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**|**PLoS One**|**v.12(12); 2017**|**PMC5720815

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- Abstract
- Introduction
- Preparations
- SDE driven by Poisson process
- Noise induces complete synchronization
- Numerical examples
- Conclusions
- Supporting information
- References

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PLoS One. 2017; 12(12): e0188632.

Published online 2017 December 7. doi: 10.1371/journal.pone.0188632

PMCID: PMC5720815

Jun Ma, Editor^{}

School of Aeronautics, Northwestern Polytechnical University, Xi’an, Shaanxi, China

Lanzhou University of Technology, CHINA

* E-mail: nc.ude.upwn@nawf

Received 2017 September 7; Accepted 2017 November 11.

Copyright © 2017 Guo, Wan

This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

The different Poisson noise-induced complete synchronization of the global coupled dynamical network is investigated. Based on the stability theory of stochastic differential equations driven by Poisson process, we can prove that Poisson noises can induce synchronization and sufficient conditions are established to achieve complete synchronization with probability 1. Furthermore, numerical examples are provided to show the agreement between theoretical and numerical analysis.

Noise-induced synchronization in chaotic systems is an interesting phenomenon due to the fact that multiplicative and/or additive noises are ubiquitous in natural and synthetic systems, and up to now it has been studied by many investigators from different areas [1–3]. Lin and his co-workers [4,5] have presented some sufficient conditions of complete synchronization between two unidirectionally coupled chaotic systems disturbed by Gaussian white noise. Later, Xiao and his co-workers have analyzed the effect of Gaussian white noise in bidirectionally coupled piecewise linear chaotic systems [6, 7] and global coupled dynamical network which consists of many nodes. Cao et al. studied the complete synchronization linear stochastic coupled network and gave the sufficient conditions for complete synchronization of network under adaptive control [8]. It has been found that many real systems should be described by the complex dynamical networks (CDNs). The CDNs widely exist in the areas of the Internet, metabolic pathways, the World Wide Web, food-webs, ecosystems, global economic markets, social networks and neuronal network [9–20].

In previous studies, noises were usually assumed to be Gaussian cases, which have been used to approximate different kinds of stochastic perturbations with/without jumps in many situations. However, Gaussian distributions are not appropriate in some practical situations and environments while there may exist large external and/or internal fluctuations [21–28]. It is well known that in the real world, beside Brown noises, there is a very common but important kind of random noises: Poisson noises. Poisson noises which can model these large external and/or internal fluctuations have been observed in various systems such as storage systems, economic systems, biological systems and so on [29–32]. And it is widely known that Poisson process can be viewed as a sequence of independent identically distributed random pulses with the left limits and right-continuous sample paths. As a consequence, the behaviors of the dynamical systems driven by Poisson noise are very different from the stochastic systems driven by Gaussian white noise [33–40]. Thus, it is very important to investigate the dynamic behaviors, such as the synchronization phenomena, for complex networks perturbed by the Poisson process.

Inspired by above analysis, in this paper, we use the stability theory of stochastic differential equation (SDE) driven by Poisson process to analyze the effect of Poisson noise in the global coupled dynamical network.

*C*_{1}(*R*^{k}) - the space of the functions which have continuous first partial derivatives on*R*^{k}.*C*_{b}(*R*^{k}) - the space of bounded continuous functions on*R*^{k}.- ${C}_{b}^{1}\left({R}^{k}\right)$ -the subspace of
*C*_{b}(*R*^{k}) constituted by the functions have continuous first partial derivatives. - ${C}_{1}^{0}\left({R}^{k}\times {R}_{+}\right)$ - class of functions
*V*(*x*,*t*) which have continuous first partial derivatives on*R*^{k}×*R*_{+}except possibly at the point*x*= 0.

A function *V*(*x*,*t*) is said to be positive definite (in Lyapunov’s sense) in *R*^{k} × *R*_{+}, *S*_{h} = {*x* *R*^{k}: |*x*| < *h*, *h* > 0}, if

- (C1)
*V*(0,*t*) = 0,*t**R*_{+}, - (C2)
*V*(*x*,*t*) ≥*W*(*x*),*x**S*_{h},*t**R*_{+}, where*W*(0) = 0,*x*≠ 0,*W*(*x*) > 0.

Obviously, it is negative definite if −*V*(*x*,*t*) is positive definite.

A function *V*(*x*,*t*) is said to be possessed of an infinitesimal upper limit, if

$$\underset{x\to 0}{\mathrm{lim}}\underset{t>0}{\mathrm{sup}}V\left(x,t\right)=0.$$

(1)

The trivial solution *X*(*t*) 0 of the differential equation is said to be

- (D1) stochastically stable, if for any
*t*_{0}≥ 0,*ε*> 0,

$$\underset{{X}_{0}\to 0}{\mathrm{lim}}P\left\{\underset{t\ge {t}_{0}}{\mathrm{sup}}\left|X\left(t,{X}_{0},{t}_{0}\right)\right|>\epsilon \right\}=0.$$

(2)

- (D2) global stochastic asymptotically stable, if it is stochastically stable and also for
*t*_{0}≥ 0,*X*_{0}*R*^{k},

$$P\left\{\underset{t\to \infty}{\mathrm{lim}}X\left(t,{X}_{0},{t}_{0}\right)=0\right\}=1.$$

(3)

Consider a SDE of the following form:

(4)

where *b*(*x*,*t*): *R*^{k} × *R*_{+} → *R*^{k}; *σ*(*x*,*t*): *R*^{k} × *R*_{+} → *R*^{k} × *R*^{m}, *X*(*t*) is a process with values in *R*^{k}, the coefficients *b*(*x*,*t*) and *σ*(*x*,*t*) are Borel measurable functions. *P*(*t*) is a m-dimensional Poisson process with parameter *λ* = (*λ*_{1},*λ*_{2},…,*λ*_{m})^{T} and *P*(0) = 0 with probability 1.

Now, we introduce the Ito’s formula of the SDE with Poisson process [33–35]:

Let ${v}_{i}^{m}$ be a *m*-dimensional vector with 1 in the *i* th component and zeros elsewhere and {*X*(*t*)} is the solution process of Eq (1), then ∀*V* *C*_{1}(*R*^{k} × *R*_{+}),

$$\begin{array}{l}V\left(X\left(t\right),t\right)=V\left(X\left(s\right),s\right)+{\displaystyle {\int}_{s}^{t}\frac{\partial V\left(X\left(u\right),u\right)}{\partial u}}du+{\displaystyle \sum _{i=1}^{k}{\displaystyle {\int}_{s}^{t}\frac{\partial V\left(X\left(u\right),u\right)}{\partial {x}_{i}}}}{b}_{i}(X\left(u\right),u)du\\ \phantom{\rule{4.5em}{0ex}}+{\displaystyle \sum _{i=1}^{m}{\displaystyle {\int}_{s}^{t}\left[V\left(X\left(u-\right)+\sigma \left(X\left(u-\right),u\right){v}_{i}^{m},u\right)-V\left(X\left(u-\right),u\right)\right]}}d{P}_{i}\left(u\right).\end{array}$$

(5)

Let $\tilde{A}$ be the extended weak infinitesimal operator of the process {*X*(*t*,*X*_{0},*t*_{0})}, and $\forall V\in {C}_{b}^{1}\left({R}^{k}\times {R}_{+}\right)$, we define the operator 𝒟:

$$\begin{array}{l}\mathcal{D}V\left(x,t\right)\cong \frac{\partial V\left(x,t\right)}{\partial t}+{\displaystyle \sum _{i=1}^{k}{b}_{i}\left(x,t\right)}\frac{\partial V\left(x,t\right)}{\partial {x}_{i}}\\ \phantom{\rule{4em}{0ex}}+{\displaystyle \sum _{i=1}^{m}{\lambda}_{i}}\left[V\left(x+\sigma \left(x,t\right){v}_{i}^{m},t\right)-V\left(x,t\right)\right],\end{array}$$

(6)

Now, we introduce the **Global stochastic asymptotic stability theorem**:

Suppose that there exists a function *V*(*x*,*t*) which satisfies the following conditions:

- (C6) $\forall V\left(x,t\right)\in {C}_{1}^{0}\left({R}^{k}\times {R}_{+}\right)$ is a positive definite function with an infinitesimal upper limit,
- (C7) 𝒟
*V*(*x*,*t*), (*x*,*t*) ∈*R*^{k}×*R*_{+}is a negative definite function.

Then, the solution to Eq (4) is global stochastic asymptotically stable [35].

Consider the global coupled dynamical network as follows:

$${\dot{x}}_{i}=f\left({x}_{i}\right)+c{\displaystyle \sum _{i=1}^{n}{a}_{ij}{x}_{j}},i=1,2,\dots n.$$

(7)

*x*_{i} = (*x*_{i1},*x*_{i2}…..*x*_{in})^{T} *R*^{n}(*i* = 1,2) are state vectors, *f* = (*f*_{1},*f*_{2},*f*_{3}……*f*_{n})^{T}: *R*_{n} → *R*_{n} is a nonlinear function describing the dynamic of an isolated node, and *c* is a positive constant which describes the coupling strength. Here, we consider the global coupled dynamical network with *a*_{ii} = −(*n*−1) and *a*_{ii} = 1 (*i* ≠ *j*). In fact, the chaotic systems are usually disturbed by noise. Therefore, for system (7), we consider the following model:

$${\dot{x}}_{i}=f\left({x}_{i}\right)+c{\displaystyle \sum _{i=1}^{n}{a}_{ij}{x}_{j}+}{d}_{i}{\xi}_{i}\left(t\right){\displaystyle \sum _{i=1}^{n}{a}_{ij}{x}_{j}},i=1,2,\dots n$$

(8)

where positive constant *d*_{i} is the noise strength and *ξ*_{1},*ξ*_{2}…*ξ*_{n} are Poisson noises.

Moreover, in order to achieve the theoretical result, we also require the function *f* satisfies the following assumption:

For any *x* = (*x*_{1},*x*_{2}…..*x*_{n})^{T} *R*^{n}, *y* = (*y*_{1},*y*_{2}…..*y*_{n})^{T} *R*^{n}, there exists a positive constant *l* satisfying

(*x*−*y*)^{T}[*f*(*x*, *t*) − *f*(*y*, *t*)] ≤ *l*(*x*−*y*)^{T}(*x* − *y*).

(9)

Assumption 1 is usually called global Lipschitz condition, and *l* is called Lipschitz constant. For the continuous smooth chaotic systems, it is difficult to find constant *l*, such as Rössler system. However, we can find the constant *l* by inequality proof for some well-known piecewise linear chaotic systems, such as the famous Chua’s circuits [41], the cellular neural network (CNN) neural model [42], and so on.

Actually, the global coupled dynamical network (8) are said to achieve complete synchronization, if *x*_{1} = *x*_{2} = = *x*_{n} → *m*(*t*), as *t* → ∞. Here, *m*(*t*) *R*^{n} is called as synchronization manifold and satisfies $\dot{m}=f(m)$. However in this paper, we use the $M\left(t\right)={\displaystyle \sum _{i=1}^{n}\frac{{x}_{i}}{n}}$ instead of synchronization manifold *m*(*t*) [41].

Obviously, *M*(*t*) satisfies the following equation:

$$\dot{M}\left(t\right)=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}f\left({x}_{i}\right)}\triangleq G\left(x\right)$$

(10)

Define the synchronization errors *e*_{i}(*t*) = *x*_{i}(*t*) − *M*(*t*)(*i* = 1,2…*n*), then, one has the error dynamics

$${\dot{e}}_{i}=f\left({x}_{i}\right)-G\left(x\right)-cn{e}_{i}-n{d}_{i}{\xi}_{i}\left(t\right){e}_{i}+{\displaystyle \sum _{i=1}^{n}{d}_{i}{\xi}_{i}\left(t\right){e}_{i}},$$

(11)

Here one should notice that *e*_{i}(*t*)(*i* = 1,2…*n*) satisfy the following condition:

$$\sum _{i=1}^{n}{e}_{i}}=0.$$

(12)

The above Eq (11) can be written as a matrix form

$$\dot{E}\left(t\right)=F(x)+cC\left(E\right)+H\left(E\right)\dot{P}\left(t\right).$$

(13)

Here $E\left(t\right)=\left(\begin{array}{l}{e}_{1}\\ \vdots \\ {e}_{n}\end{array}\right)$, $F(x)=\left(\begin{array}{l}f\left({x}_{1}\right)-G\left(x\right)\\ \vdots \\ f\left({x}_{n}\right)-G\left(x\right)\end{array}\right)$, $C\left(E\right)=c\left(\begin{array}{l}-n{e}_{1}\\ \vdots \\ -n{e}_{n}\end{array}\right)$, $H\left(E\right)=\left(\begin{array}{cccc}\left(1-n\right){d}_{1}{e}_{1}& {d}_{2}{e}_{2}& \dots & {d}_{n}{e}_{n}\\ {d}_{1}{e}_{1}& \left(1-n\right){d}_{2}{e}_{2}& \cdots & {d}_{n}{e}_{n}\\ \vdots & \vdots & \ddots & \vdots \\ {d}_{1}{e}_{1}& {d}_{2}{e}_{2}& \cdots & \left(1-n\right){d}_{n}{e}_{n}\end{array}\right)$, and $\dot{P}(t)=\left(\begin{array}{l}{\xi}_{1}(t)\\ \vdots \\ {\xi}_{n}(t)\end{array}\right)$.

Due to Assumption 1, and the theory of SDE driven by Poisson process [35], one can easily verify that the error Eq (13) possesses a global unique solution denoted by *E*(*t*,*t*_{0},*E*_{0}), for any initial condition. Obviously, *E*(*t*,0,0) 0 is a trivial solution of error system (13). Moreover, after introducing the synchronization errors *e*_{i}(*t*)(*i* = 1,2*n*), the complete synchronization problem of coupled systems (8) can be translated into the stability problem of the trivial solution of system (13), i.e. the complete synchronization of coupled system (8) corresponds to $\underset{t\to \infty}{\mathrm{lim}}\Vert E\left(t\right)\Vert =0$ with probability 1. Here, ‖•‖ stands for Euclidean norm.

In what follows, we will give sufficient conditions for the complete synchronization of coupled system (8) with probability 1.

We show that the error Eq (13) can be written as

(14)

Now, according to the **Global stochastic asymptotic stability theorem** in Part 3, we choose the positive function

(15)

where *E*^{T}(*t*) denotes the transpose of *E*(*t*).

By using the infinitesimal operator of SDE driven by a Poisson process to Eq (15) along with system (14). We have

$$\begin{array}{l}\mathcal{D}V\left(E\left(t\right)\right)=\raisebox{1ex}{$\partial V\left(E\left(t\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\partial E\left(t\right)$}\right.\left[F(x)+cC\left(E\right)\right]+{\displaystyle \sum _{i=1}^{n}{\lambda}_{i}}\left[V\left(E\left(t\right)+H\left(E\right){v}_{i}^{n}\right)-V\left(E\left(t\right)\right)\right]\\ \phantom{\rule{4.25em}{0ex}}=\left(\begin{array}{cccc}{e}_{1}^{T}& {e}_{2}^{T}& \dots & {e}_{n}^{T}\end{array}\right){\left(f\left({x}_{1}\right)-f\left(m\left(t\right)\right),f\left({x}_{2}\right)-f\left(m\left(t\right)\right)\cdots f\left({x}_{n}\right)-f\left(m\left(t\right)\right)\right)}^{T}\\ \phantom{\rule{4.25em}{0ex}}+c\left(\begin{array}{cccc}{e}_{1}^{T}& {e}_{2}^{T}& \dots & {e}_{n}^{T}\end{array}\right){\left(-n{e}_{1},-n{e}_{2}\cdots -n{e}_{n}\right)}^{T}\\ \phantom{\rule{4.25em}{0ex}}+{\displaystyle \sum _{i=1}^{n}{\lambda}_{i}}\left[V\left(E\left(t\right)+H\left(E\right){v}_{i}^{n}\right)-V\left(E\left(t\right)\right)\right]\\ \phantom{\rule{4.25em}{0ex}}={\displaystyle \sum _{i=1}^{n}{e}_{i}^{T}}\left[f\left({x}_{i}\right)-f\left(m\left(t\right)\right)\right]-nc{\displaystyle \sum _{i=1}^{n}{e}_{i}^{T}{e}_{i}}\\ \phantom{\rule{4.25em}{0ex}}+\frac{1}{2}{\displaystyle \sum _{i=1}^{n}{\lambda}_{i}}\left[{\left(E\left(t\right)+H\left(E\right){v}_{i}^{n}\right)}^{T}\left(E\left(t\right)+H\left(E\right){v}_{i}^{n}\right)-{E}^{T}\left(t\right)E\left(t\right)\right]\\ \phantom{\rule{4.25em}{0ex}}\le \left(l-nc\right){\displaystyle \sum _{i=1}^{n}{e}_{i}^{T}{e}_{i}}+\frac{1}{2}{\displaystyle \sum _{i=1}^{n}{\lambda}_{i}\left[\left({n}^{2}-n\right){d}_{i}^{2}-2n{d}_{i}\right]}{e}_{i}^{T}{e}_{i}.\end{array}$$

(16)

Obviously, if the coupling strength *c*, noise strength *d*_{i}, and constants *l*, *λ*, satisfy the inequality

$$\left(l-nc\right)+\frac{1}{2}{\lambda}_{i}\left[\left({n}^{2}-n\right){d}_{i}^{2}-2n{d}_{i}\right]<0,i=1,\mathrm{2\dots}n,$$

(17)

then due to the **Global stochastic asymptotic stability theorem** in Part 3, the trivial solution of system (13) is global stochastic asymptotically stable, and then the synchronization errors *e*_{i}(*i* = 1,2) converge to zero as *t* → ∞ with probability 1.

We can find that Poisson noise really has a positive effect on the complete synchronization. In the case of the noise strengths *d*_{i} = 0, the network become synchronized when *l* − *nc* < 0. From this we can see that there exists a value, all nodes of the network become synchronized when *nc* exceeds this value. In the case of the coupling strength *c* = 0, i.e., the network is only coupled by the internal noise, the whole network become synchronized as

$$l+\frac{1}{2}{\lambda}_{i}\left[\left({n}^{2}-n\right){d}_{i}^{2}-2n{d}_{i}\right]<0,i=1,\mathrm{2\dots}n.$$

(18)

We can see that the noise really have a positive effect on the synchronization. Moreover, the synchronization can be achieved by adding node number, if the global dynamical network (8) is not synchronized under fixed noise strength and coupling strength *c* ≠ 0.

In terms of the above analysis, we can easily get that the coupled system (8) can achieve complete synchronization with probability 1, when the noise strengths

$$\begin{array}{l}{d}_{i}<\frac{n{\lambda}_{i}+\sqrt{{n}^{2}{\lambda}_{i}{}^{2}-2{\lambda}_{i}\left({n}^{2}-n\right)\left(l-nc\right)}}{{\lambda}_{i}\left({n}^{2}-n\right)}\\ \phantom{\rule{1em}{0ex}}=\frac{n}{\left({n}^{2}-n\right)}+\sqrt{\frac{{n}^{2}}{{\left({n}^{2}-n\right)}^{2}}-\frac{2\left(l-nc\right)}{{\lambda}_{i}\left({n}^{2}-n\right)}}\\ \phantom{\rule{1em}{0ex}}\le \frac{2n}{\left({n}^{2}-n\right)}\le 2,\end{array}$$

(19)

and ${\lambda}_{i}\ge \frac{2\left({n}^{2}-n\right)\left(l-nc\right)}{{n}^{2}}$.

In this section, the example is provided and some numerical simulations are performed to verify the theoretical results. This part, we use Runge-Kutta methods to measure the synchronization, let *n* = 2, *d*_{1} = *d*_{2} = *d* in system (8), and we define the following quantities: error(*t*) = |*x*_{1} − *x*_{2}|, and *x*_{i} = (*x*_{i1},*x*_{i2}…..*x*_{in})^{T}, *i* = 1,2 represents the solution of Eq (8).

In this example we use the Chua’s circuits [41] which can be depicted by three-dimensional differential equation

$$\dot{x}=Dx+Tg(x),$$

(20)

where *x* = (*x*_{1},*x*_{2},*x*_{3})^{T} *R*^{3} is the state vector,

$$D=\left(\begin{array}{ccc}-a& a& 0\\ b& -b& c\\ 0& -h& 0\end{array}\right),T=\left(\begin{array}{ccc}-a& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),g\left(x\right)={\left(g\left({x}_{1}\right),g\left({x}_{2}\right),g\left({x}_{3}\right)\right)}^{T},$$

(21)

in which *a* = *G*/*C*_{1},*h* = *G*/*C*_{2},*c* = 1/*C*_{2},*d* = 1/*L*,*G* is resistance, *C*_{1},*C*_{2} are capacitors, *L* is the inductor, *x*_{1},*x*_{2} denote the volt-age across *C*_{1},*C*_{2}, respectively, *x*_{3} is the current through *L*. In particular, the static nonlinearity of Chua’s diode is the piecewise linear curve given by

(22)

where *m*_{0},*m*_{1},*B* are the parameters.

On the basis of the Ref. 36, under the above parameters system (20) is chaotic (Fig 1).

We can verify

$$\begin{array}{l}\left|g\left({x}_{1}\right)-g\left({y}_{1}\right)\right|=\left|{m}_{0}\left({x}_{1}-{y}_{1}\right)+\frac{1}{2}\left({m}_{1}-{m}_{0}\right)\left[\left(\left|{x}_{1}+B\right|-\left|{x}_{1}-B\right|\right)-\left(\left|{y}_{1}+B\right|-\left|{y}_{1}-B\right|\right)\right]\right|\\ \phantom{\rule{6.75em}{0ex}}\le \left(\left|{m}_{0}\right|+\left|{m}_{1}-{m}_{0}\right|\right)\left|{x}_{1}-{y}_{1}\right|.\end{array}$$

(23)

For any *x* *R*^{3}, *y* *R*^{3}, we have

$$\begin{array}{l}{\left(x-y\right)}^{T}\left(Dx-Dy+Tg(x)-Tg(y)\right)\le {\left(x-y\right)}^{T}D\left(x-y\right)+\left|{\left(x-y\right)}^{T}T\left(g(x)-g(y)\right)\right|\\ \phantom{\rule{15.5em}{0ex}}\le {\left|\left(x-y\right)\right|}^{T}\left[\left(\left|{m}_{0}\right|+\left|{m}_{1}-{m}_{0}\right|\right)\left|T\right|+D\right]\left|\left(x-y\right)\right|\\ \phantom{\rule{15.5em}{0ex}}\le {\lambda}_{m}{\left(x-y\right)}^{T}\left(x-y\right).\end{array}$$

(24)

Take the parameters and noise intensity matrices as

(25)

where

$$\left|T\right|=\left(\begin{array}{ccc}7& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),D=\left(\begin{array}{ccc}-7& 7& 0\\ 0.35& -0.35& 0.5\\ 0& -7& 0\end{array}\right),$$

and *λ*_{m} ≈ 33.0731 is the maximum eigenvalue of |*T*| − *D*. So we can acquire the constant *l* ≈ 33.0731 in Assumption 1.

By the theoretical results in Sec. 4, *λ*_{1} = 30, *λ*_{2} = 50, the two coupled systems are synchronized with probability 1 with the noise strengths 0.5838 < *d* < 1.416. Therefore, we compute the error(*t*) with different *d* to find the agreement with the theoretical results.

The simulation results are shown in Fig 2 with *d* = 0.7, and Fig 3 with *d* = 1.3.

In terms of the simulation results, we can see that error(*t*) converges to zero with time increasing. The individual systems achieve complete synchronization because the coupled term vanishes. Some chaotic attractors after synchronization achievement are shown in Fig 4.

Unlike the Brown process whose almost all sample paths are continuous, the Poisson process is a jump process and has the sample paths which are right-continuous and have left limits. Therefore, it should be pointed out that there is a great difference between the stochastic integral with respect to the Brown process and the one with respect to the Poisson process.

In this paper, we apply the stability theory of SDE driven by Poisson process to study the complete synchronization of the global coupled dynamical network perturbed by different Poisson noises, and sufficient conditions of the complete synchronization with probability 1 are established. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach. This paper presents a globally stochastic asymptotic synchronization criterion for complex networks perturbed by the Poisson noise. We conclude that Poisson noise can induce the complete synchronization in the global coupled dynamical network on actual situations.

The authors received no specific funding for this work.

Data Availability

All relevant data are within the paper and its Supporting Information files.

1. Lai C., Zhou C.. Synchronization of chaotic maps by symmetric common noise. Europhysics. Lett. 1998; 43: 376–380. doi: 10.1209/epl/i1998-00368-1

2. Pikovsky A. S.. Comment on "Chaos, Noise, and Synchronization". Phys. Rev. Lett. 1994; 73: 29–31. [PubMed]

3. Herzel H., Freund J.. Chaos, noise, and synchronization reconsidered. Physical Review E. 1995; 52: 3238–3241. doi: 10.1103/PhysRevE.52.3238 [PubMed]

4. Lin W., Chen G.. Using white noise to enhance synchronization of coupled chaotic systems. Chaos. 2006; 16, 013134
doi: 10.1063/1.2183734
[PubMed]

5. Lin W., He Y.. Complete Synchronization of the noise-perturbed chua’s circuits. Chaos. 2005; 15, 023705
doi: 10.1063/1.1938627
[PubMed]

6. Xiao Y., Xu W., Li X., Tang S.. The effect of noise on the complete synchronization of two bidirectionally coupled piecewise linear chaotic systems. Chaos. 2009; 19, 013131
doi: 10.1063/1.3080194
[PubMed]

7. Xiao Y., Tang S., Xu Y.. Theoretical analysis of multiplicative-noise-induced complete synchronization in global coupled dynamical network. Chaos. 2012; 22, 013110
doi: 10.1063/1.3677253
[PubMed]

8. Cao J., Wang Z., Sun Y.. Synchronization in an array of linearly stochastically coupled networks with time delays. Physica A. 2007; 385: 718–728.

9. Wang X.. Complex networks: topology, dynamics and synchronization. International Journal of Bifurcation and Chaos. 2002; 12: 885–916. doi: 10.1142/ S0218127402004802

10. Wang X.. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 2002; 49: 54–62.

11. Zhang X., Liu F, Wang W. Two-phase dynamics of p53 in the DNA damage response. Proceedings of the National Academy of Sciences of the United States of America. 2011; 108: 8990–8995. doi: 10.1073/pnas.1100600108
[PubMed]

12. Zhang X., Liu F., Cheng Z, Wang W.. Cell fate decision mediated by p53 pulses. Proceedings of the National Academy of Sciences of the United States of America. 2009; 106: 12245–12250. doi: 10.1073/pnas.0813088106
[PubMed]

13. Wang S., Wang W., Liu F.. Propagation of firing rate in a feed-forward neuronal network. Physical Review Letters. 2006; 96, 018103
doi: 10.1103/PhysRevLett.96.018103
[PubMed]

14. Ma J., Tang J.. A review for dynamics of collective behaviors of network of neurons. Science China Technological Sciences. 2015; 58: 2038–2045.

15. Qin H., Ma J., Jin W., Wang C.. Dynamics of electric activities in neuron and neurons of network induced by autapses. Science China Technological Sciences. 2014; 57: 936–946. doi: 10.1007/s11431-014-5534-0

16. Lv M., Wang C., Ren G., Ma J., Song X.. Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dynamics. 2016; 85: 1479–1490. doi: 10.1007/ s11071-016-2773-6

17. Shiraki Y., Kabashima Y.. Cavity analysis on the robustness of random networks against targeted attacks: Influences of degree-degree correlations. Physical Review E. 2010; 82, 036101
doi: 10.1103/PhysRevE.82.036101
[PubMed]

18. Qian Y.. Emergence of self-sustained oscillations in excitable Erdos-Renyi random networks. Physical Review E. 2014; 90, 032807
doi: 10.1103/PhysRevE.90.032807
[PubMed]

19. Yu Q., Cui X., Zheng Z.. Minimum Winfree loop determines self-sustained oscillations in excitable Erdös-Rényi random networks. Scientific Reports. 2017; 7, 5746
doi: 10.1038/s41598-017-06066-6
[PMC free article] [PubMed]

20. Qian Y., Liu F., Yang K., Zhang G., Yao C., Ma J.. Spatiotemporal dynamics in excitable homogeneous random networks composed of periodically self-sustained oscillation. Scientific Reports. 2017; 7, 11885
doi: 10.1038/s41598-017-12333-3
[PMC free article] [PubMed]

21. Xu Y., Gu R., Zhang H., Xu W., Duan J.. Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Physical Review E. 2011; 83, 056215
doi: 10.1103/PhysRevE.83.056215
[PubMed]

22. Gihman I., Skorokhod A.V.. Stochastic Differential Equations. Springer-Verlag, Berlin, 1972.

23. Zeng C., Wang H.. Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems. Ecological Modelling. 2012; 233: 52–58. doi: 10.1016/ j.ecolmodel. 2012.03.025

24. Han Q., Yang T., Zeng C., Wang H., Liu Z., Fu Y., et al.
Impact of time delays on stochastic resonance in an ecological system describing vegetation. Physica A. 2014; 408: 96–105. doi: 10.1016/j.physa.2014.04.015

25. Zeng C., Han Q., Yang T., Wang H., Jia Z.. Noise-and delay-induced regime shifts in an ecological system of vegetation. Journal of Statistical Mechanics Theory & Experiment. 2013; 10, P10017
doi: 10.1088/1742-5468/2013/03/P03003

26. Zeng C., Zhang C., Zeng J., Luo H., Zhang H., Long F., et al.
Noises-induced regime shifts and -enhanced stability under a model of lake approaching eutrophication. Ecological Complexity. 2015; 22:102–108. doi: 10.1016/j.ecocom.2015.02.005

27. Samhouri J. F., Andrews K. S., Fay G., Harvey C. J., Hazen E. L., Hennessey S. M., et al.
Defining ecosystem thresholds for human activities and environmental pressures in the California Current. Ecosphere. 2017; 8, e01860
doi: 10.1002/ ecs2.1860

28. Zeng J., Zeng C., Xie Q., Guan L., Dong X., Yang F.. Different delays-induced regime shifts in a stochastic insect outbreak dynamics. Physica A. 2016; 462: 1273–1285. doi: 10.1016/j.physa.2016.06.115

29. Xu Y., Feng J., Li J., Zhang H.. Stochastic bifurcation for a tumor-immune system with symmetric Lévy noise, Physica A. 2013; 392: 4739–4748. doi: 10.1016/ j.physa.2013.06.010

30. Xu Y., Feng J., Li J.J., Zhang H.Q.. Lévy noise induced switch in the gene transcriptional regulatory system. Chaos. 2013; 23, 013110
doi: 10.1063/1.4775758
[PubMed]

31. Wu J., Xu Y., Ma J.. Lévy noise improves the electrical activity in a neuron under electromagnetic radiation. PLoS ONE. 2017; 12, e0174330
doi: 10.1371/journal.pone.0174330
[PMC free article] [PubMed]

32. Xu Y., Pei B., Li Y.. Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise. Mathematical Methods in Applications and Science. 2015; 38: 2120–2131. doi: 10.1002/mma.3208

33. Garcia M. A., Griego R. J.. An elementary theory of stochastic differential equation by a poisson process, Commun. Statist. Stochastic Models. 1994, 10 (2), 335–336.

34. Kushner H.J.. Stochastic Stability and Control. Academic Press, New York, 1967.

35. Ting Y.. The stability theory of stochastic differential equation driven by a poisson process. Soochow Journal of Mathematics. 1999; 25:145–165.

36. Gao Z., Zhang Y.. Limit theorems for a supercritical Poisson random indexed branching process. Journal of Applied Probability. 2016; 53: 307–314. doi: 10.1017/ jpr.2015.27

37. Yin C., Yuen K.. Optimality of the threshold dividend strategy for the compound Poisson model. Statistics & Probability Letters. 2011; 81: 1841–1846. doi: 10.1016/ j.spl.2011.07.022

38. Yin C., Wang C. The perturbed compound Poisson risk process with investment and debit Interest. Methodology and Computing in Applied Probability. 2010; 12: 391–413. doi: 10.1007/s11009-008-9109-z

39. Yin C., Wen Y., Zhao Y.. On the optimal dividend problem for a spectrally positive Lévy process. ASTIN Bulletin. 2014; 44: 635–651. doi: 10.1017/asb.2014.12

40. Yin C., Yuen K. Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory. Front. Math. China. 2014; 9: 1453–1471.

41. Chua L. O., Wu C. W., Huang A., Zhong G.. A universal circuit for studying and generating chaos. I. Routes to chaos, IEEE Trans. Circuits Syst, I: Fundam. Theory Appl. 1993; 40: 732–744. doi: 10.1109/81.246149

42. Zou F., Nossek J. A.. Bifurcation and chaos in cellular neural networks. IEEE Trans. Circuits Syst, I: Fundam. Theory Appl. 1993; 40: 166–173. doi: 10.1109/81.222797

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