Research paper
Collective behavior of a nearest neighbor coupled system in a dichotomous fluctuating potential

https://doi.org/10.1016/j.cnsns.2020.105499Get rights and content

Highlights

  • The stability and collective behavior of a nearest neighbor coupled system in a dichotomous fluctuating potential driven by a periodic source are investigated.

  • The stability criterion is only related to the noise parameters and system potential parameter, and is unrelated to the number of particles and coupling strength.

  • The synchronization criterion is related to the number of particles, coupling strength, noise parameters and system potential parameter.

  • In the stationary regime, all particles’ displacements synchronously move with their mean field, and the output amplitudes are bounded and show stochastic resonance phenomenon; In the non-stationary regime, the output amplitudes are unbounded, and the distribution functions of all particles’ displacements are characterized by power-law distributions and are collectively synchronized.

Abstract

In this study, the collective and stochastic resonance (SR) behavior of a stochastic nearest neighbor coupled system in a dichotomous fluctuating potential driven by a periodic force are investigated. The exact conditions of stability, synchronization and SR of the system are given analytically and verified by a numerical algorithm. We find that the final dynamic behaviors of the system are the results of the synergy of the coupled system, noise and external periodic force: the stationary regime criterion is only related to the noise parameters and system potential parameter; the synchronization criterion is related to the number of particles, coupling strength, noise parameters and system potential parameter. Furthermore, for the stationary regime, all particles synchronously move with their mean field and show SR phenomenon. For the non-stationary regime, the power-law distribution is observed when the synchronization criteria is satisfied.

Introduction

The study of complex systems has been recognized as a new scientific discipline in recent years that is strongly rooted in the advances that have been made in diverse fields ranging from physics to anthropology [1]. Stability of complex system is a fundamental problem [2], [3]. For example, the security and reliability of the power grids are mainly reflected in the stability of the system [4]. In the four-rotor aircraft control system, the stability control of the attitude angle is the key to the stable flight of the aircraft to ensure its safety [5].

Synchronization is a basic and universal collective behavior of complex systems. Systems as diverse as clocks, singing crickets, cardiac pacemakers, firing neurons, and applauding audiences exhibit a tendency to operate in synchrony [6], [7]. Historically, the dynamics of synchronization has been widely investigated in the fields of natural science, social science, and engineering science [7], and various types of synchronization of different systems have emerged [8], [9], [10], [11]. There exist many benefits of having synchronization in some engineering applications, such as secure communication, chaos generators design, chemical reactions, biological systems, information science, and so on. Thus it is of great importance to provide the analytic results of synchronization problems. However, the theoretical research of synchronization lags behind compared with phenomenon observation and experimental research of synchronization.

Moreover, noises effect is inevitable for the real systems in the fields of biology, physics, chemistry and engineering. More and more investigations show that noise plays an important role in the dynamic behaviors of stochastic systems such as stability, synchronization, stochastic resonance (SR) and so on in different ways [12], [13], [14]. Nowadays, the investigations of complex systems have also been extended to coupled stochastic systems. For the environmental fluctuation of a coupled stochastic system, while early investigations considered additive white noise [13], [15], [16], [17], more recent works have considered multiplicative colored noise [14], [18], [19].

Recent investigations have found that the introduction of multiplicative colored noise makes the linear system to have certain nonlinearity and produces various positive effects in the linear systems [18], [20]. As one of the most important colored noises, the dichotomous noise is fairly realistic for physical, engineering, and biological systems [21]. The linear system excited by dichotomous noise creates favorable conditions for the in-depth study of its stochastic dynamics because it can be solved analytically. However, so far, the investigations for stability and synchronization of complex stochastic systems are most focused on the case of Gaussian white noise [13], [16]. The investigations for SR of stochastic systems excited by multiplicative colored noise are most focused on low-dimensional systems [22], [23], [24], [25]. Few studies have investigated the interaction mechanism between the signal, noise and system of high-dimensional coupled system excited by multiplicative colored noise because it is difficult to solve.

In Ref. [26], Yang et al. introduced the globally coupled overdamped Langevin equations in a dichotomous fluctuating potential driven by a periodic force and studied the SR and synchronization using the exact steady-state solutions. The long-time collective behavior of a globally coupled system consists of N harmonic oscillators with random damping coefficient modeled as dichotomous noise was also investigated by Lai et al. [27]. These studies have described the collective behavior of the coupled system through the mean field of all particles’ displacements and provide the stability conditions for the mean field rather than the particles’ displacements. However, the stability of the mean field is different from the stability of all particles’ displacements, and directly analyzing the stability of all particles’ displacements using an analytical method is difficult.

In general, the coupling of a real system is not global. In the locally coupled system, the particle usually only interacts with two (or more) neighbor particles. The nearest neighbor coupled systems is a common and important network model, which has been widely studied [14], [15], [16]. In this study, we investigate the nearest neighbor coupled particles in a randomly switching potential driven by a periodic force and discuss their collective and SR behaviors. We rigorously derive the stability conditions of the first and second moments (i.e., the stationary regime criterion) and the synchronization criterion of the first moments of particles’ displacements using the moment method. We verify these analytic conclusions through numerical simulations on the basis of stochastic Taylor expansion. We find that the stability of all particles’ displacements in the proposed stochastic coupled system is equivalent to the stability of their mean field and is independent of the coupling strength and number of particles. The increasing coupling between particles and noise correlation rate lead to the gradual transition from asynchronous to collective synchronous, the increasing number of particles and noise intensity make synchronization difficult. In summary, the system topology cannot completely control the synchronization of the system, but the noise can also have a great impact on the synchronization. When the synchronization criteria is satisfied, the average amplitude of the coupled particles is synchronous with the average amplitude of the average field, and they all show the same SR phenomenon.

Furthermore, in some complex systems, various power-law distributions have been frequently observed and studied [1], [26], [28], [29], [30]. In this study, we also analyze the distribution function of particles’ displacements using a numerical algorithm and find that each of the particle’s displacement presents a power law distribution under the excitation of strong noise. The distribution function of coupled particles’ displacements also has the synchronization phenomenon.

The rest of this paper is organized as follows. Section 2 proposes the model of the stochastic nearest neighbor coupled system under consideration and formulates some preliminaries. Section 3 presents the main results for the stationary regime and synchronization criteria of the stochastic nearest neighbor coupled system. Section 4 conducts numerical simulations to verify the proposed conclusions and analyzes the synchronization of the distribution functions of the particles’ displacements. Section 5 provides the conclusions.

Section snippets

Preliminaries

A generic example of a stochastic linear system subjected to a periodic sinusoidal driving force and a varying force caused by a random fluctuating monostable potential can described by the following stochastic differential equation [31]:x˙=(a+ξ)x+A0sin(Ωt),where ξ(t) represents a multiplicative color noise. This linear system with such kind of a fluctuating potential barrier has many applications [26], [31], [32], such as problems of fluctuating barrier crossing in chemistry [33], enzimatic

Stability and collective behavior of the deterministic nearest neighbor coupled system

In this section, we firstly discuss the stability and collective behavior (i.e., synchronization) of the deterministic nearest neighbor coupled system (4), which can be rewritten as the following vector form:X˙=UX+F(t),where, X=[x1,x2,,xN]T, F(t)=[A0,,A0]Tsin(Ωt), and the coefficient matrix U is a N × N symmetric tridiagonal matrix:U=TN{(a+ε),(a+2ε),ε}.

Moreover, in order to discuss the synchronization of system (4) in detail, we analyze the deviation between the adjacent particles’

Numerical simulations and discussions

In this section we will verify the main results given in Section 3 by the numerical algorithm based on the stochastic Taylor expansion. Supposing the nth particle’s displacement xn(t) at time t=kh is xn[k], n=1,2,,N;k=1,2,,K, where K is the number of samples. Then the single numerical iteration steps for the stochastic nearest neighbor coupled system (2) can be written as follows:{x1[k+1]=[1(a+ε)hΓ(h)]x1[k]+εhx2[k]A0Ω[cos((k+1)hΩ)cos(khΩ)],xn[k+1]=[1(a+2ε)hΓ(h)]xn[k]+εhxn1[k]+εhxn+1[k]

Conclusions

Understanding the roles of the system network topology and noise on the collective behavior of a complex system has potential applications in structural engineering, power systems, biological networks, electrical/electronic systems and music. In this study, we investigate a stochastic nearest neighbor coupled system driven by a periodic force in a randomly switching potential and derive the output first and second moments in detail. We find that the final dynamic behaviors of the system are the

CRediT authorship contribution statement

Lu Zhang: Writing - original draft, Software, Formal analysis. Ling Xu: Software. Tao Yu: Writing - review & editing. Li Lai: Writing - review & editing. Suchuan Zhong: Writing - original draft, Validation, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This work was supported by the National Natural Science Foundation of China (Grants No. 11401405, No. 11501386).

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