1. IntroductionStochastic resonance (SR) can be considered as a synchronization of noise-induced hopping between the potential wells with a weak periodic forcing.[1–7] For a given nonlinear system, the time-scale matching condition for SR depends on the noise intensity and the period of the driving force. Therefore, noise, nonlinear potential, and periodic driving force have an important effect on the occurrence of SR. For example, Ai and Liu[7] found that an optimal stochastic potential may induce a doubly stochastic resonance (DSR). When the intensity of the stochastic potential is too large, the effect of the double well diminishes and the SR phenomenon disappears. However, most of studies on SR only involve the stochastic systems subjected to one periodic force. Recently, it is found that the harmonic mixing signal plays a constructive role in the context of vibrational resonance or controlling SR.[8–20] Landa and McClintock[8] found the vibrational resonance in an over-damped bistable system subjected to two periodic fields. Gitterman[9] showed that an additional periodic field has control over the shift of the resonance frequency, which is desirable in electronic devices or lasers. Chizhevsky[11,12] reported the experimental evidence of suppression of vibrational higher-order harmonics in a bistable vertical-cavity surface-emitting laser driven by two harmonic signals with very different frequencies. Yang et al.[14] investigated the response to the low-frequency of a bistable system perturbed by a low-frequency and a high-frequency excitation. Gammaitoni et al.[15,16] proposed the concept of controlling SR by using the periodic mixing signal and developed a general theoretical framework based on a rate equation approach. Schmid et al.[17] used the harmonic mixing signal to enhance or suppress the nonlinear SR in an over-damped bistable system. Grigorenko et al.[18] investigated the response of a bistable system with a frequency mixing force and reported the SR through the experiment of an iron garnet thin film. Jin et al.[19] studied the phenomenon of SR in an asymmetric bistable system subjected to the multiplicative and additive white noises and two periodic fields. Machura et al.[20] explored transport properties of the Brownian particles in a symmetric potential driven by the time-periodic biharmonic signals. They demonstrated that the symmetric driving cannot distinguish the direction in the frictionless case. In the case of over-damped motion, the antisymmetric driving leads to zero current. Hence, the study of the SR in nonlinear systems driven by two periodic fields and noise is of great signification.
Generally, most of the dynamical systems using in the study of SR are restricted to over-damped regimes, then the inertial term can be neglected and the theoretical analysis is simplified. However, for intermediate and small damping, the inertial effects should be considered and the under-damped dynamical systems are modeled. The SR phenomenon in the under-damped nonlinear systems and the effects of damping on dynamical behaviors have attracted attention from the scientists of different research fields.[21–31] Du et al.[21] and Wu et al.[22] found the damping induced stochastic multi-resonance in an under-damped bistable system with delay and without delay, respectively. Saikia[23] showed that the average input energy displays a peaking behavior with the damping coefficient. That is, damping plays a crucial role in the nature of SR in the multistable system. Lindner et al.[25] investigated the non-equilibrium dynamics of massive damped Brownian particles in an asymmetric periodic potential driven by colored noise. They demonstrated that the finite inertia can enhance the noise-induced transport in a ratchet potential and that for this transport there exhibits an optimal temperature of the embedding bath. Kang et al.[26,27] explored the SR phenomenon in an under-damped Duffing oscillator by using the moment method under the weak noise limit. Jin et al.[29,30] studied the noise-induced resonances in an under-damped periodic potential driven by colored noise and correlated noise, respectively. Xu et al.[31] implemented a numerical investigation of SR in the under-damped bistable system subjected to a weak asymmetric dichotomous noise and a periodic signal. Therefore, an under-damped bistable system is adopted to demonstrate the effects of the harmonic mixing signal on SR in this paper.
The objective of this investigation is to analyze SR phenomenon at fundamental and higher-order harmonic in an under-damped bistable system driven by harmonic mixing signal and Gaussian white noise. Using the LRT, the analytic expressions of the spectral amplification are derived at the fundamental and higher-order harmonic. The effects of the system parameters on the spectral amplification are discussed in Section 2. In Section 3, the system responses, PSD and SNR are calculated to demonstrate the SR and verify the theoretical results. Finally, some conclusions are drawn in Section 4.
2. Spectral amplificationThe model of concern is an under-damped particle moving in a bistable potential system, which is driven by Gaussian white noise and a harmonic mixing signal
where
γ is the damping coefficient,
denotes a symmetric bistable potential function with a potential barrier
,
D denotes the noise intensity, the noise term
is Gaussian white noise with zero mean and autocorrelation function as follows:
The harmonic mixing signal G(t) is given by
where
A is the fundamental amplitude,
ε is the ratio of the second harmonic amplitude to
A, and
Ω is the frequency of signal.
Let
, equation (1) can be rewritten as follows:
In the absence of any excitation (i.e.,
, equation (
4) has one unstable equilibrium state
and two stable equilibrium states
for
.
The corresponding Fokker–Planck equation of Eq. (4) is found to be
where
.
According to Eq. (5), the quasi-stationary probability density is obtained as follows:
where the modified potential function
and
N is the normalization constant.
From Eq. (6), the transition rates out of
can be obtained under the adiabatic limit condition[1]
where
W0 stands for Kramers rate of Eq. (
1) driven only by noise, and the expression can be given by
[26]
Using the two-state theory of SR,
is defined as the occupation probabilities of the stable states
. Then, the master equation for the occupation probabilities
is found to be
Based on the definition of time-dependent mean value
and Eq. (9), the differential equation for
can be derived as
For the weak signal limit, the expansion of Eq. (7) reads
Substituting Eq. (
11) into Eq. (
10) leads to the analytical solution in the stationary limit
where the amplitudes
Mn bring in an additional phase lag
for each Fourier component.
Then, the spectral amplification
(i = 1,2,3) at the i-th harmonic are obtained within the LRT[16]
where
To understand the above theoretical results (13)–(15), one can turn to a case study, where the system parameters are chosen as a = b = 1 and A = 0.01. Figures 1–3 show the behaviors
of the spectral amplification
(i = 1,2,3) as a function of D for different γ, ε, and Ω. The curves of
(i = 1, 2 display a bell shape with the increase in D and SR exists. In Fig. 1, the value of
(
) increases and the position of the peak shifts to the left side with decreasing γ. Especially, the curve of
shows two maximums and one minimum with increasing D in Fig. 1(c). That is, both of noise-induced suppression and resonance are found in the spectral amplification at the third harmonic. While the noise-induced suppression disappears as the damping γ is increased.
In Fig. 2, the value of
decreases with the increase in ε, while the value of
increases with the increase of ε (see Figs. 2(a) and 2(b)). This phenomenon happens because the parameter ε denotes the ratio of the second harmonic amplitude to the fundamental one. Thus, with the increase of ε, the second term in the harmonic mixing signal has a great effect on SR. From Fig. 2(c), the value of the peak increases with increasing ε in the case of
and the noise-induced suppression disappears for ε = 0.5. In other words, when the ratio of the second harmonic amplitude to the fundamental one is small, the interaction between the two parts of the harmonic mixing signal is weak. The first part of harmonic mixing signal plays a major role in the occurrence of SR for
.
In Fig. 3, the values of
(i = 1,2) increase and the positions of the peak shift to the left side with decreasing Ω. It is observed that the optimal noise intensity
for SR increases with increasing Ω. The bimodal structure is observed in the curve of
for small Ω. When Ω = 0.1, the noise-induced suppression disappears in Fig. 3(c). For the generation of the third harmonic, the effect of resonance and suppression is really small compared with that at first and second harmonics.
3. Numerical resultsTo better understand the influences of harmonic mixing signal on SR and verify the above theoretical results, the numerical results of Eq. (1) are presented by using the fourth-order Runge–Kutta algorithm in this section. In the following analysis, the PSD and SNR are calculated to measure the SR phenomena. It is convenient to quantify SR phenomenon by the intensity of a peak in the PSD and the resonance peak in the SNR.
3.1. Power spectral densityFigure 4 shows the time history and the phase trajectory at four different levels of noise intensity. In Fig. 4(a), the particle stays in the right potential well for weak noise intensity D = 0.02. With the increase in noise intensity (i.e., D = 0.12 in Fig. 4(c)), the particle switches almost periodically between the right and the left potential well. Namely, the synchronization between noise-induced hopping and the weak signal takes place for a suitable dose of noise intensity. If the noise intensity continues to increase (i.e., D = 0.3 in Fig. 4(d)), the noise-induced hopping events between the potential wells will happen more frequently, where the statistical synchronization vanishes and the SR effect is weakened.
The PSD of the mean value
is plotted to characterize the SR at first and second harmonic in Figs. 5 and 6. From Figs. 5 and 6, there are two peaks appearing at
and
, respectively. That is, the SR phenomenon happens at first and second harmonic, which is observed also from the theoretical results (see Figs. 1–3). The intensity of the peak in PSD decreases with increasing γ as shown in Fig. 5. Compared with Fig. 5(a), the positions of peaks in Fig. 6(a) are changed by choosing different Ω. It is seen from Fig. 6 that the intensity of the peak in PSD increases with the increase in A.
3.2. Signal-to-noise ratioIn the following analysis, the signal-to-noise ratio (SNR) is introduced as a characteristic measure of SR. The SNR is calculated by
dB, where S and N are the amplitudes of the signal and noise background respectively. The S is determined directly from the power spectrum at the driving frequency of the external periodic signal, and N is the average of these amplitudes corresponding to 10 frequency values around the driving frequency in the power spectrum. Here, the SNR at the Ω and 2Ω are defined as SNR1 and SNR2, respectively. The influences of noise intensity D, signal amplitude A, driving frequency Ω, and the ratio of the second harmonic amplitude to the fundamental one ε on the SNR1 and SNR2 are discussed in Figs. 7–9. It is seen that the curves of SNR1 vs. D exhibit a minimum first and a maximum later as shown in Fig. 7(a). That is, both of the noise-induced suppression and resonance are found in the curves of SNR1. From Fig. 7(b), the noise-induced suppression disappears and only the noise-induced resonance appears in the curves of SNR2 vs. D for the case of A = 0.2. It is found that the values of SNR1 and SNR2 increase with the increase of A, which coincide with the results shown in Fig. 6.
The effects of Ω on the SNR are shown in Fig. 8. It is found that the curves of SNR1 and SNR2 display a minimum first and a maximum later with increasing D. When the values of Ω increase, the peak values of SNR1 and SNR2 decrease. Meanwhile, the positions of peak in SNR1 and SNR2 moves to right side with the increase of Ω. It is clear that the theoretical results in Fig. 3 agreet with the numerical results as shown in Fig. 8.
In Fig. 9, the SNR is a function of D for A = 0.1 with different ε. When ε = 0.5 and ε = 1.0, both of the SNR1 and SNR2 manifest the noise-induced suppression and SR at certain values of noise intensity. At the same time, the values of SNR1 are bigger than those of SNR2 for ε = 0.5 and ε = 1.0. For ε = 3.0, the curves of the SNR1 and SNR2 exhibit the suppression and resonance simultaneously. However, the values of SNR2 are bigger than those of SNR1 for this case. This phenomenon can be explained as the amplitude of
is bigger than that of
. When ε continues to increase (i.e., ε = 5.0), both of the noise-induced suppression and resonance in SNR1 disappear (see Fig. 9(a)). On the other hand, the SNR2 increases and shows a single-peak structure in Fig. 9(b), which indicates that the second term in the harmonic mixing signal has a dominant effect on SR in this case. That is, SR phenomenon can only be found at 2Ωwhen ε takes a big value. From Fig. 9, the value of SNR1 decreases and the value of SNR2 increases with the increase in ε, which is consistent with the theoretical results shown in Fig. 2.
