trapped and deformable potential systems

The harmonically trapped potential system is [224]

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta\sin
x=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),\quad\omega_{0}^{2}>0,\;\beta>0.$ (37)

Its potential is $V(x)=(1/2)\omega_{0}^{2}x^{2}-\beta\cos x$. For different
values of $\beta$, the potential may have one well, three wells and even more
odd wells, as shown in Fig. 2(k).

Another interesting system exhibiting vibrational resonance is [225]

$\ddot{x}+\delta\dot{x}+\frac{V_{0}\left(1-r^{2}\right)^{2}}{\pi}\frac{\sin\left(\pi
x\right)\left[2r+\left(1+r^{2}\right)\cos(\pi
x)\right]}{\left[1+r^{2}+2r\cos(\pi
x)\right]^{3}}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (38)

with an asymmetrical deformable (Remoissenet-Peyrard) potential

$V(x)=\frac{V_{0}\left(1-r^{2}\right)^{2}\left[1-\cos(2\pi
x)\right]}{(2\pi)^{2}\left[1+r^{2}+2r\cos(\pi x)\right]^{2}}.$ (39)

With different values of $r$, the potential presents different asymmetric
deformable potentials, as shown in Fig. 2(l). When $r=0$, the system turns to
a periodic potential system. The potential described in Eq. (39) is commonly
employed when examining one-dimensional atomic chains.

####  3.1.7 Parametric oscillators

An oscillator with one or more parameters varying with time is a parametric
oscillator, also called a parametrically excited oscillator. The parametric
oscillator constitutes a very important class of nonlinear systems. The
parametric factor may appear in the inertia, the damping, or the stiffness
term.

A bistable system with linear stiffness modulated by a low-frequency signal
and excited by an external high-frequency signal is [99]

$\ddot{x}+\delta\dot{x}+\left(\omega_{0}^{2}x+s_{\mathrm{L}}(t)\right)x+\beta
x^{3}=s_{\mathrm{H}}(t).$ (40)

In Eq. (40), the frequency $s_{\mathrm{H}}(t)$ is far greater than both the
frequency of $s_{\mathrm{L}}(t)$ and the natural frequency $\omega_{0}$.

The model of a controllable parametrically excited buckled beam is [226]

$\ddot{x}+\delta\dot{x}+\left(\omega_{0}^{2}+s_{\mathrm{H1}}(t)\right)x+\beta
x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H2}}(t).$ (41)

The oscillator is of Mathieu-Duffing type. The natural frequency of the
oscillator is adjusted by a high-frequency signal and the oscillator is
subjected to a low-frequency signal and another high-frequency signal
simultaneously.

Besides, a cantilever beam parametrically excited is described by [227]

$\left(1+\alpha_{1}x^{2}\right)\ddot{x}+2\mu_{1}\dot{x}+\mu_{2}\left|\dot{x}\right|\dot{x}+\alpha_{2}x\dot{x}^{2}+\left(\omega_{0}^{2}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)\right)x+\alpha_{3}x^{3}=0\,.$
(42)

A van der Pol-Mathieu-Duffing oscillator with nonlinear damping and two
external harmonic excitations with two widely different frequencies is given
by [228]

$\ddot{x}+\gamma\left(x^{2}-1\right)\dot{x}-\omega_{0}^{2}\left(1+h\cos\omega_{\mathrm{p}}t\right)x+\beta
x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (43)

The model in Eq. (43) can describe the dynamics of a micro-electro-mechanical
system.

Not only the linear stiffness but also the damping can be parametrically
varied with time, such as the nonlinear vibrational resonance in the van der
Pol-Duffing oscillator with a parametric damping [100]

$\ddot{x}+\gamma(1+s_{\mathrm{L}}(t))({x^{2}}-1)\dot{x}-\omega_{0}^{2}x+\alpha{x^{3}}=s_{\mathrm{H}}(t).$
(44)

The role of periodically parametric damping on vibrational resonance was
studied also in a modulated signal excited asymmetric mixed Rayleigh-Liénard
oscillator as follows [190]

$\ddot{x}+({a_{0}}+s_{\mathrm{L}}(t))\dot{x}+{b_{1}}x\dot{x}+{b_{2}}{{\dot{x}}^{2}}+{b_{3}}{{\dot{x}}^{3}}-{E_{0}}-x+c{x^{3}}=({d_{1}}+s_{\mathrm{H}}(t))s_{\mathrm{L}}(t).$
(45)

####  3.1.8 Position-dependent mass oscillators

Another important kind of nonlinear system related to vibrational resonance is
the position-dependent mass oscillator. The model with varying mass is usually
encountered in many fields such as aeronautical and astronautical engineering,
astronomy, aerology, offshore engineering, civil engineering, condensed matter
physics, semiconductor heterogeneous structures, etc.

A Duffing oscillator with position-dependent mass is described by [164]

$m(x)\ddot{x}-{m^{2}}(x)\gamma\lambda
x{\dot{x}^{2}}+\alpha\dot{x}+{m^{2}}(x)\gamma\omega_{0}^{2}x+\beta{x^{3}}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(46)

Herein, $m(x)$ is a position-dependent function. Roy-Layinde et al. used Eq.
(46) to study the dynamics of a gas bubble.

Another position-dependent mass oscillator is [229]

$m(x)\ddot{x}+\frac{1}{2}m^{\prime}(x){\dot{x}^{2}}+\alpha\dot{x}+m(x)\omega_{0}^{2}x+\beta{x^{3}}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(47)

that describes vibrational resonance in the model of a NH3 molecule. In Eq.
(46) and Eq. (47), not only the mass but also the damping and stiffness are
position-dependent.

The following nonlinear oscillator with position-dependent mass and phase-
dependent damping is used to model the current through a Josephson junction
[23]

$m(x)\ddot{x}+\frac{1}{2}m^{\prime}(x){\dot{x}^{2}}+\gamma(x)\dot{x}+{V_{0}}\sin(kx)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(48)

There is another interesting position-dependent mass oscillator which is given
by [230]

$(1+\mu{x^{2}})\ddot{x}+\mu
x{\dot{x}^{2}}+\alpha\dot{x}+\omega_{0}^{2}x=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(49)

that describes a particle that moves on a rotating-parabola system. By some
calculations and simplifications, Eq. (49) can be transformed into a quintic
oscillator which is also a parametrically excited system. Vibrational
resonance in a Mathews-Lakshmanan oscillator similar to Eq. (49) was also
investigated in [231].

####  3.1.9 Nonlinearly damped oscillators

As the name implies, a nonlinearly damped oscillator has a nonlinear damping.
Apparently, the systems in Eqs. (36), (42)-(49) are all nonlinearly damped
oscillators. In addition to these, there are other nonlinearly damped systems
that exhibit vibrational resonance. We summarize them here.

The nonlinearly damped oscillator with a rough potential is given by [18]

$\ddot{x}-\mu\left(1-x^{2}+\nu
x^{4}\right)\dot{x}+\frac{\mathrm{d}V}{\mathrm{d}x}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)$
(50)

with a potential

$V(x)=\frac{1}{2}\omega_{0}^{2}x^{2}+\frac{1}{4}\beta x^{4}+\frac{1}{6}\gamma
x^{6}+\epsilon(\cos\omega_{1}x+\sin\omega_{2}x).$ (51)

The model comes from a set of quasihydrodynamic equations and describes a two-
fluid magnetized plasma oscillator of motion. When $\epsilon=0$, the potential
is smooth and has been given in Eq. (33). When $\epsilon\neq 0$, the potential
is perturbed and increases roughly with the increase of $\epsilon$.

The Toda oscillator in the presence of a dual-frequency forcing is [19]

$\ddot{x}+k_{0}(1+\varepsilon\cos
x)\dot{x}+\mathrm{e}^{x}-1=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (52)

The Toda potential is given by $V(x)=\mathrm{e}^{x}-x+1$. The Toda oscillator
has abundant dynamic behaviors such as bifurcations, chaos, nonlinear
resonance, etc. This model comes from the study of different types of lasers.
It has been also applied to DNA, and molecular dynamics of muscle contraction
[232].

Anharmonic plasma oscillation induced by bi-harmonic forces is expressed as
[233]

$\ddot{x}+\varepsilon\left(1+x^{2}\right)\dot{x}+\frac{\mathrm{d}V(x)}{\mathrm{d}x}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(53)

The potential function here is

$V(x)=\frac{1}{2}\omega_{0}^{2}x^{2}+\frac{1}{3}\alpha x^{3}+\frac{1}{4}\beta
x^{4}.$ (54)

This physical model refers to high density plasma interactions caused by high
frequency electromagnetic waves. The model in Eq. (53) is also used to study
the ghost-vibrational resonance when a multi-frequency signal with the form
$\sum\limits_{i=1}^{nf}{{f_{i}}\cos({\omega_{i}}+\Delta{\omega_{0}})}t$
instead of the slow-varying signal $s_{\mathrm{L}}(t)$ [105].

A beam with boundary conditions of two ends fixed and prestressed is modeled
by [234]

$\ddot{x}+\left(\beta_{0}+\beta_{2}\dot{x}^{2}\right)\dot{x}+x+\gamma_{2}x^{2}+\gamma_{3}x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(55)

The nonlinear damping is the nonlinear dissipation of the Rayleigh type. The
background of the biharmonic excitation comes from wind (slow-varying
excitation) and earthquake (fast-varying excitation), or flow of people
crossing the same bridge (slow-varying excitation) and vehicles crossing the
bridge (fast-varying excitation).

A base-excited, tilted cantilever beam is given as [235]

$\displaystyle\ddot{x}\\!\\!+\\!\\!2{\mu_{1}}\dot{x}\\!\\!+\\!\\!{\mu_{2}}{{\dot{x}}^{3}}\\!\\!+\\!\\!x\\!\\!+\\!\\!{\alpha_{1}}{x^{3}}\\!\\!+\\!\\!{\alpha_{2}}({x^{2}}\ddot{x}\\!\\!+\\!\\!x{{\dot{x}}^{2}})\\!\\!$
$\displaystyle=\\!\\!$ $\displaystyle(\eta\cos\beta\\!\\!-\\!\\!\lambda
x\sin\beta)(\\!\\!-\\!\\!{{\ddot{z}}_{p}}),$ (56a) $\displaystyle{z_{p}}$
$\displaystyle=$ $\displaystyle
s_{\mathrm{L1}}(t)+s_{\mathrm{L2}}(t)+s_{\mathrm{H}}(t),$ (56b)

where $z_{p}$ is the base excitation. The terms $z_{p}\sin\beta$ and
$z_{p}\cos\beta$ are two displacements in two mutually perpendicular
directions.

Vibrational resonance occurs in the nonlinear Rayleigh-Plesset oscillator
[236]

$\displaystyle\ddot{x}+\dot{x}[\alpha_{0}-\alpha_{1}x+\alpha_{2}x^{2}-\alpha_{3}x^{3}+\alpha_{4}x^{4}]+\eta\dot{x}^{2}[1-x+x^{2}]$
$\displaystyle\quad\quad-x\\{\beta-[s_{\mathrm{H}}(t)+\varepsilon]s_{\mathrm{L}}(t)\\}+x^{2}\\{\gamma-[s_{\mathrm{H}}(t)+\varepsilon]s_{\mathrm{L}}(t)\\}$
$\displaystyle\quad\quad-\gamma x^{3}+\lambda
x^{4}=[s_{\mathrm{H}}(t)+\varepsilon]s_{\mathrm{L}}(t).$ (57)

The model is used for the study of the dynamics of a gas bubble under the
excitations of complex parametric and external forces.

There is also a dynamical equation of a charged bubble excited by a modulated
acoustic field and vibrating in a liquid given by [189]

$\displaystyle\ddot{x}$ $\displaystyle=$ $\displaystyle{\psi_{1}}{\psi_{2}}$
(58a) $\displaystyle{\psi_{1}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left[{\frac{{{z_{1}}}}{{{x^{3\Gamma}}}}(1\\!\\!+\\!\\!{z_{2}}\dot{x})\\!\\!-\\!\\!\frac{{{{\dot{x}}^{2}}}}{2}(3\\!\\!-\\!\\!\frac{{\dot{x}}}{c})+\frac{{{z_{3}}}}{{{x^{4}}}}(1\\!\\!-\\!\\!\frac{{3\dot{x}}}{c})-\frac{{{z_{4}}\\!\\!+\\!\\!{z_{5}}\dot{x}}}{x}\\!\\!-\\!\\!{\Delta_{1}}(t)\\!\\!-\\!\\!{\Delta_{2}}(t)}\right]$
(58b) $\displaystyle{\psi_{2}}$ $\displaystyle=$
$\displaystyle{\left[{(1-\frac{{\dot{x}}}{c})x+{z_{10}}}\right]^{-1}}$ (58c)

where $\Delta_{1}t$ and $\Delta_{2}t$ are the functions contained in the
parametric modulated excitations

$\displaystyle{\Delta_{1}}(t)$ $\displaystyle=$
$\displaystyle(1+\frac{{\dot{x}}}{c})\left[{{z_{6}}+{z_{7}}(1+G\sin(\Omega
t))\sin(\omega t)}\right],$ (59a) $\displaystyle{\Delta_{2}}(t)$
$\displaystyle=$ $\displaystyle x\left[{{z_{8}}(1+G\sin(\Omega t))\cos(\omega
t)+{z_{9}}G\sin(\omega t)\cos(\Omega t)}\right].\ $ (59b)

Under a dual-frequency excitation, the gyroscope system mounted on a vibrating
base is given by [237]

$\ddot{x}+c_{1}\dot{x}+c_{2}\dot{x}^{3}+\alpha^{2}\frac{\left(1-\cos
x\right)^{2}}{\sin^{3}x}-\left[\beta+s_{\mathrm{H}}(t)\right]\sin
x=s_{\mathrm{L}}(t).$ (60)

The system has a nonlinear damping and a parametric excitation, and the
potential function may be present in a double-well or a single-well form.

Another typical damping is the resistance force expressed by the signum
function. The vibrational resonance is studied in a nonlinear system with a
signum nonlinearity [238]

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta\mathrm{sgn}(x)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(61)

The sign $\mathrm{sgn}(\bullet)$ is the signum function. Here, the nonlinear
damping depends on the displacement of the coordinate. In fact, the nonlinear
damping that depends on the velocity of the coordinate is much more widely
used especially in mechanical devices. The nonlinear damping
$\mathrm{sgn}(\dot{x})$ is usually viewed as a simple model of the dry
friction [239]. It is significant to study vibrational resonance in friction
systems due to the fact that a fast-varying excitation can quench the
vibration and can be viewed as an antiresonance. There are many mathematical
models belonging to different friction damping [240]. Further, the high-
frequency excitation has a very important role on the friction induced
mechanical vibration. It can change the properties of the stiffness, bias, and
smoothness of the system [7], [241]-[243]. Vibrational resonance in much more
complex friction systems is almost blank. Investigating works in this area
will contribute to nonlinear vibration and control significantly.

####  3.1.10 Coupled oscillators

Vibrational resonance and antiresonance in unidirectionally coupled overdamped
oscillators have been studied under low-frequency and high-frequency
excitations in different oscillators, as described in [244]

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
x-x^{3}+s_{\mathrm{H}}(t),$ (62a) $\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle y-y^{3}+\gamma x+s_{\mathrm{L}}(t),$ (62b)

and mutually coupled overdamped oscillators

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
x-x^{3}+\gamma(x-y)+s_{\mathrm{L}}(t),$ (63a) $\displaystyle\dot{y}$
$\displaystyle=$ $\displaystyle y-y^{3}+\gamma(y-x)+s_{\mathrm{H}}(t).$ (63b)

An anharmonic coupled overdamped oscillator subjected to both low-frequency
and high-frequency excitations is [245]

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
a_{1}x-b_{1}x^{3}+\gamma xy^{2}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (64a)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle
a_{2}y-b_{2}y^{3}+\gamma x^{2}y.$ (64b)

The system is viewed as a dynamical system modeling the competition between
two different species. Here, $a_{1}\neq a_{2}$ and the potential function is

$V(x,y)=-\frac{1}{2}a_{1}x^{2}+\frac{1}{4}b_{1}x^{4}-\frac{1}{2}a_{2}y^{2}+\frac{1}{4}b_{2}y^{4}-\frac{1}{2}\gamma
x^{2}y^{2}.$ (65)

If $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$, $\gamma>0$, the potential presents a
four-well shape. The system has nine or five fixed points corresponding to
$\gamma^{2}<b_{1}b_{2}$ and $\gamma^{2}>b_{1}b_{2}$, respectively. In the
study of this kind of oscillators, one usually chooses $a_{1}\neq a_{2}$, and
the potential has an asymmetric potential structure. Vibrational resonance of
the coupled system in Eq. (65) was also studied when the excitation is in the
amplitude modulated form [186].

The chemical reaction model between four molecules is described by [28]

$\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle-2k_{1}x^{2}y^{2}+k_{2}x(1-x-y)+(F+s_{\mathrm{H}}(t))s_{\mathrm{L}}(t),$
(66a) $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle
k_{1}x^{2}y^{2}-k_{3}y(1-x-y).$ (66b)

This oscillator has abundant nonlinear dynamics such as hysteresis,
vibrational resonance, multistability and chaos.

Vibrational antiresonance is investigated in the following coupled nonlinear
oscillators [16]

$\displaystyle\ddot{x}_{1}+2\gamma_{1}\dot{x}_{1}-2g_{1}x_{2}+\beta_{1}x_{1}^{3}-\omega_{1}^{2}x_{1}$
$\displaystyle=$ $\displaystyle s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (67a)
$\displaystyle\ddot{x}_{2}+2\gamma_{2}\dot{x}_{2}-2g_{2}x_{1}+\beta_{2}x_{2}^{3}-\omega_{2}^{2}x_{2}$
$\displaystyle=$ $\displaystyle s_{\mathrm{H}}(t).$ (67b)

Vibrational resonance and its extend application in signal transmission are
investigated in a one-way coupled bistable system [17]

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
a_{1}x_{1}-b_{1}x_{1}^{2}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (68a)
$\displaystyle\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
a_{1}x_{i}-b_{1}x_{i}^{2}+\epsilon x_{i-1},\quad i=2,3,\cdots,n.$ (68b)

In Eq. (68), when the signal $s_{\mathrm{L}}(t)$ is added on all elements
$x_{i}$, there is a vibrational resonance phenomenon. When the signal
$s_{\mathrm{L}}(t)$ is added only on $x_{1}$, it is a signal transmission
phenomenon.

Ferroelectric liquid crystal is also modeled as a coupled system [246]

$\dot{x}_{i}=x_{i}-x_{i}^{3}+\sum_{j=1}^{n}J_{ij}(x_{j}-x_{i})+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)+x_{i}w_{i},\quad
i=1,2,\cdots,n,$ (69)

where $J_{ij}$ means the coupling strength and $w_{i}$ is the local liquid
crystal polymer interaction. Vibrational resonance appears in this physical
system as well.

####  3.1.11 Neural models and complex networks

Vibrational resonance has garnered increasing attention within various neural
models and complex networks. Below, we provide a list of some of these models.

The single FitzHugh-Nagumo model

$\displaystyle\varepsilon\dot{x}$ $\displaystyle=$ $\displaystyle
x-\frac{1}{3}x^{3}-y,$ (70a) $\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle x+a+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (70b)

is a typical system exhibiting vibrational resonance [66]-[73]. In Eq. (70b),
without the excitations, $a$ is usually considered as the bifurcation
parameter. Specifically, when $a>1$, the FitzHugh-Nagumo model has a stable
equilibrium; whereas when $a<1$, the system undergoes a supercritical Hopf
bifurcation, producing a stable limit cycle, and then the equilibrium point
becomes the focus of instability. The variable $\varepsilon$ means the time
scale separation parameter which is considered to be small. Under different
response patterns, the subthreshold vibrational resonance or suprathreshold
vibrational resonance will be present in the FitzHugh-Nagumo system [69]. In
addition, vibrational resonance in a modified FitzHugh-Nagumo dynamical system
has also been studied and the resonance patterns are discussed in [76]. When
the excitation in Eq. (70b) is replaced by
$[S_{1}(t)+S_{2}(t)][S_{L}(t)+S_{H}(t)]$, where $S_{1}$ and $S_{2}$ represent
two logical signals, a comprehensive analysis of logical vibrational resonance
has been carried out in detail in [201].

Vibrational resonance also occurs in small-word neuronal networks with
different types of synapses [74], [80], [86], [90], [97]

$\displaystyle\varepsilon\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}-\frac{1}{3}x_{i}^{3}-y_{i}-I_{i}^{\mathrm{syn}},$ (71a)
$\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}+a+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (71b) where
$I_{i}^{\mathrm{syn}}$ is the synapse coupling expressing the synaptic
current. For the case of an electrical synapse coupling,
$I_{i}^{\mathrm{syn}}$ is $\displaystyle
I_{i}^{\mathrm{syn}}=g_{\mathrm{syn}}\sum_{j}C_{\mathrm{e}}(i,j)(x_{i}-x_{j}),$
(71c) and for case of the chemical synapse coupling, $I_{i}^{\mathrm{syn}}$ is
$\displaystyle I_{i}^{\mathrm{syn}}$ $\displaystyle=$ $\displaystyle
g_{\mathrm{syn}}\sum_{j}C_{\mathrm{c}}(i,j)s_{j}(x_{i}-x_{\mathrm{syn}}),$
(71d) $\displaystyle\dot{s}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon}\alpha(x_{j})(1-s_{j})-\frac{s_{j}}{\tau_{\mathrm{syn}}},\quad\alpha(x_{j})=\frac{\alpha_{0}}{1+\exp(-x_{j}/x_{\mathrm{shp}})},$
(71e)

where $\tau_{\mathrm{syn}}=1/\delta$ represents the decay rate of the synapse.
Usually, the chemical synaptic coupling is much more efficient on the low-
frequency signal transmission than that of the electrical coupling due to its
selective coupling.

Vibrational resonance is analyzed in small-world neuronal networks that have
spike-timing-dependent plasticity [81]. This kind of networks are described as

$\displaystyle\varepsilon\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}-\frac{1}{3}x_{i}^{3}-y_{i}+I_{i}^{\mathrm{syn}}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(72a) $\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}+a-b_{i}y_{i},$ (72b) with $\displaystyle
I_{i}^{\mathrm{syn}}=-\sum_{j=1,\;(j\neq
i}^{N}g_{ij}C_{ij}s_{j}(t)(x_{i}-x_{\mathrm{syn}}),\quad\dot{s}_{j}=\alpha(x_{j})(1-s_{j})-\beta
s_{j},$ (72c) and $\displaystyle\alpha(x_{j})$ $\displaystyle=$
$\displaystyle\alpha_{0}/\left(1+\mathrm{e}^{-x_{j}/x_{\mathrm{shp}}}\right),$
(72d) $\displaystyle g_{ij}^{\mathrm{ex}}$ $\displaystyle=$ $\displaystyle
g_{ij}^{\mathrm{ex}}+\Delta g_{ij}^{\mathrm{ex}},\quad\Delta
g_{ij}^{\mathrm{ex}}=g_{ij}^{\mathrm{ex}}F(\Delta t),$ (72e) $\displaystyle
F(\Delta t)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}A_{+}\exp(-\left|\Delta
t\right|/\tau_{+}),&\mathrm{if}\;\Delta t>0,\\\ -A_{-}\exp(-\left|\Delta
t\right|/\tau_{-}),&\mathrm{if}\;\Delta t>0,\\\ 0,&\mathrm{if}\;\Delta
t=0.\end{array}\right.$ (72i)

It was found that the inhibitory synapses may weaken vibrational resonance in
this model.

Vibrational resonance was also studied in a feedforward neuron network in [87]

$\displaystyle\varepsilon\dot{x}_{i,j}$ $\displaystyle=$ $\displaystyle
x_{i,j}-\frac{1}{3}x_{i,j}^{3}-y_{i,j}+I_{i,j}^{\mathrm{syn}}(t),$ (73a)
$\displaystyle\dot{y}_{i,j}$ $\displaystyle=$ $\displaystyle
x_{i,j}+a_{i,j}-y_{i,j}+S_{i,j}(t),$ (73b) $\displaystyle
I_{i,j}^{\mathrm{syn}}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{N_{\mathrm{syn}}}g_{\mathrm{syn}}\alpha(t-{t_{i-1,k}})(V_{i,j}-V_{\mathrm{syn}}),$
(73c)

where, $\alpha(t)=(t/\tau)\mathrm{e}^{-t/\tau}$. While in a feedforward neuron
network with unreliable synapses [82]

$\displaystyle\varepsilon\dot{x}_{i,j}$ $\displaystyle=$ $\displaystyle
x_{i,j}-\frac{1}{3}x_{i,j}^{3}-y_{i,j}+I_{i,j}^{\mathrm{syn}},$ (74a)
$\displaystyle\dot{y}_{i,j}$ $\displaystyle=$ $\displaystyle
x_{i,j}+a+I_{i,j},\quad
I_{i,j}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)+\varphi_{i,j},$ (74b)
$\displaystyle I_{i,j}^{\mathrm{syn}}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k=1}^{N}G(i,j;k,j-1)[E_{\mathrm{syn}}-x_{i,j}],$
(74c) $\displaystyle G(i,j;k,j\\!\\!-\\!\\!1)\\!\\!$ $\displaystyle\leftarrow$
$\displaystyle\\!\\!G(i,j;k,j\\!\\!-\\!\\!1)\\!\\!+\\!\\!J(i,j;k,j\\!\\!-\\!\\!1)h(i,j;k,j\\!\\!-\\!\\!1),$
(74d)

and in a heterogeneous scale free neuron network [83]

$\displaystyle\varepsilon\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}-\frac{1}{3}x_{i}^{3}-y_{i}+\sum_{j}g_{ij}(x_{j}-x_{i}),$ (75a)
$\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}+a_{i}+I_{\mathrm{ex}},\quad\langle
a_{i}\rangle=a_{0},\quad\langle(a_{i}-a_{0})(a_{j}-a_{0})\rangle=\delta_{ij}\sigma^{2}.$
(75b)

Different network connections are important for weak signal propagation based
on the vibrational resonance mechanism.

Vibrational resonance can also be caused by a special heterogenous aperiodic
fast-varying signal in a FitzHugh-Nagumo neuronal parallel array without
connections between the neurons [89]

$\displaystyle\varepsilon\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}-\frac{1}{3}x_{i}^{3}-y_{i}+s(t),$ (76a) $\displaystyle\dot{y}_{i}$
$\displaystyle=$ $\displaystyle x_{i}+a+S_{i}(t),$ (76b)

or with locally coupled neurons

$\displaystyle\varepsilon\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}-\frac{1}{3}x_{i}^{3}-y_{i}+g[x_{i+1}+x_{i-1}-2x_{i}]+s(t),$ (77a)
$\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle x_{i}+a+S_{i}(t),$
(77b)

where $g$ is the strength of the coupling. Herein, $s(t)$ represents the slow-
varying harmonic signal, and $S_{i}(t)$ denotes the heterogeneous aperiodic
fast-varying signal. $S_{i}(t)$ is generated by randomly modulating the
amplitude and frequency of a sinusoidal signal. The general expression of the
aperiodic stimuli is defined as $S_{i}(\tau_{i,j-1}+t)=A_{i,j}\sin(2\pi
t/T_{i,j})$ with amplitude $A_{i,j}$ and period $T_{i,j}$. Both $A_{i,j}$ and
$T_{i,j}$ obey a certain stochastic distribution.

Besides the standard FitzHugh-Nagumo neuron models, vibrational resonance was
also studied in a modified FitzHugh-Nagumo neuron model [247], where the model
equation is described by

$\displaystyle\varepsilon\dot{x}$ $\displaystyle=$ $\displaystyle
x-\frac{{{x^{3}}}}{3}-y-{k_{1}}\rho(\varphi)x+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(78a) $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle x+a-by,$ (78b)
$\displaystyle\dot{\varphi}$ $\displaystyle=$
$\displaystyle{k_{2}}x-0.5\varphi+{\varphi_{ext}},$ (78c)

$-{k_{1}}\rho(\varphi)x$ is the modified term, and $\varphi_{ext}$ is a high-
low-frequency electromagnetic radiation. They also considered vibrational
resonance in the coupled modified FitzHugh-Nagumo models, i.e.,

$\displaystyle\varepsilon{{\dot{x}}_{1}}$ $\displaystyle=$
$\displaystyle{x_{1}}-\frac{{x_{1}^{3}}}{3}-{y_{1}}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)-{k_{1}}\rho({\varphi_{1}}){x_{1}}+g({x_{1}}-{x_{2}}),$
(79a) $\displaystyle{{\dot{y}}_{1}}$ $\displaystyle=$
$\displaystyle{x_{1}}+a-b{y_{1}},$ (79b) $\displaystyle{{\dot{\varphi}}_{1}}$
$\displaystyle=$
$\displaystyle{k_{2}}{x_{1}}-0.5{\varphi_{1}}+{\varphi_{ext}},$ (79c)
$\displaystyle\varepsilon{{\dot{x}}_{2}}$ $\displaystyle=$
$\displaystyle{x_{2}}-\frac{{x_{2}^{3}}}{3}-{y_{2}}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)-{k_{1}}\rho({\varphi_{2}}){x_{2}}+g({x_{2}}-{x_{1}}),$
(79d) $\displaystyle{{\dot{y}}_{2}}$ $\displaystyle=$
$\displaystyle{x_{2}}+a-b{y_{2}},$ (79e) $\displaystyle{{\dot{\varphi}}_{2}}$
$\displaystyle=$
$\displaystyle{k_{2}}{x_{2}}-0.5{\varphi_{2}}+{\varphi_{ext}}.$ (79f)

Vibrational resonance is analyzed in a synthetic gene network [91]

$\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle\frac{1+x^{2}+\alpha\sigma x^{4}}{[1+x^{2}+\sigma
x^{4}][1+y^{4}]}-\gamma_{x}x,$ (80a) $\displaystyle\tau_{y}\dot{y}$
$\displaystyle=$ $\displaystyle\frac{1+x^{2}+\alpha\sigma
x^{4}}{[1+x^{2}+\sigma
x^{4}][1+y^{4}]}-\gamma_{y}y+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (80b)

It is interesting that the ratio of the amplitude to the frequency of the
fast-varying excitation is always a definite constant when the optimal
vibrational resonance occurs in this system.

The Hodgkin-Huxley neuron model excited by a biharmonic signal is studied in
the equation [248]

$\displaystyle C_{m}\dot{V}$ $\displaystyle=$
$\displaystyle-\left[g_{k}n^{4}(V-V_{K})+g_{\mathrm{Na}}m^{3}h(V-V_{\mathrm{Na}})+g_{1}(V-V_{\mathrm{l}})\right]$
(81a)
$\displaystyle\quad+I_{\mathrm{aut}}+I_{0}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
$\displaystyle\dot{m}$ $\displaystyle=$
$\displaystyle\alpha_{m}(1-m)-\beta_{m}m,$ (81b) $\displaystyle\dot{n}$
$\displaystyle=$ $\displaystyle\alpha_{n}(1-n)-\beta_{n}n,$ (81c)
$\displaystyle\dot{h}$ $\displaystyle=$
$\displaystyle\alpha_{h}(1-h)-\beta_{h}h.$ (81d)

In Eqs. (81), $C_{m}$ is the cell capacitance, and $V$ is the membrane
potential. The parameters $g_{k}$, $g_{\mathrm{Na}}$ and $g_{l}$ represent the
maximum conductances of the potassium, the sodium and leak currents,
respectively. In addition, $V_{K}$, $V_{\mathrm{Na}}$ and $V_{\mathrm{l}}$
stand for the potassium, the sodium, and the leakage reversal potentials in
turn. $I_{0}$ represents the stimulus current. Based on the vibrational
resonance mechanism, the inhibitory autapse significantly enhances the low-
frequency subthreshold signal, despite its typically suppressive role in
neuronal dynamics.

A neuron model related to the dynamics of Na+ and K+ is described in detail in
[249]

$C\dot{V}=-\left(I_{\mathrm{Na}}+I_{\mathrm{K}}+I_{\mathrm{sLeak}}+I_{\mathrm{pump}}+I_{\mathrm{sd}}\right)+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(82)

where $C$ is the soma capacitance and $V$ is the membrane potential.
$I_{\mathrm{Na}}$ and $I_{\mathrm{K}}$ represent the sum of Na+ currents and
the sum of K+ currents, respectively. $I_{\mathrm{sLeak}}$,
$I_{\mathrm{pump}}$ and $I_{\mathrm{sd}}$ represent the leakage current, the
current induced by the Na+ and K+ pump, and the axial current in turn. The
continuous changes of the K+ and Na+ concentrations will cause and enhance
vibrational multi-resonances.

Multiple instances of vibrational resonance were explored in various neuron
systems, including a single Hindmarsh-Rose neuron, a modified Hindmarsh-Rose
neuron, and coupled Hindmarsh-Rose neuron systems with or without an electric
field, as reported in [78], [250]. The different models are listed below.

The single Hindmarsh-Rose model is

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
y-ax^{3}+bx^{2}-z+I_{0}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (83a)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle c-dx^{2}-y,$ (83b)
$\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle r[s(x+1.56)-z],$ (83c)

where $x$ is the membrane potential, $y$ is the recovery variable, $z$ is the
adaptation current, and $I_{0}$ represents a direct current stimulation.
Multiple vibrational resonances have been observed in this single Hindmarsh-
Rose neuron model.

For coupled Hindmarsh-Rose neurons without electric field, the dynamical model
is

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
y_{1}-ax_{1}^{3}+bx_{1}^{2}-z_{1}+g_{1}(x_{1}-x_{2})+s_{\mathrm{L}}(t),$ (84a)
$\displaystyle\dot{y}_{1}$ $\displaystyle=$ $\displaystyle
c-dx_{1}^{2}-y_{1},$ (84b) $\displaystyle\dot{z}_{1}$ $\displaystyle=$
$\displaystyle r[s(x_{1}+1.56)-z_{1}],$ (84c) $\displaystyle\dot{x}_{2}$
$\displaystyle=$ $\displaystyle
y_{2}-ax_{2}^{3}+bx_{2}^{2}-z_{2}+g_{2}(x_{2}-x_{1})+s_{\mathrm{H}}(t),$ (84d)
$\displaystyle\dot{y}_{2}$ $\displaystyle=$ $\displaystyle
c-dx_{2}^{2}-y_{2},$ (84e) $\displaystyle\dot{z}_{2}$ $\displaystyle=$
$\displaystyle r[s(x_{2}+1.56)-z_{2}].$ (84f)

Herein, the coupling between the two neurons are $g_{1}(x_{1}-x_{2})$ and
$g_{2}(x_{2}-x_{1})$ with coupling strength $g_{1}$ and $g_{2}$. The improved
Hindmarsh-Rose neuron model with external electric field is

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
y-ax^{3}+bx^{2}-z+I_{0}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (85a)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle c-dx^{2}-y+RE,$ (85b)
$\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle r[s(x+1.56)-z],$ (85c)
$\displaystyle\dot{E}$ $\displaystyle=$ $\displaystyle my+E_{\mathrm{ext}},$
(85d)

where $R$ is the radius size of the cell, $E$ is the intensity of the uniform
electric field, $m$ is a parameter corresponding to the polarization property,
and $E_{\mathrm{ext}}$ is the external electric field.

For coupled Hindmarsh-Rose neurons with electric field, the model is

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
y_{1}-ax_{1}^{3}+bx_{1}^{2}-z_{1}+s_{\mathrm{L}}(t)+g_{1}(x_{1}-x_{2}),$ (86a)
$\displaystyle\dot{y}_{1}$ $\displaystyle=$ $\displaystyle
c-dx_{1}^{2}-y_{1}+RE_{1},$ (86b) $\displaystyle\dot{z}_{1}$ $\displaystyle=$
$\displaystyle r[s(x_{1}+1.56)-z_{1}],$ (86c) $\displaystyle\dot{E}_{1}$
$\displaystyle=$ $\displaystyle my_{1}+E_{\mathrm{ext1}},$ (86d)
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle
y_{2}-ax_{2}^{3}+bx_{2}^{2}-z_{2}+s_{\mathrm{H}}(t)+g_{2}(x_{2}-x_{1}),$ (86e)
$\displaystyle\dot{y}_{2}$ $\displaystyle=$ $\displaystyle
c-dx_{2}^{2}-y_{2}+RE_{2},$ (86f) $\displaystyle\dot{z}_{2}$ $\displaystyle=$
$\displaystyle r[s(x_{2}+1.56)-z_{2}],$ (86g) $\displaystyle\dot{E}_{2}$
$\displaystyle=$ $\displaystyle my_{2}+E_{\mathrm{ext2}}.$ (86h)

Multiple vibrational resonances are prominent regardless of the presence of an
electric field. Moreover, the electric field appears to attenuate the impact
of multiple vibrational resonances in the single Hindmarsh-Rose neuron.
Conversely, it enhances the effect of multiple vibrational resonances in the
case of bidirectional coupling between two Hindmarsh-Rose neurons.

####  3.1.12 Circuit systems

Chua’s circuit is a renowned nonlinear model known for its complex dynamical
behaviors. Vibrational resonance has been observed through simulation or
experimentation in numerous studies involving Chua’s circuit. Below, we
outline the Chua’s circuit models that have been extensively discussed in the
context of vibrational resonance.

The model of a single Chua’s circuit is [251]

$\displaystyle C_{1}\dot{v}_{1}$ $\displaystyle=$
$\displaystyle(1/R)(v_{2}-v_{1})-f(v_{1}),$ (87a) $\displaystyle
C_{2}\dot{v}_{2}$ $\displaystyle=$ $\displaystyle(1/R)(v_{1}-v_{2}+i_{L}),$
(87b) $\displaystyle L\dot{i}_{L}$ $\displaystyle=$ $\displaystyle-
v_{2}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (87c) where $\displaystyle
f(v_{1})=G_{b}v_{1}+0.5(G_{a}-G_{b})\left[\left|v_{1}+BP_{1}\right|-\left|v_{1}-BP_{1}\right|\right],$
(87d)

and its $n$-coupled form

$\displaystyle C_{1}\dot{v}_{1}^{(i)}$ $\displaystyle=$
$\displaystyle(1/R)(v_{2}^{(i)}-v_{1}^{(i)})-f(v_{1}^{(i)}),$ (88a)
$\displaystyle C_{2}\dot{v}_{2}^{(i)}$ $\displaystyle=$
$\displaystyle(1/R)(v_{1}^{(i)}-v_{2}^{(i)}+i_{L}^{(i)}),$ (88b)
$\displaystyle L\dot{i}_{L}^{(i)}$ $\displaystyle=$ $\displaystyle-
v_{2}+\delta_{i}[s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)]+\varepsilon_{i}\left(v_{1}^{(i-1)}-i_{L}^{(i)}R_{C}\right).$
(88c)

There are two stable equilibria for the single Chua’s circuit. Vibrational
resonance appears when the response switches between the two stable
equilibrium state. In the coupled Chua’s circuits, the low-frequency signal
propagation is discussed for a wide range of coupling parameter by carrying
out a vibrational resonance analysis. The response amplitude at the low
frequency of the $i$-th circuit increases with its size $i$, eventually
reaching saturation.

The modified Chua’s circuit model is [103], [252]

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle\alpha y-\alpha
F(x)+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (89a) $\displaystyle\dot{y}$
$\displaystyle=$ $\displaystyle x-y+z,$ (89b) $\displaystyle\dot{z}$
$\displaystyle=$ $\displaystyle-\beta y,$ (89c) where $\displaystyle
F(x)=\epsilon x+\epsilon A\mathrm{sgn}(x)-\epsilon
A\sum_{j=0}^{n-1}\left[\mathrm{sgn}(x+2jA)+\mathrm{sgn}(x-2jA)\right]$ (89d)
or $\displaystyle F(x)=\epsilon x-\epsilon
A\sum_{j=0}^{n-1}\left[\mathrm{sgn}(x+(2j+1)A)+\mathrm{sgn}(x-(2j+1)A)\right]$
(89e) with
$\displaystyle\mathrm{sgn}(x)=\left\\{\begin{array}[]{rl}1,&\mathrm{if}\;x>0\\\
0,&\mathrm{if}\;x=0\\\ -1,&\mathrm{if}\;x<0\end{array}\right.$ (89i)

and $\alpha$, $\beta$, $\epsilon$, $A>0$ and $n\geq 1$. Both expressions of
$F(x)$ represent sawtooth functions, yet they have distinct equilibrium points
and breakpoints. Typically, the number of resonance peaks corresponds to the
number of breakpoints.

The piecewise linear nonautonomous Murali-Lakshmanan-Chua circuit is [200]

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle y-h(x),$ (90a)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle-\beta(1+v)y-\beta
x+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (90b) where $\displaystyle
h(x)=\left\\{\begin{array}[]{ll}bx+(a-b),&\mathrm{if}\;x>1\\\
ax,&\mathrm{if}\;\left|x\right|<1\\\
bx-(a-b),&\mathrm{if}\;x<-1\end{array}\right.$ (90f)

Both vibrational resonance and logical vibrational resonance occurs in this
kind of Chua’s circuit.

In addition to the Chua’s circuit mentioned above, there are various models
within the Chua’s circuit family [253]. The dynamics induced by fast-varying
and slow-varying excitations in these variants of Chua’s circuits may reveal
novel vibrational resonance phenomena with practical implications.

####  3.1.13 Energy harvesting system

Vibrational resonance occurs in an energy harvesting system which is governed
by [148], [149]

$\displaystyle\ddot{x}+2\delta\dot{x}-kx\left(1-x^{2}\right)-\chi y$
$\displaystyle=$ $\displaystyle s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (91a)
$\displaystyle\dot{y}+\lambda y+\kappa\dot{x}$ $\displaystyle=$ $\displaystyle
0,$ (91b)

where $x$ is the transverse displacement of the beam, while $\delta$ and $k$
describe the damping ratio the stiffness, respectively. Regarding the other
parameters, $y$, $\chi$, $\kappa$ denote the voltage across the load resistor,
the piezoelectric coupling strength of the mechanical equation, and the
piezoelectric coupling strength of the electrical equation, respectively.
Additionally, $\lambda\propto 1/RC$ represents the reciprocal of the
electrical circuit, where $R$ denotes the load resistance and $C$ represents
the capacitance.

Another study on vibrational resonance involves a tri-stable energy harvester
connected to a standard rectifier circuit, as discussed in [150]

$\displaystyle\ddot{x}+\delta\dot{x}+\frac{{dV}}{{dx}}+\chi y$
$\displaystyle=$ $\displaystyle{s_{L}}(t)+{s_{H}}(t),$ (92a)
$\displaystyle\dot{y}+I$ $\displaystyle=$ $\displaystyle\dot{x},$ (92b)

where $x$, $\delta$, $\chi$ and $y$ have the same meaning as in Eq. (89),
while $I$ represents the current supplied to the standard rectifier circuit.
The expression for $V(x)$ remains identical to the one presented in Eq. (33).”

Besides the external form, both slow-varying and fast-varying excitations may
also manifest in parametric form within an energy harvesting system, as
illustrated in the model proposed by [151]

$\displaystyle\ddot{x}+2\delta\dot{x}-kx\left(1-x^{2}\right)-\chi
y+[s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)]x$ $\displaystyle=$ $\displaystyle 0,$
(93a) $\displaystyle\dot{y}+\lambda y+\kappa\dot{x}$ $\displaystyle=$
$\displaystyle 0,$ (93b)

Here, the parametric excitation is the axial load onto the piezoelectric
buckled beam harvester.

Investigation of vibrational resonance in an energy harvesting system holds
promise for enhancing the efficiency of converting vibrational energy into
electrical energy, offering significant engineering applications.

####  3.1.14 Laser system

The optimal Bloch vibrational resonance model is [24], [26], [254]

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle-\omega_{0}y,$ (94a)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle\omega_{0}x+\chi zf(t),$
(94b) $\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle-\chi yf(t),$
(94c) $\displaystyle f(t)$ $\displaystyle=$ $\displaystyle
f_{0}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (94d)

where $x^{2}+y^{2}+z^{2}=1$, $\omega_{0}$ is the transition frequency of the
atom, $\chi$ and $f_{0}$ are constants. Vibrational resonance in vertical-
cavity surface-emitting lasers are also studied mainly experimentally [162],
[255]-[258]. In laser systems, vibrational resonance has the capability to
amplify not just the weak low-frequency signal at the fundamental frequency,
but also to enhance the response at subharmonic or superharmonic frequencies
by appropriately selecting a high-frequency signal.

###  3.2 Models with maps

Depending on the complexity of the map, the response exhibits not only the
conventional vibrational resonance phenomenon but also a broader range of
additional dynamical phenomena.

Vibrational resonance was analyzed in the one-dimensional Bellows map [259]

$x_{n+1}=\frac{rx_{n}}{1+x_{n}^{b}}+s_{\mathrm{L}}(n)+s_{\mathrm{H}}(n),$ (95)

which describes the dynamical evolution of the population density of an
organism, and it was analyzed also in the two-dimensional Rulkov map

$\displaystyle x_{n+1}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{1+x_{n}^{2}}+y_{n}+s_{\mathrm{L}}(n)+s_{\mathrm{H}}(n),$
(96a) $\displaystyle y_{n+1}$ $\displaystyle=$ $\displaystyle y_{n}-\beta
x_{n}-\sigma,$ (96b)

that mimics the behavior of complex continuous time neuronal models.

The sine square map, which characterizes the hybrid optical bistable
interferometer, and the sine circle map under biharmonic signals are
respectively defined as follows [260]

$x_{n+1}=A\sin^{2}\left(x_{n}-b\right)+s_{\mathrm{L}}(n)+s_{\mathrm{H}}(n)$
(97)

and

$y_{n+1}=y_{n}+\mu\sin(y_{n})+s_{\mathrm{L}}(n)+s_{\mathrm{H}}(n).$ (98)

These two kinds of maps exhibit vibrational resonance induced by high-
frequency excitation.

Vibrational resonance was analyzed in detail in coupled neuronal maps in [79]

$\displaystyle x_{n+1}$ $\displaystyle=$ $\displaystyle
a_{i}/\left(1+x_{n}^{2}\right)+y_{n}+s_{\mathrm{L}}(n)+s_{\mathrm{H}}(n),$
(99a) $\displaystyle y_{n+1}$ $\displaystyle=$ $\displaystyle y_{n}-\beta
x_{n}-\gamma,$ (99b)

and in small-word networks

$\displaystyle x_{i,n+1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!a_{i}/\left(1\\!\\!+\\!\\!x_{i,n}^{2}\right)\\!\\!+\\!\\!y_{i,n}\\!\\!+\\!\\!\varepsilon\sum_{j}C(i,j)\left(x_{j,n}\\!\\!-\\!\\!x_{i,n}\right)\\!\\!+\\!\\!s_{\mathrm{L}}(n)\\!\\!+\\!\\!s_{\mathrm{H}}(n),$
(100a) $\displaystyle y_{i,n+1}$ $\displaystyle=$ $\displaystyle
y_{i,n}-\sigma_{i}x_{i,n}-\beta_{i}x_{n}$ (100b)

and in modular networks

$\displaystyle x_{I,i,n+1}$ $\displaystyle=$
$\displaystyle\alpha/\left(1+x_{I,i,n}^{2}\right)+y_{I,i,n}+I_{I,i,n}^{\mathrm{syn}}+s_{\mathrm{L}}(n)+s_{\mathrm{H}}(n),$
(101a) $\displaystyle y_{I,i,n+1}$ $\displaystyle=$ $\displaystyle
y_{I,i,n}-\beta x_{I,i,n}-\gamma,$ (101b) where $\displaystyle
I_{I,i,n}^{\mathrm{syn}}=\varepsilon_{\mathrm{intra}}\sum_{j}A_{I}(i,j)\left(x_{I,j,n}\\!\\!-\\!\\!x_{I,i,n}\right)\\!\\!+\\!\\!\varepsilon_{\mathrm{inter}}\sum_{j}B_{I,J}(i,j)\left(x_{J,j,n}\\!\\!-\\!\\!x_{I,i,n}\right).$
(101c)

The delayed discrete Rulkov neuron model is [40]

$\displaystyle x_{n+1}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{1+x_{n}^{2}}+y_{n}+s_{\mathrm{L}}(n)+s_{\mathrm{H}}(n),$
(102a) $\displaystyle y_{n+1}$ $\displaystyle=$ $\displaystyle y_{n}-\beta
x_{n-\tau}-\gamma.$ (102b)

Vibrational resonance in the above Rulkov neuron model has also been explored.

###  3.3 Models with fractional differential systems

Three commonly used definitions for the fractional-order derivative are the
Riemann-Liouville, Caputo, and Grünwald-Letnikov definitions [261] (see
Appendix for details). Vibrational resonance has been investigated in certain
systems involving fractional-order derivatives. The presence of fractional-
order damping may lead to new vibrational resonance patterns. We outline these
systems below.

There are different fractional Duffing oscillators, such as it in the
overdamped version [48], [112]

$\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}^{2}x+\beta
x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (103)

or in the underdamped version

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}^{2}x+\beta
x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (104)

or with both external and intrinsic fractional-order damping terms [49]

$\frac{\mathrm{d}^{\beta}x}{\mathrm{d}t^{\beta}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}^{2}x+bx^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(105)

The fractional Duffing oscillator with asymmetric bistable potential is [64]

$\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}-\omega_{0}^{2}x+ax^{2}+bx^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(106)

Vibrational resonance is analysed also in a quadratic oscillator [65]

$\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}-\mu
x+x^{2}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (107)

The fractional Mathieu-Duffing oscillator is [61]

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\left(\omega_{0}^{2}+s_{\mathrm{H}}(t)\right)x+\beta
x^{3}=s_{\mathrm{L}}(t).$ (108)

There are two kinds of quintic oscillators with fractional-order damping, as
shown in [53], [62]

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}^{2}x+\beta
x^{3}+\gamma x^{5}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (109)

or in the other form [54]

$\frac{\mathrm{d}^{\beta}x}{\mathrm{d}t^{\beta}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}^{2}x+\beta
x^{3}+\gamma x^{5}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (110)

Under the excitations of a fast-varying parametric signal and a slow-varying
external signal, the model of an oscillator with two fractional-order damping
terms is given by [50]

$\frac{\mathrm{d}^{q}x}{\mathrm{d}t^{q}}+\delta\frac{\mathrm{d}^{p}x}{\mathrm{d}t^{p}}+\left(\omega_{0}^{2}+s_{\mathrm{H}}(t)\right)x+\beta
x\left|x\right|^{r-1}=s_{\mathrm{L}}(t).$ (111)

The fractional-order system with the periodic potential is [51]

$\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\cos
ax+\frac{1}{2}\Delta\cos 2ax=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (112)

Coupled fractional anharmonic oscillators are given as [52]

$\displaystyle\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}$
$\displaystyle=$ $\displaystyle a_{1}x-b_{1}x^{3}+\gamma
xy^{2}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (113a)
$\displaystyle\frac{\mathrm{d}^{\alpha}y}{\mathrm{d}t^{\alpha}}$
$\displaystyle=$ $\displaystyle a_{2}y-b_{2}y^{3}+\gamma x^{2}y.$ (113b)

###  3.4 Models with delayed differential systems

There are some different delayed models corresponding to vibrational
resonance. A very simple example system is the overdamped bistable oscillator

$\dot{x}+\omega_{0}^{2}x(t-\tau)+\beta
x^{3}+r=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (114)

The cases $r=0$ and $r\neq 0$ are considered in [29], [33], [45] and [36],
respectively.

Vibrational resonance has been explored in the overdamped version of the
delayed Duffing oscillator

$\dot{x}+\omega_{0}^{2}x+\beta x^{3}+\gamma
x(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (115)

and in underdamped version

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta x^{3}+\gamma
x(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (116)

respectively [34]. In these two systems, the potential depends on the system
parameters and may present in single-well, double-well or double-hump
configuration.

The delayed harmonically trapped potential system is [43]

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta\sin
x+rx(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (117)

The delayed coupled anharmonic oscillators are described by [15], [30], [31]

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle
a_{1}x(t-\tau)-b_{1}x^{3}+\delta xy^{2}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(118a) $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle
a_{2}y-b_{2}y^{3}+\delta x^{2}y$ (118b)

and the globally coupled oscillators are given as [32]

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
x_{1}-x_{1}^{3}+\frac{\varepsilon}{N}\sum_{j=1}^{N}[x_{j}(t-\tau)-x_{1}]+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(119a) $\displaystyle\dot{x}_{i}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!x_{i}\\!\\!-\\!\\!x_{i}^{3}\\!\\!+\\!\\!\frac{\varepsilon}{N}\sum_{j=1}^{N}[x_{j}(t\\!\\!-\\!\\!\tau)\\!\\!-\\!\\!x_{i}]\\!\\!+\\!\\!s_{\mathrm{H}}(t),i=2,3,\cdots,N.$
(119b)

Vibrational resonance was also investigated in the same coupled oscillators
described by Eq. (118), albeit with excitations replaced by an amplitude-
modulated signal, as studied in [31].

Considering multiple time delays [35], vibrational resonance was reported in a
single Duffing oscillator

$\ddot{x}+\delta\dot{x}+\omega_{0}^{2}x+\beta
x^{3}+\frac{\gamma}{M}\sum_{m=1}^{M}x(t-m\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)$
(120)

and in coupled Duffing oscillators

$\displaystyle\ddot{x}_{1}+\delta\dot{x}_{1}+\omega_{0}^{2}x_{1}+\beta
x_{1}^{3}$ $\displaystyle=$ $\displaystyle
s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (121a)
$\displaystyle\ddot{x}_{i}\\!\\!+\\!\\!\delta\dot{x}_{i}\\!\\!+\\!\\!\omega_{0}^{2}x_{i}\\!\\!+\\!\\!\beta
x_{i}^{3}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\gamma}{M}\sum_{l=1}^{M}x_{i\\!\\!-\\!\\!1}(t\\!\\!-\\!\\!m\tau),i=2,3,\cdots,N.$
(121b)

Vibrational resonance appears in the delayed single FitzHugh-Nagumo model
[37], [38]

$\displaystyle\varepsilon\dot{x}$ $\displaystyle=$ $\displaystyle
x-\frac{1}{3}x^{3}-y,$ (122a) $\displaystyle\dot{y}$ $\displaystyle=$
$\displaystyle
x+a+K\left[y(t-\tau)-y\right]+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (122b)

and in the delayed coupled FitzHugh-Nagumo model [39]

$\displaystyle\varepsilon\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
x_{1}-\frac{1}{3}x_{1}^{3}-y_{1},$ (123a) $\displaystyle\dot{y}_{1}$
$\displaystyle=$ $\displaystyle
x_{1}+a+\frac{K}{N}\sum_{j=1}^{N}\left[y_{j}(t-\tau)-y_{1}\right]+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(123b) $\displaystyle\varepsilon\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}-\frac{1}{3}x_{i}^{3}-y_{i},$ (123c) $\displaystyle\dot{y}_{i}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!x_{i}+a+\frac{K}{N}\sum_{j=1}^{N}\left[y_{j}(t-\tau)-y_{i}\right]+s_{\mathrm{H}}(t),i=2,3,\cdots,N.$
(123d)

In [38], $\tau$ is time varying and
$\tau=\tau_{0}+\varepsilon_{0}\sin\omega_{0}t$. In [37], [39], $\tau$ is a
constant.

The delayed genetic toggle switch is [41]

$\displaystyle\dot{u}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{1+v^{\beta}}-u(t-\tau)+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(124a) $\displaystyle\dot{v}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{1+u^{\beta}}-v(t-\tau).$ (124b)

Herein, $u$ and $v$ indicate the concentrations of two considered
transcription factors. The excitations come from some factors such as the
oscillating temperature, the experimental setting variation of the chemical
inductor, etc.

The gene transcriptional regulatory system with a linear delayed term is
modeled by [42]

$\dot{x}=\frac{k_{\mathrm{f}}x^{2}}{x^{2}+K_{\mathrm{d}}}-k_{\mathrm{d}}x(t-\tau)+R_{\mathrm{bas}}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(125)

When the delayed terms are in nonlinear form, the dynamical equation is

$\dot{x}=\frac{k_{\mathrm{f}}x^{2}(t-\tau)}{x^{2}(t-\tau)+K_{\mathrm{d}}}-k_{\mathrm{d}}x+R_{\mathrm{bas}}+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(126)

In the absence of time delay and external perturbations, the system’s
potential exhibits an asymmetric shape similar to the curve shown in Fig.
2(d).

Some systems also consider both fractional-order damping and time delay
simultaneously. The fractional Duffing oscillators incorporating a time delay
term are described in [59]

$\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}^{2}x+\beta
x^{3}+\gamma x(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (127)

and

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}-\omega_{0}^{2}x+\beta
x^{3}+\gamma x(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (128)

and [262]

$\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}^{2}x+ax^{2}+\beta
x^{3}+\gamma x(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (129)

The Duffing system with a generalized time delay term is [263]

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}x+\beta
x^{3}+\gamma\mathrm{D}^{\alpha}x(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$
(130)

where $\mathrm{D}^{\alpha}x(t-\tau)$ is the delayed feedback in a generalized
fractional-order differential form.

The fractional-order quintic oscillator with a linear time delay is
investigated in the equation [264]

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+\omega_{0}x+\beta
x^{3}+\gamma x^{5}+\xi x(t-\tau)=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$ (131)

The fractional Mathieu-Duffing oscillator modeled with distributed time delay
is [265]

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+[a+s_{\mathrm{H}}(t)]x+bx^{3}+\gamma\int_{-\infty}^{t}h(t-\tau)x(\tau)\mathrm{d}\tau=s_{\mathrm{L}}(t),$
(132)

while with fixed time delay is

$\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+\delta\frac{\mathrm{d}^{\alpha}x}{\mathrm{d}t^{\alpha}}+[a+s_{\mathrm{H}}(t)]x+bx^{3}+\gamma
x(t-\tau)=s_{\mathrm{L}}(t).$ (133)

The fractional Duffing oscillator with distributed time delay and excited by
multi-frequency excitations is described as [266]

$\displaystyle
m\ddot{x}(t)+c\dot{x}(t)+\delta\frac{{{d^{\alpha}}x}}{{d{t^{\alpha}}}}+kx(t)+s_{\mathrm{L1}}(t)x(t)+\beta{x^{3}}(t)$
$\displaystyle+r\int_{-\infty}^{t}{\eta(t-\tau)}x(\tau)d\tau=s_{\mathrm{L2}}(t)+s_{\mathrm{H}}(t).$
(134)

A generalized fractional Duffing-van der Pol oscillator with distributed time
delay and excited by slow-varying external signal and fast-varying parametric
signal was investigated in the following system [63]

$\displaystyle
m\ddot{x}(t)+[{c_{1}}+{c_{2}}{x^{2}}(t)]\dot{x}(t)+{k_{1}}x(t)+{\delta_{1}}D_{0,t}^{q}x(t)+{\alpha_{1}}{x^{3}}(t)$
$\displaystyle+{\lambda_{1}}\int_{-\infty}^{t}{r(t-\tau)}x(\tau)d\tau=s_{\mathrm{L}}(t)+x(t)s_{\mathrm{H}}(t),$
(135)

where $D_{0,t}^{q}x(t)$ is the fractional-order derivative with respect to the
variable $x(t)$ in the Caputo definition.

###  3.5 Models with stochastic differential systems

Due to the significant influence of noise in various scenarios, we consider in
this subsection the equation of vibrational resonance in the presence of
noise.

####  3.5.1 Monostable systems

Vibrational resonance in a monostable system subjected to multiplicative noise
in [134]

$\dot{x}+\omega_{0}^{2}x\xi(t)+\beta
x^{3}+r=\eta(t)+s_{\mathrm{L1}}(t)+s_{\mathrm{L2}}(t)+s_{\mathrm{H}}(t).$
(136)

There are a fast-varying excitation, two slow-varying excitations, and two
random excitations in the above equation, where $\xi(t)$ and $\eta(t)$ are
uncorrelated Gaussian white noises and $r$ is the bias parameter.

####  3.5.2 Bistable systems

The bistable system excited by both a slow- and fast-varying harmonic signals
and a Gaussian white noise is given by [125], [130]

$\dot{x}+\omega_{0}^{2}x+\beta
x^{3}=s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)+\xi(t).$ (137)

The coupled piecewise linear system is a symmetric bistable system [124]

$\displaystyle\dot{x}_{i}=f(x_{i})+g(x_{i})\xi_{i}(t)+\frac{D}{2d}\sum_{j\in
nn(i)}(x_{j}-x_{i})+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)$ (138a) with
$\displaystyle\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\sigma_{m}^{2}\delta_{ij}\delta(t-t^{\prime}),$ (138b)
$\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-G_{b}x-(G_{a}-G_{b})B_{\mathrm{p}}&\mathrm{if}\;x\leq-{B_{p}},\\\
-G_{a}x&\mathrm{if}\;\left|x\right|<B_{\mathrm{p}},\\\
-G_{b}x+(G_{a}-G_{b})B_{\mathrm{p}}&\mathrm{if}\;x\geq
B_{\mathrm{p}}.\end{array}\right.$ (138f)

The piecewise linear system can be viewed as an approximation of a typical
bistable system. A similar topic was also investigated in the conventional
globally coupled bistable system

${\dot{x}_{i}}={x_{i}}-x_{i}^{3}+\frac{\varepsilon}{N}\sum\limits_{j=1}^{N}{({x_{j}}-{x_{i}})}+\sqrt{2D}{\xi_{i}}(t)+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t).$
(139)

The effects of the noise and the bi-harmonic signal on the logical response of
the system are investigated in the model [143]

$\dot{x}+\omega_{0}^{2}x(t)+\beta
x^{3}(t)=I_{1}+I_{2}+r+\xi(t)+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t),$ (140)

where $I_{1}$ and $I_{2}$ are two logical inputs, and $\xi(t)$ is a Gaussian
white noise.

Vibrational ratchet which is very similar to vibrational resonance was
investigated in the system [144]-[146]

$\dot{x}=-V^{\prime}(x)+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)+\xi(t),\quad
V(x)=d(1-\cos x).$ (141)

The model describes the dynamics of a Brownian particle diffusing on a one-
dimensional periodic substrate. The case with

$V(x)=\sum_{n=1}^{\infty}a_{n}\cos nx+\sum_{n=1}^{\infty}b_{n}\sin nx$ (142)

has also been explored. The transport of the particle presents a vibrational
ratchet effect with the cooperation of the system and the excitations.
Certainly, vibrational resonance can also occur in this kind of systems.

####  3.5.3 The Schmitt trigger circuit

The Schmitt trigger circuit is [267]

$U_{\mathrm{out}}=\mathrm{sgn}\left[\Delta
U-s_{\mathrm{L}}(t)-s_{\mathrm{H}}(t)-\xi(t)\right],$ (143)

where $\Delta U=\frac{{{R_{1}}}}{{{R_{1}}+{R_{2}}}}{U_{out}}$ is the threshold
value. $U_{out}$ is either the positive or negative voltage $\pm{U_{0}}$ and
adjusted by external signals. Here, $\xi(t)$ is a colored noise.

####  3.5.4 Neural models and complex networks

The role of noise on vibrational resonance was discussed in the FitzHugh-
Nagumo model by adding the noise term $\xi(t)$ to the right-side of Eq. (70b)
[66] and also in the FitzHugh-Nagumo model with different coupling [75]

$\displaystyle\varepsilon\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}-\frac{1}{3}x_{i}^{3}-y_{i}-I_{i}^{\mathrm{syn}},$ (144a)
$\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle
x_{i}+a+s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)+\xi(t).$ (144b)

Here, $I_{i}^{\mathrm{syn}}$ represents different kinds of coupling.

The feed-forward multilayer neural network corresponding to the Hodgkin-Huxley
neuron model is governed by [84]

$\displaystyle C_{m}\dot{V}_{i,j}$ $\displaystyle=$
$\displaystyle-\left[g_{k}n_{i,j}^{4}\left(V_{i,j}-V_{k}\right)+g_{\mathrm{Na}}m_{i,j}^{3}h_{i,j}\left(V_{i,j}-V_{\mathrm{Na}}\right)+g_{l}\left(V_{i,j}-V_{l}\right)\right]$
(145a)
$\displaystyle\quad+I_{0}-I^{\mathrm{syn}}_{i,j}+a_{i,j}\left[s_{\mathrm{L}}(t)+s_{\mathrm{H}}(t)\right],$
$\displaystyle\dot{x}_{i,j}$ $\displaystyle=$
$\displaystyle\alpha_{x}(1-x_{i,j})-\beta_{x}x_{i,j}+\xi_{i,j}x_{i,j},\quad
x_{i,j}=m_{i,j},n_{i,j},h_{i,j},$ (145b) where $\displaystyle\alpha_{m_{i,j}}$
$\displaystyle=$ $\displaystyle
0.1\left(V_{i,j}+40\right)/\left(1-\mathrm{e}^{-(V_{i,j}+40)/10}\right),$
(145c) $\displaystyle\beta_{m_{i,j}}$ $\displaystyle=$ $\displaystyle
4\mathrm{e}^{-(V_{i,j}+65)/18},$ (145d) $\displaystyle\alpha_{n_{i,j}}$
$\displaystyle=$ $\displaystyle
0.01\left(V_{i,j}+55\right)/\left(1-\mathrm{e}^{-(V_{i,j}+55)/10}\right),$
(145e) $\displaystyle\beta_{n_{i,j}}$ $\displaystyle=$ $\displaystyle
0.125\mathrm{e}^{-(V_{i,j}+65)/80},$ (145f) $\displaystyle\alpha_{h_{i,j}}$
$\displaystyle=$ $\displaystyle 0.07\mathrm{e}^{-(V_{i,j}+65)/20},$ (145g)
$\displaystyle\beta_{h_{i,j}}$ $\displaystyle=$ $\displaystyle
1/\left(1+\mathrm{e}^{-(V_{i,j}+35)/10}\right),$ (145h)

and $\xi_{i,j}(t)$ is a Gaussian white noise.

##  4 Performance metrics of vibrational resonance

In this section, we primarily provide performance metrics for different types
of vibrational resonance. These frequently employed metrics encompass response
amplitude and gain factor for harmonic component vibrational resonance, cross-
correlation coefficient and bit error rate for binary aperiodic signal-induced
vibrational resonance, cross-correlation coefficient and spectrum
amplification factor for frequency-modulated signal-induced vibrational
resonance, as well as success probability for logical vibrational resonance,
among others

###  4.1 Response amplitude

The response amplitude is used in measuring the response at a harmonic
frequency. We label the response amplitude as $Q$. We extract the harmonic
$\omega$ from the time series $x(t)$ by the Fourier series. The sine component
$Q_{\mathrm{s}}$ and cosine component $Q_{\mathrm{c}}$ of $x(t)$ at the
frequency $\omega$ are respectively calculated by

$Q_{\mathrm{s}}=\frac{2}{mT}\int\limits_{0}^{mT}x(t)\sin(\omega
t)\text{d}t,\quad
Q_{\mathrm{c}}=\frac{2}{mT}\int\limits_{0}^{mT}x(t)\cos(\omega t)\text{d}t,$
(146)

where $m$ is a large enough positive number. Without a special explanation,
$x(t)$ is a steady time series where the initial transient response has been
removed. For a discrete series, the Fourier coefficients are

$Q_{\mathrm{s}}=\frac{2\Delta t}{mT}\sum\limits_{i=1}^{mT/\Delta
t}x(t_{i})\sin(\omega t_{i}),\quad Q_{\mathrm{c}}=\frac{2\Delta
t}{mT}\sum\limits_{i=1}^{mT/\Delta t}x(t_{i})\cos(\omega t_{i}).$ (147)

In Eq. (147), $\Delta t$ is the time step for the numerical calculations, and
$T$ is the period of the low-frequency signal which is equal to $2\pi/\omega$.
As a result, the response amplitude $Q$ at the frequency $\omega$ is
calculated by

$Q=\sqrt{Q_{\mathrm{s}}^{2}+Q_{\mathrm{c}}^{2}}/A\,$ (148)

where $Q$ indicates the amplification of the weak low-frequency signal after
is passes through the nonlinear system. Specifically, the low-frequency signal
signal is improved $Q$ times by the vibrational resonance.

Consider the system (16) with

$s_{\mathrm{L}}(t)=A\cos(\omega t),\quad s_{\mathrm{H}}(t)=B\cos(\Omega
t),\quad\omega\ll\Omega,$ (149)

$\omega_{0}^{2}<0$ and $\beta>0$. By numerical simulations, a typical
vibrational resonance is shown in Fig. 3(a). With varying the amplitude $B$,
the resonance on the $Q-B$ curve is clearly shown. Moreover, when we change
the frequency $\Omega$, we also find a vibrational resonance phenomenon, as
given in Fig. 3(b). The critical value $B_{\mathrm{c}}$ at which $Q$ becomes
maximum and the corresponding maximum value $Q_{\mathrm{max}}$ are indicated
in Fig. 3(a) for $\omega=0.1$. The two signals in Eq. (149) are always used in
the following analysis unless otherwise stated.

Figure 3: Vibrational resonance induced by changing the amplitude $B$ or the
frequency $\Omega$ of the fast-varying auxiliary signal in the overdamped
bistable system of Eq. (16). (a) The response amplitude $Q$ versus the
amplitude $B$ presents vibrational resonance for four fixed values of
$\omega$. (b) The response amplitude $Q$ versus the frequency $\Omega$
presents vibrational resonance for four fixed values of $B$. The parameters
are $\omega_{0}^{2}=-1$, $\beta=1$, $A=0.1$, $\Omega=4$ in (a), and
$\omega=0.1$ in (b).

The output of the time series corresponding to the critical amplitude
$B_{\mathrm{c}}=2.8$ for which $Q=Q_{\mathrm{max}}=6.4$ is shown in Fig. 4 to
present an intuitive understanding. While in Fig. 3 is given along with
$s_{\mathrm{L}}(t)$ and $6.4(=Q_{\mathrm{max}})s_{\mathrm{L}}(t)$. In this
figure, we find that the response amplitude at the low-frequency $\omega$ is
greatly enhanced. It agrees with the results in Fig. 3. It presents the
occurrence of vibrational resonance once again in the time domain. As
ilustrated by the results, the enhancement of the low-frequency characteristic
signal is observed in both Fig. 3 and Fig. 4.

Figure 4: Comparison of the time series of the optimal vibrational resonance
response and the input low-frequency signal. The parameters are
$\omega_{0}^{2}=-1$, $\beta=1$, $A=0.1$, $\omega=0.1$, $B=2.8$, $\Omega=4$.

The response may vary depending on the initial conditions when a system has
more than one stable state. Taking an asymmetric bistable system in an
overdamped version as an example, the potential function is given in Eq. (22).
The two harmonic excitations are $A\cos(\omega t)$ and $B\cos(\Omega t)$,
respectively, where $\Omega\gg\omega$. The influence of the initial conditions
on the response is illustrated in Fig. 5. Clearly, if the initial conditions
differ, the response may also differ. Regarding the asymmetric bistable
system, the difference is easily understandable. The tilt of the potential
function causes the particle to move only in a high or a low potential well
when the excitation is weak and there is no cross-well motion.

Figure 5: Different initial conditions induce the change in the modal of
vibrational resonance in the overdamped asymmetric bistable system with
potential of Eq. (22). (a) The response amplitude $Q$ versus the amplitude $B$
of the fast-varying auxiliary signal presents single resonance. The initial
condition is $x(0)=-1$. (b) The response amplitude $Q$ versus the amplitude
$B$ of the fast-varying auxiliary signal presents double resonance. The
initial condition is $x(0)=1$. (c) The phase portrait oscillates in the left
potential well under the conditions $x(0)=-1$ and $B=0.9$. (d) The phase
portrait oscillates in the right potential well under the conditions $x(0)=1$
and $B=0.9$. Other parameters are $\omega_{0}^{2}=-1$, $\alpha=1$, $\beta=1$,
$A=0.1$, $\omega=0.1$, $\Omega=3$.

Next, consider the FitzHugh-Nagumo neuron model in Eq. (70) [73], which is an
excitable system subjected to biharmonic signals. The curve of $Q$ versus $B$
and some time series are depicted in Fig. 6. To display the periodic
relationship between the signal $s_{L}(t)$ and the output $x(t)$ better, we
give $20s_{L}(t)$ as a comparison in the figure. At the critical value
$B=0.058$, as shown in Fig. 6(a), $Q_{max}=3.47$, the input signal $s_{L}(t)$
and the output $x(t)$ achieve the optimal synchronization. The weak slow-
varying signal is amplified remarkably, and the spikes appear just in a half
cycle of the slow-varying signal. For a smaller value of $B$, such as
$B=0.008$ in the figure, almost no spike emerges. For a larger value of $B$,
e.g., $B=0.075$, the spikes occur frequently. Although the output of the
FitzHugh-Nagumo neuron model has a different form with the ordinary bistable
system, the vibrational resonance still occurs when the system is excited by
appropriate slow- and fast-varying signals. In fact, the vibrational resonance
in the FitzHugh-Nagumo neuron model is complicated. It will produce the
subthreshold and suprathreshold vibrational resonance. The bifurcation
parameter $a$ also influences the vibrational resonance pattern. In Fig. 6,
the subthreshold vibrational resonance is shown. For other cases, a detailed
discussion can be found in the work of [69].

Figure 6: The response of the FitzHugh-Nagumo neuron system of Eq. (70) to the
slow-varying characteristic signal is influenced by the amplitude $B$ of the
fast-varying auxiliary signal. (a) The response amplitude $Q$ versus the
signal amplitude $B$ presents vibrational resonance. (b)-(d) The time series
of the response corresponding to different values of the signal amplitude $B$.
The parameters are $\varepsilon=0.01$, $a=1.05$, $A=0.01$, $\omega=0.1$,
$\Omega=5$, and $B=0.008$, $0.058$, $0.075$ in (b), (c), (d), respectively.

###  4.2 Gain factor

The phenomenon of vibrational resonance results from the interaction between
the slow characteristic signal, the nonlinear system, and the fast auxiliary
signal. However, a strong resonance can still occur if the system is solely
driven by a single characteristic signal. This raises the question: What is
the role of the fast auxiliary signal? Is it indispensable for amplifying the
weak, slowly varying characteristic signal in a nonlinear system?

To reflect the role of the fast auxiliary signal on vibrational resonance
directly, Chizhevsky and Giacomelli defined the gain factor of the response
amplitude in their work [127], [214]. The gain factor $G_{\mathrm{VR}}$ is
calculated by

$G_{\mathrm{VR}}=\frac{Q_{\mathrm{L}}(\omega)}{Q_{0}(\omega)}.$ (150)

Herein, $Q_{\mathrm{L}}(\omega)$ and $Q_{0}(\omega)$ represent the response
amplitude at the frequency of the slow-varying characteristic signal in the
presence and absence of the fast-varying auxiliary signal, respectively.
$Q_{\mathrm{L}}(\omega)$ and $Q_{0}(\omega)$ are calculated using Eq. (148).
We continue to use the overdamped system in Eq. (16) as an example. The curve
of $G_{\mathrm{VR}}$ is depicted in Fig. 7. The effect of the fast-varying
auxiliary signal is clearly illustrated in the figure. Due to the presence of
the auxiliary signal, similar to the results shown in Fig. 3, the weak slow-
varying characteristic signal is notably amplified.

Figure 7: The gain factor $G_{\mathrm{VR}}$ versus the amplitude $B$ of the
fast-varying auxiliary signal also presents vibrational resonance. All
simulation parameters are the same as that in Fig. 3(a).

In fact, to investigate the role of the fast-varying auxiliary signal on
vibrational resonance, one can consider introducing the concept of the gain
factor as described in Eq. (150). Moreover, the response amplitude $Q(\omega)$
can be substituted with various other metrics such as signal-to-noise ratio
(SNR), spectral amplification factor, cross-correlation coefficient, and so
on, as deemed suitable.

###  4.3 Cross-correlation coefficient

When investigating vibrational resonance induced by an aperiodic binary
signal, a commonly used metric is the cross-correlation coefficient between
the system output and the input of a slowly-varying signal, as discussed in
[110], [115], [116]. The cross-correlation coefficient is defined by

$C_{sx}=\frac{\sum_{j=1}^{n}\left[s(j)-\bar{s}\right]\left[x(j)-\bar{x}\right]}{\sqrt{\sum_{j=1}^{n}\left[s(j)-\bar{s}\right]^{2}\sum_{j=1}^{n}\left[x(j)-\bar{x}\right]^{2}}},$
(151)

where $\bar{s}$ and $\bar{x}$ are the average values of the input aperiodic
signal and the output time series, respectively. The numerical value of
$C_{sx}$ lies in $[-1,1]$. When vibrational resonance occurs, the cross-
correlation coefficient usually reaches a maximum. Nevertheless, even though
the cross-correlation coefficient is a large enough value, vibrational
resonance does not necessarily occur. Specifically, the cross-correlation
coefficient is the necessary but not sufficient condition for vibrational
resonance. Thus, there is a need to establish novel metrics for assessing
aperiodic vibrational resonance. Furthermore, recent studies have revealed
that in cases where the system is subjected to a single aperiodic signal, the
most pronounced aperiodic resonance can also manifest at the inflection point,
where the correlation coefficient reaches its minimum, as reported in [268].

Here, we consider the system described by

$\dot{x}+\omega_{0}^{2}x+\beta x^{3}=s(t)+B\mathrm{sgn}(\cos\Omega t)$ (152)

where $s(t)$ is an aperiodic binary signal defined in Eq. (6). The fast-
varying auxiliary signal is a periodic square signal with amplitude $B$. In
Fig. 8, the cross-correlation coefficient corresponding to the amplitude of
the fast-varying auxiliary signal is presented. At the maximal value
$C_{\mathrm{max}}$, i.e., $B=B_{\mathrm{c}}$, the response achieves the
strongest aperiodic vibrational resonance. The response corresponding to the
optimal aperiodic vibrational resonance is given in Fig. 9. The amplification
of the weak aperiodic signal is evident.

Figure 8: The cross-correlation coefficient $C_{sx}$ versus the amplitude $B$
of the fast-varying auxiliary signal presents aperiodic vibrational resonance
in the system of Eq. (152). The parameters are $\omega_{0}^{2}=-1$, $\beta=1$,
$A=0.3$, $T=20$ and $\Omega=\pi$. Figure 9: Time series of the optimal
aperiodic vibrational resonance caused by the binary signal in the system of
Eq. (152). The thick line (red color) is the aperiodic binary characteristic
signal. The thin line (blue color) is the output corresponding to the optimal
aperiodic vibrational response. The parameters are $\omega_{0}^{2}=-1$,
$\beta=1$, $A=0.3$, $B=1.5$, $T=20$ and $\Omega=\pi$.

###  4.4 Bit error rate

The bit error rate is another commonly used metric to quantify the correctness
of the output-input. The resonance occurs when the bit error rate reaches the
minimum. The bit error rate has been successfully used in the stochastic
resonance research [269]-[271]. Chizhevsky and Giacomelli introduced the bit
error rate to quantify the occurrence of vibrational resonance induced by an
aperiodic binary signal [110]. The bit error rate is the ratio of wrong
received bits to the total number of transmitted bits. Specifically, it is the
number of the received error bits divided by the total number of all
transferred bits during the considered time interval. Usually, the bit error
rate is a percentage number. As a result, it is important to define the
correct or wrong received bits of the output. Here, there is a little
difference to the conventional calculation of the bit error rate. As is well
known, the response amplitude does not equal the input signal level in
general. Hence, we consider that the output is correct if the absolute value
of the discrete response value is greater than the given threshold value
$A_{th}$, and vice versa. Apparently, the threshold value $A_{th}$ is
important. We still use the system in Eq. (152) as an example and give two
values to plot the curves of the bit error rate versus the signal amplitude
$B$ in Fig. 10. Apparently, with the increase of the amplitude $B$, the
minimal value of the bit error rate appears. It indicates the occurrence of
aperiodic vibrational resonance. The bit error rate has the same effect as the
cross-correlation coefficient. For the case $A_{th}=0$, it indicates that the
received bit is correct if the output and the input signal have the same sign.
For this case, the bit error rate achieves the minimum at $B=1.85$. It is
different from the optimal value $B_{c}$ in Fig. 8. We also know that $B=1.85$
can not make the aperiodic vibrational resonance reach the strongest resonance
output from the time series in Fig. 10. In addition, if we choose $A_{th}=A$,
implies that the output amplitude is enhanced in comparison to the input
level. Regarding to $A_{th}=A$, the critical value of $B$ corresponding to the
minimal of the bit error rate is $B=1.55$. Compared with the result $B=1.5$ in
Fig. 8, there is only a minor error. This shows that using an error rate to
measure resonance is also effective as long as the appropriate threshold is
selected. The time series in Fig. 10 corresponding to $B=1.55$ illustrates the
feasibility of using the bit error rate as the measure of aperiodic
vibrational resonance again.

Figure 10: The bit error rate versus the amplitude $B$ of the fast-varying
auxiliary signal presents aperiodic vibrational resonance in the system of Eq.
(152) (left) and the optimal vibrational resonance response corresponding to
the threshold value $A_{th}=A$ (right, top) and the threshold value
$A_{th}=0$(right, bottom), respectively. All simulation parameters are the
same as those in Fig. 8

.

###  4.5 Spectrum amplification factor

Consider the system [119]

$\dot{x}-\omega_{0}^{2}x+\beta x^{3}=A\cos\left(\pi\gamma t^{2}+2\pi
ft\right)+B\cos\left(k\pi\gamma t^{2}+2k\pi ft\right),$ (153)

which is used to investigate vibrational resonance induced by the linear
frequency modulated signal. In this equation, $A\cos\left(\pi\gamma t^{2}+2\pi
ft\right)$ is the characteristic signal $s_{\mathrm{L}}(t)$,
$B\cos\left(k\pi\gamma t^{2}+2k\pi ft\right)$ is the auxiliary signal
$s_{\mathrm{H}}(t)$ with $k$ is a positive number and $k\gg 1$. The
instantaneous frequency of the auxiliary signal is much greater than that of
the characteristic signal. We provide the plot of the characteristic signal,
i.e., the linear frequency modulated signal, in Fig. 11(a) and its
corresponding spectrum in Fig. 11(b). Unlike the periodic signal, the spectrum
of the linear frequency modulated signal is continuous. The spectrum of a
periodic signal is discrete in some specific frequencies. In fact, when the
frequency of the signal is varying with time, it is better to analyze the
spectrum by using time-frequency analytical tools, such as the short-time
Fourier transform [272], the fractional Fourier transform [273], and the
empirical mode decomposition [274]-[276], etc.

Figure 11: The linear frequency modulated signal considered in the system of
Eq. (153). (a) Time series of the input linear frequency modulated signal, (b)
Spectrum of the input linear frequency modulated signal. The parameters are
$A=0.1$, $\gamma=1$, $f=1$.

For studying vibrational resonance induced by the linear frequency modulated
signal, Jia et al. [119] proposed the spectrum amplification factor. At first,
they used the cross-correlation coefficient to measure aperiodic vibrational
resonance. Then, by a new spectrum amplification factor, they found that the
new index is better for describing the vibrational resonance performance. The
spectrum amplification factor $\eta$ is defined by

$\eta=\frac{{\displaystyle{\frac{1}{f_{\mathrm{end}}-f_{\mathrm{start}}}}}{\displaystyle{\int_{f_{\mathrm{start}}}^{f_{\mathrm{end}}}s_{x}\left(f_{\mathrm{in}}\right)\mathrm{d}f_{\mathrm{in}}}}}{{\displaystyle{\frac{1}{f_{\mathrm{end}}-f_{\mathrm{start}}}}}{\displaystyle{\int_{f_{\mathrm{start}}}^{f_{\mathrm{end}}}s_{u}\left(f_{\mathrm{in}}\right)\mathrm{d}f_{\mathrm{in}}}}}=\frac{{\displaystyle{\sum_{i=1}^{n}s_{x}(i)}}}{{\displaystyle{\sum_{i=1}^{n}s_{u}(i)}}},$
(154)

where $s_{u}(\bullet)$ is the spectrum of the input linear frequency modulated
signal, and $s_{x}(\bullet)$ is the spectrum of the system output. The
spectrum amplification factor is an appropriate measure to quantify
vibrational resonance in the case of driving forces considered in Eq. (153).

For the system Eq. (153), the spectrum amplification factor and the cross-
correlation coefficient are given in Fig. 12(a) and Fig. 12(b), respectively.
Apparently, there is only one maximal value corresponding to the peak value of
the curve in Fig. 12(a). Although these two kinds of curves can prove the
occurrence of vibrational resonance, in Fig. 12(b), the peak value of the
curve does not correspond to the maximal value of the considered index
$C_{sx}$. Due to the time modulation of the frequency, this kind of
vibrational resonance is essentially an aperiodic vibrational resonance. It
may occur at the peak of the $C_{sx}-B$ curve. For the spectrum amplification
factor, the peak value of the $\eta-B$ curve corresponds to the maximal
response of the system, i.e., the aperiodic vibrational resonance. Besides,
the spectrum amplification parameter can measure the degree of the
amplification of the input characteristic signal. From the value of the
optimal amplitude of the fast-varying auxiliary signal, in Fig. 12(a), at
$B=0.42$, $\eta$ achieves the peak value $12.68$. In Fig. 12(b), at $B=0.46$,
the peak value is $C_{sx}=0.8729$. That is to say, the optimal amplitudes of
$B$ obtained from the two measures are approximately equal. To This method
will be detailed in Section 5.4.

Figure 12: Different characterizations of aperiodic vibrational resonance
induced by the linear frequency modulated signal in the system of Eq. (153).
(a) The spectrum amplification factor $\eta$ versus the amplitude $B$ of the
fast-varying auxiliary signal presents aperiodic vibrational resonance. (b)
The cross-correlation coefficient $C_{sx}$ versus the amplitude $B$ of the
fast-varying auxiliary signal presents aperiodic vibrational resonance. The
parameters are $\omega_{0}^{2}=300$, $\beta=300$, $A=0.1$, $\gamma=1$, $f=1$,
$k=10$.

The response of the system corresponding to strong aperiodic vibrational
resonance is illustrated in Fig. 13. To enhance clarity, we display the time
series within the interval $[0,12]$ across four subplots. Upon comparison of
the time series of $x(t)$, $s(t)$, and $12.68s(t)$, it becomes evident that
the weak characteristic signal experiences significant amplification.
Furthermore, the curve $12.68s(t)$ exhibits a magnitude nearly identical to
that of the curve $x(t)$. This observation aligns with the peak value of the
$\eta-B$ curve depicted in Fig. 12(a).

Figure 13: Time series of the response when the optimal aperiodic vibrational
resonance occurs in system of Eq. (153). The parameters are $A=0.1$,
$\gamma=1$, $f=1$, $B=0.42$, $k=10$.

###  4.6 Success probability

We define $P$ as the success probability, which serves as the index to measure
the logical vibrational resonance

$P=\frac{\text{The correct turns}}{\text{The total turns}}.$ (155)

The specific process for obtaining the success probability $P$ is given as
follows.

  1. 1.

Randomly arrange an array of four possible logical states (0,0), (0,1), (1,1),
(1,0) and place it into the nonlinear system.

  2. 2.

Drive the system by each logical state with the same time. Choose the total
turns in a large number in the calculation. The turns mean all trajectories
for the calculation.

  3. 3.

Analyze the output based on the logical calculation truth table to obtain the
success probability.

For the logical $\bf OR$, $\bf AND$, $\bf NOR$, $\bf NAND$, $\bf XOR$, $\bf
NXOR$, the logical calculation truth values are given in Table 1.

Table 1: The logical calculation truth table. Input set($I_{1}$,$I_{2}$) | Value of ($I_{1}+I_{2}$) | OR | AND | NOR | NAND | XOR | NXOR
---|---|---|---|---|---|---|---
(0,0) | $-1$ | 0 | 0 | 1 | 1 | 0 | 1
(0,1)/(1,0) |  0 | 1 | 0 | 0 | 1 | 1 | 0
(1,1) |  1 | 1 | 1 | 0 | 0 | 0 | 1

In the absence of noise, the logical computation in the system described by
Eq. (140) involves two harmonic signals: $A\cos(\omega t)$ and $B\cos(\Omega
t)$. Typically, the system output realizes $\bf OR$, $\bf NOR$, $\bf AND$, and
$\bf NAND$ logical calculations for $r>0$ and $r<0$, respectively. However,
the bistable system cannot achieve $\bf XOR$ and $\bf NXOR$ calculations. To
accomplish these operations using the response of a nonlinear system, it is
suitable to employ a triple-well system [277]. Notably, the logical response
exhibits a resonance region when varying the two harmonic signals, as depicted
in Fig. 14. This phenomenon is logical vibrational resonance. In the
simulation results shown in Fig. 14, $I_{1},I_{2}=-0.5$ corresponds to logic
$0$, and $I_{1},I_{2}=0.5$ corresponds to logic $1$. In this figure, for a
weak slow-varying signal and a strong fast-varying signal, i.e., $B\gg A$, the
logical vibrational resonance occurs. It is noteworthy that the same
phenomenon occurs for a weak high-frequency signal and a strong low-frequency
signal, i.e., $A\gg B$. This behavior is distinct from conventional
vibrational resonance, where the amplitude of the slow-varying signal is
typically very weak. In summary, appropriately chosen harmonic signals have a
positive impact on the logical response of a nonlinear system.

Figure 14: There is an obvious logical vibrational resonance region
corresponding to the $\bf OR$ logic operation in the two dimensional plane of
the success probability with both $A$ and $B$. The parameters are
$\omega_{0}^{2}=-2$, $\beta=4$, $r=0.5$, $\omega=1$, $\Omega=20$.

To illustrate the logical vibrational resonance further, we show a time series
plot in Fig. 15. In Figs. 15(a), (b), (d), the time series present a poor
logical output. In Figs. 15(c), corresponding to the strong resonance (bright
color region) in Fig. 14, the time series presents an excellent logical
output.

Figure 15: Time series of the logic input $I_{1}+I_{2}$ (thick line in red
color) and logical output (thin line in blue color) of the system of Eq. (140)
for different values of the amplitude $B$ of the fast-varying auxiliary
signal. The parameters are $\omega_{0}^{2}=2$, $\beta=4$, $r=0.5$, $A=0.1$,
$\omega=1$, $\Omega=20$, (a) $B=0$, (b) $B=2$, (c) $B=6$, (d) $B=12$.

##  5 Theoretical formulation of vibrational resonance

###  5.1 The linear response theory

The method of direct separation of motions either alone or in combination with
other methods, serves as a valuable technique for analyzing the effects of a
high-frequency excitation within a theoretical framework [278]-[291]. In this
subsection, the method of direct separation of motions serves as the primary
theoretical basis. Additionally, unless otherwise specified, we will
consistently employ $A\cos(\omega t)$ and $B\cos(\Omega t)$ as the two
excitations acting on the system.

####  5.1.1 Vibrational resonance in the bistable system

Taking the overdamped Duffing oscillator system Eq. (16) as an example, we
introduce the method of direct separation of motions in detail. Letting

$x(t)=X(t)+\Psi(t),$ (156)

where, $X(t)$ is a slow variable with period $2\pi/\omega$, and
$\Psi(\tau=\Omega t)$ is a fast variable with period $2\pi$. Assuming the
average of $\Psi$ over the period is
$\Psi_{\mathrm{av}}=(1/2\pi)\int_{0}^{2\pi}\Psi\mathrm{d}\tau=0$, the
equations for $X(t)$ and $\Psi(t)$ are

$\displaystyle\dot{X}\\!\\!+\\!\\!\left(\omega_{0}^{2}+3\beta\Psi_{\mathrm{av}}^{2}\right)X+\beta\left(X^{3}+\Psi_{\mathrm{av}}^{3}\right)+3\beta
X^{2}\Psi_{\mathrm{av}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!A\cos(\omega t),$ (157a)
$\displaystyle\dot{\Psi}\\!\\!+\\!\\!\omega_{0}^{2}\Psi\\!\\!+\\!\\!3\beta
X^{2}\left(\Psi\\!\\!-\\!\\!\Psi_{\mathrm{av}}\right)\\!\\!+\\!\\!3\beta
X\left(\Psi^{2}\\!\\!-\\!\\!\Psi_{\mathrm{av}}^{2}\right)\\!\\!+\\!\\!\beta\left(\Psi^{3}\\!\\!-\\!\\!\Psi_{\mathrm{av}}^{3}\right)\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!B\cos(\Omega t),$ (157b)

where $\Psi_{\mathrm{av}}^{n}=(1/2\pi)\int_{0}^{2\pi}\Psi^{n}\mathrm{d}\tau$.

As $\Psi$ is fast varying, we neglect all nonlinear terms in Eq. (157b). Then,
the solution of the linear equation in the limit of $t\to\infty$ is given by

$\displaystyle\Psi=C\cos(\tau+\phi),\quad
C=\frac{B}{\sqrt{\Omega^{2}+(\omega_{0}^{2})^{2}}},\,\quad\phi=\tan^{-1}\left(-\frac{\Omega}{\omega_{0}^{2}}\right).$
(158)

For $\Omega\gg\omega_{0}^{2}$, we can neglect the $(\omega_{0}^{2})^{2}$ in
the above equation and approximate the amplitude of $\Psi$ as $g=B/\Omega$.
Since $\Omega\gg\omega$, in the interval $[t,t+2\pi/\Omega]$, $X(t)$ can be
viewed as a constant. Using the $\Psi$ given in Eq. (158), we find
$\Psi_{\mathrm{av}}=0$, $\Psi_{\mathrm{av}}^{2}=C^{2}/2$ and
$\Psi_{\mathrm{av}}^{3}=0$. Then, the equation for the slow variable (157a)
becomes

$\dot{X}+\mu X+\beta X^{3}=A\cos(\omega t),$ (159)

where $\mu=\frac{3}{2}\beta C^{2}+\omega_{0}^{2}$.

For $\omega_{0}^{2}<0$ and $\beta>0$ (bistable case), the equilibrium points
of Eq. (159) when $A=0$ are $X^{*}=0$ and
$X^{*}_{\pm}=\pm\sqrt{-\mu/\beta}\,$. In this scenario, a slow oscillation
occurs around the stable equilibrium points. The parameters $B$ and $\Omega$
are typically considered as control parameters for vibrational resonance. By
adjusting these parameters, the equilibrium points and their stability can be
naturally altered.

To determine the response amplitude at the frequency $\omega$, we define
$Y=X-X^{*}$ as the deviation of $X$ from $X^{*}$, where $X^{*}$ represents the
stable equilibrium of the equivalent system (159). Then, we can derive the
following expression

$\dot{Y}+\omega_{\mathrm{r}}Y+3\beta X^{*}Y^{2}+\beta Y^{3}=A\cos\omega t,$
(160)

where $\omega_{\mathrm{r}}=\mu+3\beta X^{*2}$. For a weak characteristic
signal $A\cos\omega t$ with $A\ll 1$, $|Y|\ll 1$, making it reasonable to
neglect all nonlinear terms in the equation. Then, the steady solution of the
corresponding linear equation becomes $Y=A_{\mathrm{L}}\cos(\omega t+\phi)$,
where

$\displaystyle A_{\mathrm{L}}$ $\displaystyle=$