The existence of inner synchronized stationary distribution for stochastic coupled systems on networks

Journal Pre-proof The existence of inner synchronized stationary distribution for stochastic coupled systems on networks Sen Li, Huan Su, Xiaohua Ding

PII: DOI: Reference:

S0378-4371(19)31608-5 https://doi.org/10.1016/j.physa.2019.122828 PHYSA 122828

To appear in:

Physica A

Received date : 1 June 2019 Revised date : 29 August 2019 Please cite this article as: S. Li, H. Su and X. Ding, The existence of inner synchronized stationary distribution for stochastic coupled systems on networks, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.122828. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

Journal Pre-proof Highlights

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1. The model we consider is special that there is an ideal subsystem that free from stochastic perturbation and others experience stochastic perturbations that have no relationship with their states. 2. The existence of inner synchronized stationary distribution for stochastic coupled systems is studied in theory for the first time based on the theory of the existence for stationary distribution. 3. Different from the existing research methods on inner synchronization problem (that are Kronecker product method and linear matrix inequality), we investigate the existence of inner synchronized stationary distribution by graph theory and Lyapunov method.

*Manuscript Click here to view linked References

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The existence of inner synchronized stationary distribution for stochastic coupled systems on networks Sen Li, Huan Su∗, Xiaohua Ding

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Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China

Abstract

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This paper is concerned with the existence of inner synchronized stationary distribution for stochastic coupled systems on networks (SCSNs), which is the first time to consider this problem. Compared with the existing results on inner synchronization problem, we study the problem based on Lyapunov method and Kirchhoff’s Matrix Tree Theorem in graph theory without utilizing Kronecker product method and Linear matrix inequalities, which simplifies some complex analysis and avoids difficulties. Then some new sufficient conditions are presented to guarantee the existence of inner synchronized stationary distribution for SCSNs. These conditions show that the existence domain of inner synchronized stationary distribution has a close relationship with stochastic perturbation intensity. And when stochastic perturbation vanishes, inner synchronized stationary distribution will become complete synchronization. To illustrate the practicability of theoretical results, an application about stochastic coupled oscillators is given with a numerical example being carried out. Keywords: inner synchronized stationary distribution; stochastic coupled systems; graph theory; Lyapunov method; stochastic coupled oscillators

1. Introduction

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During the past decades, stochastic coupled systems on networks (SCSNs) have received much attention since they can be widely applied in various domains encompassing biology, physics, engineering and so forth (see [1, 2, 3, 4, 5, 6, 7]). Up to now, there have been a multitude of researching results on SCSNs concerning the issues of stability [8, 9, 10], synchronization [11, 12, 13] and periodicity [14]. Particularly, synchronization, as an important topic due to wide applications in secure communication, cryptography and so on, has attracted the attention of many scholars. And numerous significant results have been obtained. For example, in [15], Li et al. reported the adaptive pinning synchronization problem of stochastic complex dynamical networks based on algebraic graph theory and Lyapunov method. In [16], synchronization of stochastic coupled systems via intermittent control was investigated in terms of discrete-time state observations. Via constructing a suitable Lyapunov-Krasovskii functional containing some novel triple integral terms with sufficient information about the actual sampling pattern, the sampled-data synchronized control problem for complex dynamical networks with time-varying coupling delay was studied in [17]. It should be pointed out that the above-mentioned results on the stochastic synchronization are mainly concentrated on stochastic perturbation intensity that related to the state of error system. However, this does not Corresponding author. Tel.: +86 0631 5678533; fax: +86 0631 5687572. Email address: [email protected] (Huan Su)

Preprint submitted to Elsevier

August 29, 2019

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always happen in our real life. In some situations, systems may experience some stochastic external inputs and the intensity of perturbation has no relationship with the state of error system, that is, stochastic perturbation intensity is a constant. For example, in power systems, it is unavoidable to encounter bad weather, which will cause the situation where the external disturbances have no relationship with the state of power systems. Due to this kind of stochastic perturbation, the frequency always fluctuates near a fixed value. In this case, it is very difficult for the power systems to achieve complete frequency synchronization. Acebron et. al investigated synchronized stationary states in the Kuramoto model for oscillator populations in [18], which said that synchronization error fluctuates around zero stationarily because that stochastic perturbations destroy the equilibrium for error system. Although synchronized stationary distribution has been studied in the physics model, to the best of our knowledge, until now, there are few results on synchronized stationary distribution of SCSNs. In most of literature, for example [15, 16, 17], synchronization is realized by studying the stability of the equilibrium for error system. Similarly, synchronized stationary distribution of SCSNs can be investigated in virtue of researching the existence of stationary distribution of error system. Recently, the existence of stationary distribution that can be seen as a weak stability state has drawn increasing attention from researchers in different areas [19]. And a large number of important results have been reported. Specific results are as follows: in [20, 21], Zou et al. presented the existence of stationary distribution for a noisy predator-prey system both theoretically and numerically; in [22, 23, 24], the authors investigated stationary distribution of stochastic epidemic models, stochastic ecological models and stochastic generalized logistic system by Lyapunov method, respectively; in [25], Mao et al. discussed stationary distribution of the solution for a population model in the form of stochastic differential equation; in [26], Zhao et al. studied stationary distribution of a stochastic phytoplankton allelopathy model under regime switching with Hasminskii’s method and Lyapunov method; in [27, 28, 29], Huang et al. investigated existence, non-existence and degenerate diffusion of steady states of Fokker-Planck equations by using level set method and Lyapunov method especially integral identity and measure estimates; then based on [27], the existence of stationary distribution for two class of SCSNs was studied in [30] and [31]. Motivated by above discussions, in this paper, we will devote to studying the existence of inner synchronized stationary distribution for SCSNs. Based on the results of graph theory, Lyapunov method and stochastic analysis techniques, the existence conditions of synchronized stationary distribution for a class of SCSNs are derived by means of studying stationary distribution of error system. It should be stressed that the existence domain has a close relationship with the stochastic perturbation intensity and when the perturbation intensity tends to zero, the considered system will achieve complete synchronization. Then, we apply the theoretical results to a stochastic coupled oscillators model in physics with a numerical simulation carried out to illustrate the effectiveness of the theoretical results. The differences of this paper compared with the previous results are that: • The model we consider is special that there is an ideal subsystem that free from stochastic perturbation and others experience stochastic perturbations that have no relationship with their states. • The existence of inner synchronized stationary distribution for SCSNs is studied in theory for the first time based on the theory of the existence for stationary distribution in [27]. • Different from the existing research methods on inner synchronization problem (that are Kronecker product method and linear matrix inequality, see [32, 33, 34]), we firstly investigate the existence of inner synchronized stationary distribution by graph theory and Lyapunov method. The rest of this paper is arranged as follows. Section 2 presents some relative preliminaries and model descriptions. In Section 3, inner synchronized stationary distribution of SCSNs is realized with sufficient conditions being given. Then Section 4 is devoted to the results of inner synchronized stationary distribution for stochastic 2

Journal Pre-proof coupled oscillators. And a corresponding numerical simulation is exploited to show the effectiveness of the theoretical results acquired in this paper in Section 5. Also, a conclusion of this paper is presented in Section 6. 2. Preliminaries and model descriptions

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The notations used throughout this paper are fairly standard. For the brevity of the following analysis, we denote Rn by the n-dimensional Euclidean space. Write | · | for the Euclidean norm of vectors or the trace norm of matrices. Let (Ω, F , F, P) be a complete probability space with a filtration F = {Ft }t≥0 satisfying the usual conditions. The superscript “T” stands for the transpose of a vector or a matrix. Moreover, the notations L = {1, 2, . . . , N} and R+ = (0, +∞) are used. And C 2 (Rn ) represents the family of all nonnegative functions V(X) on Rn which are continuously twice differentiable about X. Here, we introduce some useful concepts associated with a digraph. Let H = {A, E} be a digraph, where A = {1, 2, . . . , n} is the vertex set and E is the arc set of digraph H. A digraph is called strongly connected if for any two distinct vertices j and k in the digraph, there exists a path from vertex j to vertex k. Define the weight matrix of H as Λ = (˜η jk )n×n , where η˜ jk > 0 if there exists an arc from vertex k to vertex j, and 0 otherwise. Digraph H with weight matrix Λ is denoted by (H, Λ). The Laplacian matrix of (H, Λ) is defined P as L(H) = (p jk )n×n , where p jk = k, j η˜ jk for j = k and p jk = −˜η jk for j , k. For more details on graph theory, we refer readers to [35]. In this paper, we consider the following stochastic coupled systems (SCSs) with N (N ≥ 2) subsystems:  N   X      dxi (t) = f (xi (t)) + aik Hik (xi (t) − xk (t)) + I(t) dt,      k=1 (1)   N    X     a jk H jk (x j (t) − xk (t)) + I(t) dt + σ j dB(t), j ∈ L, j , i,   dx j (t) = f (x j (t)) + 

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where i ∈ L is a fixed subsystem which can be seen as a ideal subsystem that free from stochastic perturbation, xi (t), x j (t) ∈ Rm are the state variables, f (·) : Rm → Rm is a continuous vector-valued function o describing n (1) (2) (m) m the dynamics of subsystem, I(t) ∈ R is an external input vector, σ j = diag σ j , σ j , . . . , σ j Rm×m denotes the constant perturbation matrix of subsystem j, where σ(h) j , 0 for h = 1, 2, . . . , m, B(t) is an m-dimensional Brownian motion defined on the probability space, Hik (·), H jk (·) : Rm → Rm are the coupling functions, aik , a jk ≥ 0 indicate the coupling strength.

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Remark 1. The coupling function in the form of H jk (x j − xk ) is universal, see [36, 37, 38, 39] and the references therein. Especially, what is noticeable is that the overwhelming majority of this coupling form appeared in inner synchronization problem is mainly linear, which is disposed of by Kronecker product method [32, 33, 34]. Nevertheless, in the inner synchronization problem we consider, i.e., inner synchronized stationary distribution, the coupling function H jk (x j − xk ) can be nonlinear, which increases generality. In order to investigate the existence of inner synchronized stationary distribution for system (1), denote X ji (t) = x j (t) − xi (t) and F(X ji (t)) = f (x j (t)) − f (xi (t)). Then we get the error dynamics as follows: N N   X X dX ji (t) = F(X ji (t)) + a jk H jk (X jk (t)) − aik Hik (Xik (t)) dt + σ j dB(t), k=1

k=1

3

j ∈ L, j , i.

(2)

Journal Pre-proof We describe system (2) on digraph (H, Λ) with N − 1 vertices and each vertex represents the dynamics of X ji , for any fixed i ∈ L, j ∈ L, j , i, Λ = (˜η jk )(N−1)×(N−1) is the weight matrix and η˜ jk is the weight of arc in H from Xki to X ji . To get the main results in a more clear way, let   T T T T X(t) = X1iT , . . . , Xi−1,i , Xi+1,i , . . . , XNi ∈ Rm(N−1) , G = diag{σ1, . . . , σi−1 , σi+1 , . . . , σN } ∈ Rm(N−1)×m(N−1) ,

¯ dX(t) = F(X)dt + GdB(t). T

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 T  T T T ¯ B(t) = BT1 (t), . . . , BTi−1 (t), BTi+1 (t), . . . , BTN (t) , F(X) = F 1T (X), . . . , F i−1 (X), F i+1 (X), . . . , F NT (X) , PN in which F l (X) = F(X ji ) + k=1 (a jk H jk (X jk ) − aik Hik (Xik )). It is clear that B(t) is an m(N − 1)-dimensional Brownian motion defined on the probability space. Then system (2) can be rewritten as

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Moreover, we set D = (d sr )m(N−1)×m(N−1) = GG2 as the diffusion matrix, which is everywhere positive definite since each diagonal element of matrix G is not zero. It is claimed that inner synchronized stationary distribution of system (1) is achieved, if error system (2) exists a stationary distribution. Next, we introduce the following definitions which will be used later.

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Definition 1. [27] An unbounded, non-negative and continuous function V(x) is called a compact function in Rm(N−1) iff lim V(x) = +∞. x→∂Rm(N−1)

Definition 2. [27] Let V(X) be a C 2 compact function in Rm(N−1) where X = (X1 , X2 , . . . , Xm(N−1) )T . V(X) is called a Lyapunov function in Rm(N−1) with respect to L∗ , if there is a ρ ∈ R+ , and a constant γ > 0, called Lyapunov constant of V(X), such that L∗ V(X) ≤ −γ, X ∈ Rm(N−1) \Ωρ ,

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n o P where Ωρ = X ∈ Rm(N−1) : V(X) ≤ ρ , L∗ = d sr ∂2sr + F¯ (s) ∂ s , ∂ sr = m(N−1) s,r=1 ¯ element of F(X) for r = 1, 2, . . . , m(N − 1).

∂s =

Pm(N−1) s=1

∂ , ∂X s

F¯ (r) is the r-th

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3. Main results

∂2 , ∂X s ∂Xr

In this section, we shall give several sufficient conditions about the existence of inner synchronized stationary distribution for system (1). At the beginning, we show an assumption that helps us carry out the conclusion. 1,p p Assumption 1. [27] d sr ∈ Wloc (Rm(N−1) ), F¯ (r) ∈ Lloc (Rm(N−1) ) for any s, r = 1, 2, . . . , m(N − 1), where p > m(N − 1) is fixed.

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For more symbol descriptions and details, we refer readers to [27]. Remark 2. According to literature [27], we can see that it is a typical assumption in the stationary distribution analysis. This assumption can be satisfied for many functions, such as polynomial functions, sine (cosine) functions, and so on. In fact, if functions F, H jk and Hik in system (2) are continuously once differentiable about variables, Assumption 1 will be satisfied.

4

Journal Pre-proof Theorem 1. Let Assumption 1 hold. For any fixed i ∈ L, j ∈ L, j , i, if there exists a function V ji (x) ∈ C 2 (Rm ) such that the following conditions meet, A1. Digraph (H, Λ) is strongly connected, where Λ = (˜η jk )(N−1)×(N−1) . A2. There exist positive constants θ ji and ϑ ji such that θ ji |x|2 ≤ V ji (x) ≤ ϑ ji |x|2 .

∗

N X

2

L V ji (X ji ) ≤ −β ji |X ji | +

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A3. There are positive constants β ji , m j and a function M ji (X ji ) satisfying η˜ jk (Mki (Xki ) − M ji (X ji )) + m j ,

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for X ji ∈ Rm \ Ωρ ji , where ρ ji ∈ R+ is related to m j , Ωρ ji = {X ji ∈ Rm : V ji (X ji ) ≤ ρ ji }. then system (1) admits a inner synchronized stationary distribution. Proof. For any fixed i ∈ L, let V(X) =

N X

γ ji V ji (X ji ),

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j=1, j,i

where γ ji is the cofactor of the s-th (s = j for j < i and s = j − 1 for j > i) diagonal element of L(H). The strong connectedness of digraph H guarantees γ ji > 0. By condition A2, we can get V(X) ≥ and

j=1, j,i

N X

j=1, j,i

γ ji θ ji |X ji |2 ≥ min {γ ji θ ji }|X|2 j∈L, j,i

γ ji ϑ ji |X ji |2 ≤ max {γ ji ϑ ji }|X|2 . j∈L, j,i

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V(X) ≤

N X

Hence,

S 1 |X|2 ≤ V(X) ≤ S 2 |X|2 ,

L V(X) =

N X

∗

j=1, j,i

γ ji L V ji (X ji ) ≤ −

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where S 1 = min j∈L, j,i {γ ji θ ji } > 0, S 2 = max j∈L, j,i {γ ji ϑ ji } > 0. Moreover, considering V(X) ≥ S 1 |X|2 , we can get limX→∂Rm(N−1) V(X) = +∞, which means V(X) is a compact function according to Definition 1. By condition A3, one can make the following calculation N X

j=1, j,i

2

γ ji β ji |X ji | +

N N X X

j=1, j,i k=1,k,i

γ ji η˜ jk (Mki (Xki ) − M ji (X ji )) +

By the Kirchhoff’s Matrix Tree Theorem (Theorem 2.2 in literature [35]), we can get N N X X

j=1, j,i k=1,k,i

γ ji η˜ jk (Mki (Xki ) − M ji (X ji )) = 0.

5

N X

j=1, j,i

γ ji m j .

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L∗ V(X) ≤ − where σ = min j∈L, j,i {γ ji β ji }, C = Since

PN

N X

j=1, j,i

j=1, j,i

we have |X| >

ρ , S2

which implies ρ >

j=1, j,i

γ ji m j ≤ −σ|X|2 + C,

γ ji m j .

S 2 |X|2 ≥ V(X) = 2

N X

γ ji β ji |X ji |2 +

CS 2 . σ

N X

γ ji V ji (X ji ) >

j=1, j,i

N X

γ ji ρ ji , ρ,

j=1, j,i

Letting γ =

σρ S2

− C > 0, we can obtain

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Therefore, we have

L∗ V(X) ≤ −γ, X ∈ Rm(N−1) \ Ωρ , Ωρ = {X ∈ Rm(N−1) : V(X) ≤ ρ}.

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So V(X) is a Lyapunov function in Rm(N−1) with respect to L∗ of system (2) by Definition 2. From the above computation, we verify all the conditions of Theorem A in literature [27], which can guarantee the existence of stationary distribution for error system (2). Subsequently, we can see that system (1) admits a inner synchronized stationary distribution. It completes the proof.

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Remark 3. In the real world, because of the effects of stochastic factors, complete synchronization of system can be destructed, which will come into being a weaker form of synchronization compared with the general meaning of synchronization in most of literature. That is referred to as synchronized stationary distribution. Synchronized stationary distribution phenomenon clearly elucidates that the solution of error system undulates around zero. In [18], synchronized stationary state on a class of simple kuramoto model in physics were studied. However, to the best of our knowledge, on account of analysis difficulty, synchronized stationary distribution for relatively complex systems that further simulate reality, such as stochastic coupled systems, was rarely investigated. Hence, our results build a platform for the issue of synchronized stationary distribution analysis for complex dynamical systems.

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Remark 4. What is noteworthy is that the modality of Lyapunov function for error system (2) is V(X) = PN j=1, j,i γ ji V ji (X ji ), which has an intimate relationship with the topological structure of digraph H. This construction of Lyapunov function was first proposed by Li et al. in [35], then it was successfully used in [30]. The summing coefficients γ ji are useful for us to cope with the coupling item. Moreover, to the best of our knowledge, most existing results that studied inner synchronization problem were based on Kronecker product method and Linear matrix inequalities, see [32, 33, 34] and the references therein. This paper firstly investigates inner synchronization problem (inner synchronized stationary distribution) via graph theory without using Kronecker product method.

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Remark 5. It has been recognized that in condition A3, constant m j is closely related to perturbation matrix σ j , and so as ρ ji . In fact, the value of ρ ji determines the existence domain of inner synchronized stationary distribution. Moreover, in the proof of Theorem 1, we can see that if the perturbation matrix is a zero matrix, then C = 0, which implies that ρ = 0. Then condition A3 will become L∗ V ji (X ji ) ≤ −β ji |X ji |2 +

N X

k=1,k,i

e η jk (Mki (Xki ) − M ji (X ji )),

which is satisfied for all X ji . From the proof process, we can get L∗ V(X) ≤ 0. This shows that system (1) achieves complete synchronization. 6

Journal Pre-proof Theorem 1 is called Lyapunov-type theorem whose conditions are related to vertex Lyapunov functions V ji . However, we can not promise that this kind of function exists. Hence, on the basis of Theorem 1, we give a coefficients-type theorem that the conditions depend on the coefficients of error system. Theorem 2. For any fixed i ∈ L, X ji ∈ Rm \ Ωρ ji , ρ ji ∈ R+ , Ωρ ji = {X ji ∈ Rm : V ji (X ji ) ≤ ρ ji }, if conditions B1-B3 in the following are satisfied and Assumption 1 holds, then system (1) admits a inner synchronized stationary distribution. B1. There exist positive constants α ji and P jk such that

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X Tji F(X ji ) ≤ −α ji |X ji |2 , j ∈ L, j , i and |H jk (X jk )| ≤ P jk |X jk |, j, k ∈ L. B2. The inequality below is satisfied,

k=1,k,i

 a jk + aik + 4a jk P2jk + aik P2ik − 2a ji P ji > 0,

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2α ji −

N X 

j ∈ L, j , i.

B3. Digraph (H, Λ) is strongly connected, where Λ = (˜η jk )(N−1)×(N−1) , η˜ jk = 2a jk P2jk + aik P2ik .

k=1

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Proof. For any fixed i ∈ L, j ∈ L, j , i, take X ji ∈ Rm \ Ωρ ji , where ρ ji ∈ R+ will be determined later. We define V ji (X ji ) = |X ji |2 . Apparently, condition A2 holds. Meanwhile, we have Hii (Xii ) = 0 in terms of condition B1. Then, by condition B1 and the definition of L∗ operator, one can obtain that   N N X X   ∗ 2 T a jk H jk (X jk ) − aik Hik (Xik ) L V ji (X ji ) =|σ j | + 2X ji F(X ji ) + k=1

  N N X X   2 T a jk H jk (X jk ) + a ji |H ji (X ji )| − aik Hik (Xik ) − aii Hii (Xii ) ≤|σ j | + 2X ji F(X ji ) + k=1,k,i

2

≤|σ j | − 2α ji |X ji | + 2

2

k=1,k,i

N X

k=1,k,i

≤|σ j |2 − 2α ji |X ji |2 +

N X

k=1,k,i

k=1,k,i

2

a jk H jk (X jk ) + 2a ji P ji |X ji | − 2

2

a jk (|X ji | + |H jk (X jk )| ) +

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≤|σ j | − 2α ji |X ji | +

2X Tji

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2

N X

a jk (|X ji |2 + P2jk |X jk |2 ) +

N X

k=1,k,i

2X Tji

N X

aik Hik (Xik )

k=1,k,i

aik (|X ji |2 + |Hik (Xik )|2 ) + 2a ji P ji |X ji |2

N X

aik (|X ji |2 + P2ik |Xki |2 ) + 2a ji P ji |X ji |2 .

+2

N X

k=1,k,i

Then, based on inequality

a jk P2jk |X jk |2

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N X

k=1,k,i

≤2

N X

k=1,k,i

a jk P2jk |X ji |2

7

k=1,k,i

a jk P2jk |Xki |2 ,

Journal Pre-proof we can derive ∗

2

2

L V ji (X ji ) ≤|σ j | − 2α ji |X ji | + N X

+

k=1,k,i

aik |X ji |2 +

N X

k=1,k,i N X

k=1,k,i

2

a jk |X ji | + 2

N X

k=1,k,i

a jk P2jk |X ji |2

+2

N X

k=1,k,i

a jk P2jk |Xki |2

aik P2ik |Xki |2 + 2a ji P ji |X ji |2

k=1,k,i

N X

+

k=1,k,i

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  N N N X X X   =|σ j |2 + −2α ji + a jk + aik + 2 a jk P2jk + 2a ji P ji  |X ji |2 k=1,k,i

k=1,k,i

(2a jk P2jk + aik P2ik )|Xki |2

(3)

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  N X   2 2 =|σ j | + −2α ji + (a jk + aik + 4a jk P jk + aik Pik ) + 2a ji P ji  |X ji |2 2

k=1,k,i

+

k=1,k,i

(2a jk P2jk + aik P2ik )(|Xki |2 − |X ji |2 ) 2

= − β ji |X ji | +

N X

k=1,k,i

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N X

η jk (Mki (Xki ) − M ji (X ji )) + m j ,

PN where β ji = 2α ji − k=1,k,i (a jk + aik + 4a jk P2jk + aik P2ik ) − 2a ji P ji > 0, η jk = 2a jk P2jk + aik P2ik , Mki (Xki ) = |Xki |2 , PN P η jk (|Xki |2 − |X ji |2 ) = 0 for j = k whatever the value of η jk , we m j = |σ j |2 . In addition, because Nj=1, j,i k=1,k,i can let ( η jk , j , k, η˜ jk = 0, j = k.

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Consequently, (3) amounts to 2

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L V ji (X ji ) = −β ji |X ji | +

N X

k=1,k,i

η˜ jk (Mki (Xki ) − M ji (X ji )) + m j ,

which implies that condition A3 holds. According to B3, we can know that condition A1 holds in Theorem 1. Therefore, we verify all conditions in Theorem 1, which means that system (1) admits a inner synchronized stationary distribution. This completes the proof. 4. An application to stochastic coupled oscillators

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In this section, we will apply main results to discussing the existence of inner synchronized stationary distribution for stochastic coupled oscillators model: x¨i (t) + ϕ x˙i (t) + xi (t) + g(xi (t)) +

N X k=1

8

cik Gik (xi (t) − xk (t)) = 0, i ∈ L,

(4)

Journal Pre-proof where xi (t) ∈ R is the phase of the i-th oscillator, g(·) : R → R is a nonlinear function, cik is the coupling strength, Gik (·) : R → R is the coupling function and ϕ > 0 represents the damping coefficient. We assume that g(·) and Gik (·) are continuously once differentiable in variables. Letting yi (t) = x˙i (t) + ξxi (t), one can recast system (4) as:   dxi (t) = [yi (t) − ξxi (t)]dt,      N   X (5)   2  dy (t) = (ξ − ϕ)y (t) + (ξϕ − ξ − 1)x (t) − g(x (t)) − c G (x (t) − x (t)) dt, i ∈ L.  i i i i ik ik i k  

of

k=1

Pr e-

p ro

Assume that the i-th oscillator is an ideal oscillator that is free from stochastic perturbation. And except for the i-th oscillator, others cannot escape the influence of stochastic perturbation. Then for any fixed i ∈ L, we can rewrite system (5) in the following form   dxi (t) = [yi (t) − ξxi (t)]dt,      N    X    2  dyi (t) = (ξ − ϕ)yi (t) + (ξϕ − ξ − 1)xi (t) − g(xi (t)) − cik Gik (xi (t) − xk (t)) dt,      k=1     dx j (t) = [y j (t) − ξx j (t)]dt + r j dB(1)   j (t),     N    X    2  c jk G jk (x j (t) − xk (t)) dt + s j dB(2)   j (t), j ∈ L, j , i,  dy j (t) = (ξ − ϕ)y j (t) + (ξϕ − ξ − 1)x j (t) − g(x j (t)) − k=1

k=1

al

(6) where r j , s j , 0 are the constant perturbation intensities of oscillator j. Using the j-th subsystem to subtract the i-th subsystem, we can get the following error system:   d(x j (t) − xi (t)) =[y j (t) − yi (t) − ξ(x j (t) − xi (t))]dt + r j dB(1)  j ,   h    2    d(y j (t) − yi (t)) = (ξ − ϕ)(y j (t) − yi (t)) + (ξϕ − ξ − 1)(x j (t) − xi (t)) − (g(x j (t)) − g(xi (t))) (7)    N   X         − c jk G jk (x j (t) − xk (t)) − cik Gik (xi (t) − xk (t))  dt + s j dB(2) j ∈ L, j , i.  j , 

Jo

urn

For the following analysis, denote Xi (t) = (xi (t), yi (t))T , X ji (t) = X j (t) − Xi (t) = (x ji (t), y ji (t))T , and F ji (X ji ) = (2) F (1) ji (X ji ) + F ji (X ji ), where ! y ji − ξx ji (1) , F ji (X ji ) = (ξ − ϕ)y ji + (ξϕ − ξ 2 − 1)x ji − (g(x j ) − g(xi )) ! 0 (2) P F ji (X ji ) = . N − k=1 (c jk G jk (x jk ) − cik Gik (xik ))  T (2) U j = diag{r j , s j }, B(t) = B(1) (t), B (t) . Then system (7) becomes j j dX ji (t) = F ji (X ji )dt + U j dB(t).

(8)

Using the method we have figured out before, a digraph (H, Λ) with N − 1 vertices can be constructed for system (8) in a similar manner. Meanwhile, for Λ = (˜η jk )(N−1)×(N−1) , we can define η˜ jk , i.e., η˜ jk = 2c jk Z 2jk + c1k Z 2jk 9

Journal Pre-proof  T for j , k and η˜ jk = 0 for j = k. For any fixed i ∈ L, j ∈ L, j , i, write X(t) = X Tji (t) , F(X) = 2(N−1)×1  T  T , G = diag{U1 , . . . , Ui−1 , Ui+1 , . . . , U N } ∈ R2(N−1)×2(N−1) . Then system (8) , B(t) = BTj (t) F Tji (X ji ) 2(N−1)×1 2(N−1)×1 can be modified as dX = F(X)dt + GdB(t). (9) T

Clearly, diffusion matrix D = (d sr )2(N−1)×2(N−1) = GG2 is everywhere positive definite. Here, the following assumptions are introduced for the technical reasons.

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Assumption 2. Digraph (H, Λ) is strongly connected. Assumption 3. There is a positive constant Z jk satisfying |G jk (X jk )| ≤ Z jk |X jk |,

j, k ∈ L.

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Next, we give the following theorem which can assure the existence of inner synchronized stationary distribution for stochastic coupled oscillators model (6).

Pr e-

Theorem 3. Under Assumptions 2 and 3, for any fixed i ∈ L, X ji ∈ Rm \ Ωρ ji , ρ ji ∈ R+ , Ωρ ji = {X ji ∈ Rm : V ji (X ji ) ≤ ρ ji }, if the following conditions hold, then system (6) admit a inner synchronized stationary distribution. C1. There exists a positive constant κ such that |g(x j ) − g(xi )| ≤ κ|x j − xi |,

with 2ϕ < 3 + 2ξ, ϕ > 2ξ, κ ≤ 4ξ . C2. The following inequality is satisfied

j ∈ L, j , i,

N X ξ − (c jk + cik + 4c jk Z 2jk + cik Zik2 ) − 2c ji Z ji > 0, 4 k=1,k,i

j ∈ L, j , i.

al

Proof. Since we assume that g(·) and Gik (·) are continuously once differentiable in variables, Assumption 1 holds clearly according to Remark 2. On the basis of condition C1 and Assumption 3, we have 2 2 2 X Tji F (1) ji (X ji ) =x ji y ji − ξx ji + (ξ − ϕ)y ji + (ξϕ − ξ − 1)y ji x ji − y ji (g(x j ) − g(xi ))

(10)

urn

3 2(ϕ−ξ)

and δ2 = 4ξ . Substituting these expressions into (10) leads to

3 2 ξ(ϕ − ξ)2 2 ξ 2 2 2 2 ξx + y ji + y ji + |g(x j ) − g(xi )|2 X Tji F (1) (X ) = − ξx − ξy + ji ji ji ji 4 ji 3 8 ξ 3 3 ξ 2 ≤ − ξx2ji − ξy2ji + ξx2ji + ξy2ji + y2ji + κ2 x2ji 4 4 8 ξ 3 3 ξ ξ ≤ − ξx2ji − ξy2ji + ξx2ji + ξy2ji + y2ji + x2ji 4 4 8 8 ξ 2 ξ = − (x ji + y2ji ) = − |X ji |2 . 8 8

Jo

We let ε2 =

= − ξx2ji + (ξ − ϕ)y2ji + (ξϕ − ξ 2 )y ji x ji − y ji (g(x j ) − g(xi ))  ! 2  ξ(ϕ − ξ)  2 2 y ji  1 2 2 |g(x j ) − g(xi )|2 2 2 . ≤ − ξx ji − ξy ji + ε x ji + 2  + δ y ji + 2 ε 2 δ2

10

(11)

Journal Pre-proof And in view of |Gk (X jk )| ≤ Z jk |X jk |, condition B1 in Theorem 2 is satisfied. Furthermore, according to condition C2, it is clear that condition B2 can be verified. Assumption 2 makes condition B3 holds. Therefore, we exhibit that all conditions of Theorem 2 hold, which implies that system (6) can achieve inner synchronized stationary distribution. The proof is completed. Remark 6. In fact, nonlinear oscillator x¨(t) + ϕ x˙(t) + g(x(t)) = 0

p ro

of

is widely used in many fields. The dynamics of nonlinear oscillator have received much attention over the past few years, such as stability [9], synchronization [11] and so on. However, to the best of our knowledge, there are few researches considering the existence of inner synchronized stationary distribution of this kind of nonlinear oscillator. Because stochastic factors are indispensable, we consider nonlinear stochastic coupled oscillators. The existence theory of inner synchronized stationary distribution exhibited in our main results are resoundingly applied to this nonlinear stochastic coupled oscillators model, which explains the practical application of the theoretical results in physics. Besides, we will explore synchronized stationary distribution in the real world biological and engineering models in the future.

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5. Numerical example

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In order to verify the effectiveness and feasibility of the theoretical results, we give a numerical example of stochastic coupled oscillators model (6). Let N = 5. Fix i as 1. The functions and parameters we choose are given below: ϕ = 2.9, ξ = 1.42, κ = 0.352, G jk (x j − xk ) = 0.3(x j − xk ), Z jk = 0.5, k, j = 1, 2, 3, 4, 5. Besides, we select r2 = 4.15, r3 = 4.25, r4 = 4.35, r5 = 4.45, s2 = 4.15, s3 = 4.25, s4 = 4.35, s5 = 4.45. g(x j ) = 0.35 sin x j , j = 1, 2, 3, 4, 5. c11 = 0, c12 =0, c13 = 0, c14 = 0.02, c15 = 0.04, c21 = 0.01, c22 = 0, c23 = 0.01, c24 = 0, c25 = 0.01, c31 = 0, c32 = 0, c33 = 0, c34 = 0.02, c35 = 0.03, c41 = 0, c42 = 0.03, c43 = 0, c44 = 0, c45 = 0.02, c51 = 0, c52 = 0, c53 = 0.02, c54 = 0, c55 = 0. According to the proof process of Theorems 2 and 3, we can see that element η˜ jk of the weight matrix for digraph H can be computed by η˜ jk = 2c jk Z 2jk + c1k Z 2jk , when j , k and η˜ jk = 0, when j = k. Therefore, we obtain weighted matrix Λ = (˜η jk )4×4 as follows:

urn

  Λ = η˜ jk

   =  

4×4

 0 0.005 0.005 0.015  0 0 0.015 0.025  . 0 0 0 0.02   0.01 0 0.005 0

Jo

Obviously, digraph (H, Λ) is strongly connected. Consider error system (7) obtained by stochastic coupled oscillators on digraph H. Apparently, it can be derived that D is everywhere positive definite in R8 . By simple PN 2 calculations, we have |g(x j ) − g(x1 )| ≤ κ|x j − x1 |, |G jk (X jk )| ≤ Z jk |X jk | and 4ξ − k=2 (c jk + c1k + 4c jk Z 2jk + c1k Z1k )− 2c j1 Z j1 > 0. Therefore, we validate that all the conditions of Theorem 3 are satisfied. Furthermore, the initial values are selected as follows: x21 = 3.4, y21 = 3.6, x31 = 1.1, y31 = −1.4, x41 = −1.4, y41 = 1.1, x51 = 3.2, y51 = 3.3. We can get the figures of the distribution of sample path in the phase space in Figure 1 and the corresponding joint stationary distribution with the initial values in Figure 2. Thus, system (6) admits a inner synchronized stationary distribution according to Figures 1 and 2. Moreover, from Figures 3 and 4, we can see that the distribution of sample path in the phase space for system (7) eventually goes to zero such that 11

Journal Pre-proof

0

0

31

−5

−5

−5

0

−10 −10

5

x (t)

−5

0

5

x (t)

21

p ro

−10 −10

31

5

5

0

51

y (t)

0

41

y (t)

of

5

y (t)

5

21

y (t)

system (6) achieves complete synchronization if the perturbation matrix is a zero matrix, which is a intuitive explanation of Remark 5. Comparing Figures 1 with 5, we can observe that inner synchronized stationary distribution domain becomes larger when the integer part of stochastic perturbation intensity is changed to 10, which fully embodies the influence of perturbation intensity size on inner synchronized stationary distribution domain. These numerical results show the effectiveness of our theoretical results.

−5

−5

−5

0

−10 −10

5

−5

x (t) 41

0

Pr e-

−10 −10

5

x (t) 51

1.5 1 0.5 0 −0.5 5

al

Figure 1: The distribution of sample path in the phase space of system (7).

0 y (t)

−5

0 x (t)

1.5 1 0.5 0 −0.5 5

0 y (t)

0 y (t) 31

−5

−5

0 x (t)

5

5 −5

−5

0 x (t) 31

1.5 1 0.5 0 −0.5 5 0 y (t) −5 51

41

Jo

41

5

21

urn

21

−5

1.5 1 0.5 0 −0.5 5

−5

0 x (t) 51

Figure 2: The joint stationary distribution of system (7).

12

5

5

0

0

31

y (t)

5

21

y (t)

Journal Pre-proof

−5

−10 −10

−5

−5

0

−10 −10

5

−5

x (t)

0

0

51

of

5

y (t)

5

−5

−5

−5

0

−10 −10

p ro

−10 −10

5

31

41

y (t)

0 x (t)

21

5

x (t)

−5

0

5

x (t)

41

51

200

0

x31(t),y31(t)

x21(t),y21(t)

200

−200 −400 −600

0

20

40

t 200

−400

0

20

40

60

40

60

t

200

x51(t),y51(t)

0 −200 −400 −600

0

−200

−600

60

0

al

x41(t),y41(t)

Pr e-

Figure 3: The distribution of sample path in the phase space of system (7) without stochastic disturbance.

20

40

0

−200 −400 −600

60

t

0

20 t

Jo

urn

Figure 4: The distribution of sample path of system (7) without stochastic disturbance.

13

5

0

0

31

y (t)

5

21

y (t)

Journal Pre-proof

−5

−10 −10

−5

−5

0

−10 −10

5

−5

x (t)

0

0

51

of

5

y (t)

5

−5

−5

−5

0

−10 −10

p ro

−10 −10

5

31

41

y (t)

0 x (t)

21

5

x (t)

−5

0

5

x (t)

41

51

Figure 5: The distribution of sample path in the phase space of system (7) that the integer portion of disturbance intensity is 10.

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6. Conclusion

urn

Acknowledgments

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In this paper, based on stochastic analysis techniques, graph theory and Lyapunov method, the existence of inner synchronized stationary distribution for a class of SCSNs was studied for the first time. These results were not only obtained by theoretical analysis, but also applied to a stochastic coupled oscillators model. And a numerical simulation also supports our theoretical results. Our results obtained in this paper build a platform for the research on the existence of inner synchronized stationary distribution for SCSNs. Some interesting topics deserve further investigations. Note that in all analysis, the strong connectedness of the digraph is a fundamental assumption, we will next consider the non-strongly connected case. In addition, one may propose some more realistic but complex models, including considering the effect of Markovian switching in SCSNs. And it is entertaining to reveal that the analysis procedure in this paper can also be used to consider other models in many fields, not just stochastic coupled oscillators in physics. We will leave these research work to the future.

References

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This work was supported by the NSF of Shandong Province (Nos. ZR2018MA005, ZR2018MA020 and ZR2017MA008); the Project of Shandong Province Higher Educational Science and Technology Program of China (No. J18KA218); the Key Project of Science and Technology of Weihai (No. 2014DXGJMS08) and the Innovation Technology Funding Project in Harbin Institute of Technology (No. HIT.NSRIF.201703).

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