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The stochastic resonance (SR) characteristics of a generalized Langevin linear system driven by a multiplicative noise and a periodically modulated noise are studied (the two noises are correlated). In this paper, we consider a generalized Langevin equation (GLE) driven by an internal noise with long-memory and long-range dependence, such as fractional Gaussian noise (fGn) and Mittag-Leffler noise (M-Ln). Such a model is appropriate to characterize the chemical and biological solutions as well as to some nanotechnological devices. An exact analytic expression of the output amplitude is obtained. Based on it, some characteristic features of stochastic resonance phenomenon are revealed. On the other hand, by the use of the exact expression, we obtain the phase diagram for the resonant behaviors of the output amplitude versus noise intensity under different values of system parameters. These useful results presented in this paper can give the theoretical basis for practical use and control of the SR phenomenon of this mathematical model in future works.

The phenomenon of stochastic resonance (SR) characterized the cooperative effect between weak signal and noise in a nonlinear systems, which was originally perceived for explaining the periodicity of ice ages in early 1980s [

Nowadays, there have been considerable developments in SR, and the original understanding of SR is extended. Firstly, in the initial stage of investigation of SR, the nonlinearity system, noise and periodic signal were thought of as three essential ingredients for the presence of SR. However, in 1996, Berdichevsky and Gitterman [

The SR phenomenon driven by Gaussian noise has been investigated both theoretically and experimentally. However, the Gaussian noise is just an ideal model for actual fluctuations and not always appropriate to describe the real noisy environment. For instance, in the nonequilibrium situation, the stochastic processes describing the interactions of a test particle with the environment exhibit a heavy tailed non-Gaussian distribution. Nowadays, Dybiec et al. investigated the resonant behaviors of a stochastic dynamics system with Lévy stable noise and an isotropic

Recently, more and more scholars began to pay attention to another important class of non-Gaussian noise, the bounded noise. It should be emphasized is that the bounded noise is a more realistic and versatile mathematical model of stochastic fluctuations in applications, and it is widely applied in the domains of statistical physics, biology, and engineering in the last 20 years. Furthermore, the well-known telegraph noise, such as dichotomous noise (DN) and trichotomous noise that are widely used in the studied of SR phenomena, is a special case of bounded noise. The deepening and development of theoretical studies on bounded noise led to the fact that lots of scholars investigated the effect of bounded noise on the stochastic resonant behaviors in the special model in physics, biology, and engineering [

Since Richardson’s work in literature [

It is well-known that the normal diffusion can be modeled by a Langevin equation, where a Brownian particle subjected to a viscous drag from the surrounding medium is characterized by a friction force, and it also subjected to a stochastic force that arises from the surrounding environment. The friction constant determines how quickly the system exchanges energy with the surrounding environment. For a realistic description of the surrounding environment, it is difficult to choose a universal value of the friction constant. Indeed, in order to depict the real situation more effectively, a different value of the friction constant should be adopted. Hence, a generalization of the Langevin equation is needed, leading to the so-called generalized Langevin equation (GLE) [

Nowadays, the GLE driven by a fractional Gaussian noise (fGn) [

The overwhelming majority of previous studies of SR have related to the case where the external noise and the weak periodic force are introduced additively. However, Dykman et al. [

Due to the synergy of generalized friction kernel of a GLE and the periodically modulated noise, the stochastic resonant behaviors of a GLE can be influenced. In contrast to the case that has been investigated before, new dynamic characteristics emerge. Motivated by the above discussions, we would like to explore the stochastic resonance phenomenon in a generalized harmonic oscillator with multiplicative and periodically modulated noises. Moreover, we consider the GLE is driven by a fractional Gaussian noise and a Mittag-Leffler noise, respectively, in this paper. We focus on the various nonmonotonic behaviors of the output amplitude

The physical motivations of this paper are as follows: (1) in view of the importance of stochastic generalized harmonic oscillator (linear oscillator) with memory in physics, chemistry, and biology and due to the periodically modulated noise arising at the output of the amplifier of the optics device and radio astronomy device, to establish a physical model in which the SR can contain the effects of the two factors, the linearity of the system and the periodical modulation of the noise. (2) The second one is to give a theoretical foundation for the study of SR characteristic features of a generalized harmonic oscillator subject to multiplicative, periodically modulated noises and external periodic force. Our study shows that such a model leads to stochastic resonance phenomenon. Meanwhile, an exact analytic expression of the output amplitude is obtained. Based on it, some characteristic features of SR are revealed.

The paper is organized as follows. Section

The generalized Langevin equation (GLE) is an equation of motion for the non-Markovian stochastic process where the particle has a memory effect to its velocity. Anomalous diffusion in physical and biological systems can be formulated in the framework of a GLE that reads as Newton’s law for a particle of the unit mass (

Fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) were originally introduced by Mandelbrot and Van Ness [

Now consider the one-side normalized fBm which is a Gaussian process

It is well-known that the physical origin of anomalous diffusion is related to the long-time tail correlations. Thus, in order to model anomalous diffusion process, a lot of different power-law correlation functions are employed in (

Viñales and Despósito have introduced a novel noise whose correlation function is proportional to a Mittag-Leffler function, which is called Mittag-Leffler noise [

In this paper, we consider a periodically driven linear system with multiplicative noise and periodically modulated additive noise described by the following generalized Langevin equation:

In this paper, we assume that the external noise

First of all, we should transfer the stochastic equation (

It can be found that (

Using the Shapiro-Loginov formula (

To summarize, for the linear generalized Langevin equation (

In order to solve the closed equations (

The solutions of (

Applying the inverse Laplace transform technique, by the theory of “signals and systems,” the product of the Laplace domain functions corresponding to the convolution of the time domain functions, we can obtain the solutions

Equations (

The relationships of the input periodic signals and the output signals by the theory of “signals and systems.”

Moreover, the amplitudes

Thus, from (

In addition, we can obtain the expressions of frequency response functions

Then, the amplitudes of frequency response functions

Meanwhile, the amplitudes of the output signals are

It should be emphasized that the following inequality must hold for the sake of the stability of solutions [

When the internal noise

When the internal noise

In this section, we will perform the numerical simulations on the above analytical expression in (

From (

It can be seen from the stability condition (

The phase diagram for the stochastic resonance behaviors of the output amplitude

In the unshaded region [see the domain (0) of level = 0 which corresponds to Figure

The light shaded region (i) corresponds to the single-peak SR phenomenon [see the domain of level = 1 which corresponds to Figure

The dark shaded region (ii) corresponds to the double-peaks SR phenomenon [see the domain of level = 2 which corresponds to Figure

From Figure

In Figure

The output amplitude

The main contribution of this section is as follows: with the help of phase diagram for the SR phenomenon, we can effectively control the SR phenomenon of this generalized harmonic system in a certain range and further broaden the application scope of the SR phenomenon in physics, biology, and engineering, such as the detection of weak stimuli by spiking neurons in the presence of certain level of noisy background neural activity [

From (

It is found from the stability condition (

The phase diagram for the stochastic resonance behaviors of the output amplitude

In Figure

The output amplitude

As shown in Figure

To summarize, in this paper we explore the SR phenomenon in a generalized Langevin equation with multiplicative, periodically modulated noises, and external periodic force. Moreover, the system internal noise is modeled as a fractional Gaussian noise and a Mittag-Leffler noise, respectively. Without loss of generality, the fluctuations of system intrinsic frequency are modeled as a multiplicative dichotomous noise. By the use of the stochastic averaging method and the Laplace transform technique, we obtain the exact expression of the output amplitude

We focus on the various nonmonotonic behaviors of the output amplitude

We believe all the results in this paper not only supply the theoretical investigations of the generalized harmonic oscillator subject to multiplicative, periodically modulated noises and external periodic force but also can suggest some experimental anomalous diffusion results in physical and biological applications in the future [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Key Program of National Natural Science Foundation of China (Grant nos. 11171238 and 11601066) and Natural Science Foundation for the Youth (Grant nos. 11501386 and 11401405).