Distributed adaptive pinning control for cluster synchronization of nonlinearly coupled Lur’e networks

Accepted Manuscript

Distributed Adaptive Pinning Control for Cluster Synchronization of Nonlinearly Coupled Lur’e Networks Ze Tang, Ju H. Park, Tae H. Lee PII: DOI: Reference:

S1007-5704(16)30052-1 10.1016/j.cnsns.2016.02.023 CNSNS 3781

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Received date: Revised date: Accepted date:

6 July 2015 20 November 2015 28 February 2016

Please cite this article as: Ze Tang, Ju H. Park, Tae H. Lee, Distributed Adaptive Pinning Control for Cluster Synchronization of Nonlinearly Coupled Lur’e Networks, Communications in Nonlinear Science and Numerical Simulation (2016), doi: 10.1016/j.cnsns.2016.02.023

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Highlights • cluster synchronization issue of nonlinearly coupled Lur’e networks under the distributed adaptive pinning control strategy is considered. • An efficient distributed nonlinearly adaptive update law based on the local information of the dynamical behaviors of node is proposed.

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• Verification of presented results via a real example is given.

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Distributed Adaptive Pinning Control for Cluster Synchronization of Nonlinearly Coupled Lur’e Networks

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Ze Tang1 , Ju H. Park1∗ , Tae H. Lee1 Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyonsan 38541, Republic of Korea. ∗

To whom correspondence should be addressed; E-mail: [email protected].

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Abstract. This paper is devoted to the cluster synchronization issue of nonlinearly coupled Lur’e networks under the distributed adaptive pinning control strategy. The time-varying delayed networks consisted of identical and nonidentical Lur’e systems are discussed respectively by applying the edgebased pinning control scheme. In each cluster, the edges belonging to the spanning tree are pinned. In view of the nonlinearly couplings of the networks, for the first time, an efficient distributed nonlinearly adaptive update law based on the local information of the dynamical behaviors of node is proposed. Sufficient criteria for the achievement of cluster synchronization are derived based on Sprocedure, Kronecker product and Lyapunov stability theory. Additionally, some illustrative examples are provided to demonstrate the effectiveness of the theoretical results.

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Introduction

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Keywords: Cluster synchronization · Distributed pinning control · Nonlinearly adaptive law · Nonidentical nodes · Nonlinear couplings.

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In the past two decades, collective behaviors of complex networks and systems, such as consensus and synchronization, have drawn increasingly attention from different fields of researchers. In this regard, synchronization [1] is undoubtedly one of the most significant collective behaviors of a dynamic system because of its potential applications in practical society, such as regulation of power grid, parallel image processing, the operation of no-man air vehicle, the realization of chain detonation, and etc., see [2], [3] and references therein. The main purpose of synchronization is to achieve the coordination of events in order to force systems into unification. Until now, various types of synchronization phenomena have been investigated, such as complete synchronization [4], cluster synchronization [5], lag synchronization [6], phase synchronization [7] and impulsive synchronization [8]. While cluster synchronization requires the coupled oscillators split into subgroups called cluster, such that the oscillators synchronized with one another in the same cluster, but there is no synchronization among different clusters. For instance, Wu, Zhou and Chen [9] discussed the issue of the cluster synchronization for general networks by means of a pinning control strategy. On the other hand, many known systems could be expressed in Lur’e form [10], [11], [12], just like chaotic Chua circuit, Lorenz system and Chen system, the Goodwin model, the repressilator, the toggle switch and the swarm model. Lur’e system belongs to the nonlinear system, and they could be viewed as linear systems equipped with a nonlinearity feedback which satisfies certain sector conditions. In view of the potential applications of Lur’e systems, many literatures studied the synchronization of Lur’e systems. In paper, Wu et al. [13] discussed the problem of sampled-data control for master-slave synchronization schemes that consist of identical chaotic Lur’e systems with time delays by analyzing the corresponding synchronization error systems. Additionally, Feng et al. [14] studied the cluster synchronization problem of Lur’e networks with non-linear coupling and non-identical

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nodes by using the linear and non-linear negative feedback control schemes, the Lyapunov stability theorem and linear matrix inequality. Generally, a complex network contains large-scale dynamical systems which consist of many interacted agents communicating through the edges, we called them couplings. In order to force the whole network to be synchronized, in most of cases, controllers should be imposed on the networks. However, it is very difficult to control all of the agents because of the manufacturing costs. Under this constraint, pinning control was proposed as an efficient strategy to force all of the agents towards some desired trajectory ([9], [14]). Meanwhile, adaptive control method benefits the whole network to obtain a suitable control gain which ensures the synchronization of the network ([15], [16]). Yang et al. [16] learned the global exponential synchronization of delayed complex dynamical networks with nonidentical nodes and stochastic perturbations by combining adaptive control and impulsive control schemes. Furthermore, Pietro et al. [17] discussed the synchronization of complex networks by means of two local adaptive protocols, vertex-based and edge-based adaptive methods. Based on the control methods proposed in [17] authors in [18], [19] and [20] discussed the synchronization issue for complex networks and Lur’e networks respectively by applying the adaptive pinning methods. For instance, Delellis, Bernardo and Garofalo concerned the synchronization and consensus of networks of nonlinear systems in the Lur’e form by applying a distributed adaptive control strategy in [18]. Moreover, distributed adaptive control of synchronization in complex networks was studied by Yu et al. in [21]. Motivated by above analyses, we believe the cluster synchronization for nonlinearly coupled complex networks via distributed adaptive pinning control deserves to be further investigated. Therefore, this paper considers the cluster synchronization issue for the Lur’e networks with nonlinearly couplings and time-varying delays. The main contribution of this paper could be concluded as the following three aspects: (1) Because cluster phenomenon is one of most important collective behaviors and it has extensive application prospect, cluster synchronization for a kind of dynamical networks consisted of Lur’e systems with identical and distinct communities are studied in this paper. In this regard, we further study and extend the results for synchronization of Lur’e networks which were presented in [18]. (2) Since nonlinear phenomenon is ubiquitous in industrial manufacturing and production, we consider the coupling weights among the systems in a nonlinear form which should satisfy some given requirements. (3) In view of the nonlinearly coupled Lur’e networks, different from [18], [22], [19], [21] and [23], for the first time, an asymmetrical edge-based distributed adaptive pinning control strategy is designed in order to obtain some appropriate coupling weights where the couplings belong to the spanning tree in each cluster are pinned. And finally, the realization of (cluster) synchronization is guaranteed by only regulating the coupling weights included in a spanning tree instead of applying the feedback controllers proposed in paper [21]. The rest of the paper is organized as follows. In Section 2, we first present the mathematical model and some preliminaries. In Section 3, the cluster synchronization for nonlinearly coupled Lur’e network with identical nodes and distinct communities will be considered respectively. In Section 4, numerical simulation is given to verify the theoretical results of this paper. Finally, we make the conclusion in Section 5.

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Notations. LT denotes the transpose of the matrix L. The sign ∗ in a matrix stands for the symmetric part. R denotes the n-dimensional Euclidean space. Rn×n is the set of n × n real matrices. diag{· · · } stands for a diagonal matrix. The symbol k · k stands for the Euclid norm of the matrix or the vector. T r(L) stands for the trace of matrix L. A symmetric real matrix L is positive definite (semi-definite) if xT Lx > 0(≥ 0) for all nonzero x, then we denote this as L > 0(L ≥ 0). In stands for the identity matrix with n dimension. Let C([−τ, 0]; R) be the space of continuous functions mapping [t0 − τ, t0 ] into Rn with the norm defined by kϕk = max1≤i≤n {sups∈[t0 −τ,t0 ] |ϕ(s)|}. G = [gij ]N ×N denotes a RN ×N matrix G with elements gij for i, j = 1, 2, · · · , N. The dimension of these vectors and matrices will be cleared in the context. n

2

Model Description and Preliminaries

2.1 Network structure notation In order to analyze the cluster synchronization of the complex dynamical network which consists of Lur’e systems, we give the following assumption of the network structure.

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Consider a network with N nodes and d clusters with N > d ≥ 2. Let µi = j if node i belongs to the j−th cluster. The graph could be denoted as G = (V, E) where V is the set of nodes and E = {(i, j)|i, j ∈ V} as the Sd set T of edges. The set {V1 , V2 , · · · , Vd } is called a partition of the set of node V if Vk 6= ∅, k=1 Vk = V and Vk Vl = ∅ for k 6= l, k, l = 1, 2, · · · , d. Denote Gk = (Vk , Ek ) as the subgraph for the k−th cluster, where the set of nodes in k−th cluster is Ek = {(i, j)|(i, j) ∈ E, i, j ∈ V, µi = µj = k}. Vk = {qk−1 + 1, qk−1 + 2, · · · , qk } with q0 = 0, qd = N and satisfies qk−1 < qk for k = 1, 2, · · · , d. Furthermore, denote Tk = (Vk , Ek∗ ) be the spanning tree of the subgraph Gk for k = 1, 2, · · · , d, E/Ek∗ = {(i, j)|(i, j) ∈ E, (i, j) ∈ / Ek∗ , k = 1, 2, · · · , d}. Let Ekl = {(i, j)|(i, j) ∈ E, i, j ∈ V, µi = k, µj = l, k 6= l} be the edges between the k−th cluster and the l−th cluster.

Problem formulation

Consider the following nonlinearly coupled dynamical network consisted of Lur’e systems with time-varying delays x˙ i (t) =Aµi xi (t) + Bµi xi (t − τ (t)) + Dµi fµi (Cxi (t)) + c

N X

˜ j (t)) gij ΓH(x

(1)

j=1

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where xi (t) = [x1i (t), x2i (t), · · · , xni (t)]T ∈ Rn for i = 1, 2, · · · , N and Aµi ∈ Rn×n , Bµi ∈ Rn×n , Dµi ∈ Rn×m , C ∈ Rm×n are constant matrices. The constant c > 0 denotes the coupling strength and Γ is the inner-linking matrix which is a diagonal matrix with each elements ri ≥ 0. τ (t) is time-varying delay which satisfies τ (t) ∈ [0, τ ]. fµi : Rm → Rm is a memoryless nonlinear vector-valued function which is continuously differentiable on R, and fµi (0) = 0. G = [gij ]N ×N ∈ RN ×N is the coupling matrix that results from the connection topology and determined by the network structure, gji = gij 6= 0 if there is a connection between node i and node j(i 6= j) and gij = 0 otherwise. And we assume the matrix G satisfies PN PN ˜ gii = − j6=ij=1 gij = − j6=ij=1 gji . The nonlinear coupling function H(·) : Rn → Rn is continuous with ˜ i (t)) = [h1 (x1 (t)), h2 (x2 (t)), · · · , hn (xn (t))]T , i = 1, 2, · · · , N. The notation Vk is the set of the form:H(x i i i all nodes in the k−th cluster for k = 1, 2, · · · , d. We define the function µ : {1, 2, · · · , N } → {1, 2, · · · , d}, if node j ∈ Vk , then one has µj = k. Denote Cxi (t) = [c1 xi (t), c2 xi (t), · · · , cm xi (t)]T , fµi (Cxi (t)) = [fµi 1 (c1 xi (t)), · · · , fµi m (cm xi (t))]T , where C = [c1 , c2 , · · · , cm ]T , cj ∈ R1×n . The functions fµi j (cj xi (t)), i = 1, 2, · · · , N, j = 1, 2, · · · , m are assumed to satisfy the sector condition (see [14], [24], [25] and references therein) ωµi j ≤ f˙µi j (cj xi (t)) ≤ δµi j

(2)

x(t0 + s) = ϕ(s), s ∈ [−τ, 0],

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for all cj xi (t) ∈ R, where ωµi j , δµi j are constants for i = 1, 2, · · · , N, j = 1, 2, · · · , m, t ∈ R+ . The initial condition associated with (1) and (2) can be given by

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where ϕ(s) ∈ C([−τ, 0], R), and denote x(t; t0 , ϕ) as a solution of (1) with given initial condition (t0 , ϕ). One could know that all of the state in the coupled system (1) will be the same when they are synchronized even though with different initial conditions. In the first analysis of the cluster synchronization of Lur’e network, we will consider the identical nodes situation, that is dynamical behaviors of all nodes in each cluster are the same. For main system (1) with the identical nodes situation under the distributed adaptive pinning control schemes can be described as X X ˜ j (t)) + c ˜ j (t)) (4) x˙ i (t) =Axi (t) + Bxi (t − τ (t)) + Df (Cxi (t)) + c gij (t)ΓH(x gij ΓH(x (i,j)∈Ek∗

(i,j)∈E / k∗

where gij (t) are the couplings in spanning tree to be pinned adaptively while gij are the constant couplings for i ∈ Vk , k = 1, 2, · · · , d. One can find that a fraction of the couplings in the matrix G will be controlled. For the edge-based distributed adaptive pinning control, we design the adaptive laws as ˜ j (t)) − H(x ˜ i (t))), g˙ ij (t) = σij (xj (t) − xi (t))T Γ(H(x 4

(i, j) ∈ Ek∗ , k = 1, 2, · · · , d,

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where σij = σji are adaptive gains to be determined, and we assume gij (0) = gji (0). Definition 1. The Lur’e network is said to realize cluster synchronization with d clusters if and only if limt→+∞ kxi (t) − xj (t)k = 0 for µi = µj , for any initial values and limt→+∞ kxi (t) − xj (t)k 6= 0 for µi 6= µj , for i, j = 1, 2, · · · , N. Definition 2. [26] A nonlinear function hk (·) : R → R is said to belong to the acceptable nonlinear coupling function class (NCF), denoted by hk (·) ∈ N CF (β, z), if there exist two nonnegative scalars β and z such that hk (x) − βx satisfies the Lipschitz condition

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| (hk (x1 ) − βx1 ) − (hk (x2 ) − βx2 ) |≤ z | x1 − x2 | for all x1 , x2 ∈ R.

Remark 1. [14] Since the function hk (·) ∈ N CF (β, z) is the restriction of the oscillatory amplitude of hk (x) around the linear function βz, a nonlinear function can be made to approach a linear function by taking a large β and a small z. The nonlinear function hk (x) can therefore be decomposed into the linear part βx and the oscillatory part pk (x) = hk (x) − βx. Obviously, hk (x) satisfies

for all x1 , x2 ∈ R.

hk (x1 ) − hk (x2 ) ≤β+z x1 − x2

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β−z ≤

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Definition 3. [9] For a symmetric matrix G ∈ RN ×N , if it has the block decomposition as G = [Gkl ]dd and each block Gkl (k, l = 1, 2, · · · , d) is a zero-row-sum matrix, then G is said to belong to M1 . Furthermore, if Gkk (k = 1, 2, · · · , d) is also irreducible, then G is said to belong to M2 .

qi (X) =

p X

xTj Qi xj

+

j=1

2bTi

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Definition 4. [27] Define p X

xj + ci , i = 1, 2, · · · , m,

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Qi = QTi ∈ Rn×n , i = 0, 1, · · · , m, j = 1, 2, · · · , p, X = (x1 , x2 , · · · , xp ) F := {X ∈ Rn×p : qi (X) ≥ 0, i = 1, 2, · · · , m},

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qi (xj ) is called quadratic function and if bi = 0 and ci = 0, then it is called quadratic form. Now consider the following conditions: (C1) q0 (X) ≥ 0 ∀X ∈ FP ; m n×p (C2) ∃s ∈ Rm . + : q0 (X) − i=1 si qi (X) ≥ 0, ∀X ∈ R Method of verifying (C1) using (C2) is called S-procedure for p = 1 and called extended S-procedure for p ≥ 1.

Useful lemmas and assumption

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Lemma 1. [28] Suppose G = (Gij ) ∈ RN ×N , and it satisfies Gij = Gji , Gii = − any two vectors x = (x1 , x2 , · · · , xn )T , y = (y1 , y2 , · · · , yn )T , we have xT Gy = −

Pn

j=1,j6=i

Gij then for

N 1 XX gij (xj − xi )(yj − yi ). 2 i,j=1 j6=i

Lemma 2. [26] Let 1n = (1, 1, · · · , 1)T , M = (mij ) = In − G ∈ Rn×n , θ > 0, we have xT Gy = xT GM y ≤

1 T N 1n 1n .

1 1 T ( x GGT x + θy T M y). 2 θ 5

For all zero-row-sum matrices

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Lemma 3. [29](Jessen Inequality) For any positive definite matrix R ∈ Rn×n , scalar τ > 0 and vector function e : [0, τ ] → Rn such that the integrations concerned are well defined, then  T Z τ Z τ Z τ T e(s)ds . R τ e (s)Re(s)ds ≥ e(s)ds 0

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Lemma 4. [9](Schur Complement). The linear matrix inequality (LMI)   Q(x) S(x) >0 T S(x) R(x)

where Q(x) and R(x) are symmetric matrices and S(x) is a matrix with suitable dimensions, is equivalent to one of the following conditions: T −1 (i) Q(x) > 0, R(x) − S(x) Q(x) S(x) > 0; −1 (ii) R(x) > 0, Q(x) − S(x)R(x) S(x)T > 0. Assumption 1. Throughout this paper, we assume the coupling matrix G ∈ M2 .

Cluster synchronization for the Lur’e network

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In this section, we will present the main results of this paper. In the first subsection, the cluster synchronization for Lur’e networks with identical systems will be considered by applying S-procedure, Kronecker product and Lyapunov stability theory. And in the second subsection, conditions of cluster synchronization for coupled Lur’e network with distinct communities will be derived through the nonlinearly distributed adaptive pinning control strategy.

The Lur’e network with identical nodes

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We rewrite the coupling matrix G into a block form G = [Gkl ]dd ∈ RN ×N where each block satisfies Gkl ∈ R(qk −qk−1 )×(ql −ql−1 ) for k, l = 1, 2, · · · , d. For the network (4), we define the following matrices G(1) (t) = (1) (2) [gij (t)]N ×N , G(2) = [gij ]N ×N as ( gij (t) (i, j) ∈ Ek∗ , (1) gij (t) = 0 (i, j) ∈ / Ek∗ , ( gij (i, j) ∈ E/Ek∗ , (2) gij = 0 (i, j) ∈ / E/Ek∗ , P P (1) (1) (2) (2) and gii (t) = − j=1,j6=i gij (t), gii = − j=1,j6=i gij for i, j = 1, 2, · · · , N . Similar to matrix G = (1)

(2)

(1)

(2)

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[Gij ]d×d , we write the matrices into block form as G(1) (t) = [Gkl (t)]dd and G(2) = [Gkl ]d×d with Gkl (t), Gkl ∈ (1) R(qk −qk−1 )×(ql −ql−1 ) for k, l = 1, 2, · · · , d. Afterwards, we could get that Gkl (t) = 0 if k 6= l from the definition of matrix G(1) (t) and G = G(1) + G(2) . Then, the system (4) could be written as x˙ i (t) =Axi (t) + Bxi (t − τ (t)) + Df (Cxi (t)) + c +c

N X j=1

N X

(1)

˜ j (t)) gij (t)ΓH(x

j=1

(2)

˜ j (t)), i ∈ Vk , k = 1, 2, · · · , d. gij ΓH(x

Let si (t) = [s1i (t), s2i (t), · · · , sni (t)]T ∈ Rn (i = 1, 2, · · · , d) be the solution of an isolated node in the i−th cluster satisfying s˙ i (t) = Asi (t) + Bsi (t − τ (t)) + Df (Csi (t)) 6

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where limt→+∞ ksi (t) − sj (t)k = 6 0, for i 6= j, i, j = 1, · · · , d. (1) (1) (1) Since matrix G ∈ M2 from Assumption 1, it gives Gkk (t)T = Gkk (t) and Gkk (t) are irreducible and zero(2) row-sum matrices, and Gkl are zero-row-sum matrices for k, l = 1, 2, · · · , d. Therefore, we have N X X (1) (1) ˜ µ (t)) = ˜ µ (t)) = 0, gij (t)ΓH(s gij (t)ΓH(s j i j=1

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N d X X X (2) ˜ (2) ˜ gij ΓH(s gij ΓH(s µj (t)) = k (t)) = 0.

Rewrite system (4) with the error vector ei (t) = xi (t) − sµi (t)(i = 1, 2, · · · , N )

N X d ei (t) (1) =Aei (t) + Bei (t − τ (t)) + Dψ(Cei (t)) + c gij (t)ΓH(ej (t)) dt j=1 N X j=1

(2)

gij ΓH(ej (t)), i ∈ Vk , k = 1, 2, · · · , d,

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+c

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˜ j (t)) − H(s ˜ µ (t)) for i, j = where functions ψ(Cei (t)) = f (Cxi (t)) − f (Csµi (t)) and H(ej (t)) = H(x j 1, 2, · · · , N . With considering the error vector, we could transform the adaptive laws (5) as in error form 1 d gij (t) = σij (ej (t) − ei (t))T Γ(H(ej (t)) − H(ei (t))), dt

(i, j) ∈ Ek∗ , k = 1, 2, · · · , d.

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wj ≤

fj (cj xi (t)) − fj (cj sµi (t)) ψj (cj ei (t) = ≤ δj , cj ei (t) cj ei (t)

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that is

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On the other hand, since the sector condition (2) in Lur’e system for function f (·), we have

[ψj (cj ei (t) − wj cj ei (t)][ψj (cj ei (t) − δj cj ei (t)] ≤ 0

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for all cj ei (t) 6= 0, i = 1, · · · , N, j = 1, · · · , m and t ∈ R+ , where ψ(Cei (t)) =[ψ1 (c1 ei (t)), ψ2 (c2 ei (t)), · · · , ψm (cm ei (t))]T .

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After introducing the network model and the distributed adaptive pinning controllers, we would like to present the first main result of this paper, which guarantees the realization of the cluster synchronization for identical Lur’e networks.

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Theorem 1. Suppose that Assumption 1 holds and all of the nonlinear coupling functions hi (·) ∈ N CF (β, z) with the positive constants β > 0, z > 0 for i = 1, 2, · · · , n. If there exist positive scalars θ1 and θ2 , a symmetrical matrix Σ, a diagonal positive definite matrix S = diag{s11 , · · · , s1m , s21 , · · · , s2m , · · · , sN 1 , · · · , sN m }, and positive definite matrices Q and R such that the following conditions hold   1 ˜ ˜ + C˜ T (W ˜ + ∆)S ˜ Π11 B D Σ ⊗ Γ2  ∗  −2Q 0 0  ≤ 0, (10) Π=  ∗  ∗ −2S 0 ∗ ∗ ∗ −θ2 InN

where

˜ S∆ ˜ C, ˜ ˜ + 2τ R ˜ + c[2β(G(2) + Σ) + 1 G(2) 2 + 2z 2 (θ1 + θ2 )(1 − 1 )IN ] ⊗ Γ − 2C˜ T W Π11 =A˜ + A˜T + 2Q θ1 N 7

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˜ = IN ⊗ B, C˜ = IN ⊗ C, matrices W = diag{w1 , w2 , · · · , wm }, ∆ = diag{δ1 , δ2 , · · · , δm } and A˜ = IN ⊗ A, B ˜ ˜ ˜ ˜ Q = IN ⊗ Q, R = IN ⊗ R, W = IN ⊗ W , ∆ = IN ⊗ ∆, then cluster synchronization of the network (4) could be achieved under the distributed adaptive pinning strategy (5). Proof. For convenience notation, we denote e(t) = [eT1 (t), eT2 (t), · · · , eTN (t)]T , e˜k (t) = [ek1 (t), ek2 , · · · , ekN (t)]T ,

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˜ Ψ(Ce(t)) = [ψ(Ce1 (t))T , ψ(Ce2 (t))T , · · · , ψ(CeN (t))T ]T , ˜ k (xk (t)) = [hk (xk (t)), hk (xk (t)), · · · , hk (xk (t))]T , h 1 2 N

˜ k (sk (t)) = [hk (sk (t)), hk (sk (t)), · · · , hk (sk (t))]T , h µ1 µ2 µN p˜k (xk (t)) = [pk (xk1 (t)), pk (xk2 (t)), · · · , pk (xkN (t))]T ,

p˜k (sk (t)) = [pk (skµ1 (t)), pk (skµ2 (t)), · · · , pk (skµN (t))]T .

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Construct the Lyapunov function as

(11)

V (t) =V1 (t) + V2 (t) + V3 (t) + V4 (t), V1 (t) =

N X

eTi (t)ei (t),

i=1

V2 (t) =2

N Z X i=1

t

t−τ (t) 0

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N Z X

V3 (t) =2

−τ

i=1

1 2

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V4 (t) =

N X

eTi (s)Qei (s)ds,

t

t+ϑ

N X

i=1 j=1,j6=i

eTi (s)Rei (s)dsdϑ,

c (1) (g (t) − εij )2 , σij ij

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where matrices Q > 0 and R > 0, and εij = εji are nonnegative constants to be determined later. Then, by calculating the derivative of V (t) in (11) along the trajectories of the error system (7) with considering the distributed adaptive pinning controller (8), we have N X

eTi (t)[Aei (t) + Bei (t − τ (t)) + Dψ(Cei (t))] + 2c

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V˙ 1 (t) =2

i=1

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+ 2c

N X N X

N N X X

(1)

gij (t)ΓH(ej (t))

i=1 j=1

(2)

gij ΓH(ej (t)),

i=1 j=1

V˙ 2 (t) =2

N X i=1

V˙ 3 (t) =2

N X

eTi (t)Qei (t) − 2 τ eTi (t)Rei (t)

i=1

≤2τ

N X i=1

eTi (t)Rei (t)

N X i=1

−2

eTi (t − τ (t))Qei (t − τ (t)),

N Z X i=1

0

−τ

N Z X − 2τ ( i=1

eTi (t + s)Rei (t + s)ds 0

−τ

Z ei (t + s)ds) R(

0

T

−τ

8

ei (t + s)ds),

ACCEPTED MANUSCRIPT

V˙ 4 (t) =

N N X X

c (1) (1) (g (t) − εij )g˙ ij (t) σij ij

i=1 j=1,j6=i

=

N N X X

(1)

c(gij (t) − εij )(ej (t) − ei (t))T Γ(H(ej (t)) − H(ei (t))).

i=1 j=1,j6=i

(12)

From V˙ 4 (t) in (12), we calculate the following terms by using Lemma 1, then it gives that N N X X

(1)

i=1 j=1,j6=i

−c

N X

N X

i=1 j=1,j6=i

gij (t)(ej (t) − ei (t))T Γ(H(ej (t)) − H(ei (t))) = −2c εij (ej (t) − ei (t))T Γ(H(ej (t)) − H(ei (t))) = 2c

N X i=1

i=1 j=1

N X i=1

+2

N X

eTi (t)Dψ(Cei (t)) + 2c

i=1

i=1

− 2τ

eTi (t)Bei (t − τ (t))

N X N X

(2)

gij ΓH(ej (t))

i=1 j=1

eTi (t − τ (t))Qei (t − τ (t)) + 2c

N X

(

i=1

Z

Z ei (t + s)ds)T R(

0

M

−2

N X

εij eTi (t)ΓH(ej (t)).

AN US

eTi (t)(A + AT + 2Q + 2τ R)ei (t) + 2

(1)

gij (t)eTi (t)ΓH(ej (t)),

i=1 j=1

N X N X

Consequently, we obtain that V˙ (t) ≤

N X N X

CR IP T

c

−τ

0

N X N X

εij eTi (t)ΓH(ej (t))

i=1 j=1

(13)

ei (t + s)ds).

−τ

ED

It follows from sector condition (9) and S-procedure in Definition 4, we derive V˙ (t) ≤ 0 for all cj ei (t) 6= 0 if there exist scalars sij ≥ 0 such that V˙ (t) + S ∗ < 0 for i = 1, 2, · · · , N, j = 1, 2, · · · , m, where we select m N X X

PT

S ∗ = −2

i=1 j=1

sij [ψj (cj ei (t)) − wj cj ei (t)][ψj (cj ei (t)) − δj cj ei (t)].

CE

Therefore, we have

AC

V˙ (t) + S ∗ ≤

N X

eTi (t)(A + AT + 2Q + 2τ R)ei (t) + 2

i=1

+2

i=1

N X

eTi (t)Dψ(Cei (t)) + 2c

i=1

−2

N X i=1

− 2τ −2

N X

N X N X

eTi (t)Bei (t − τ (t))

(2)

gij ΓH(ej (t))

i=1 j=1

eTi (t − τ (t))Qei (t − τ (t)) + 2c

N X i=1

Z (

0

−τ

N X m X i=1 j=1

Z T ei (t + s)ds) R(

0

N X N X

εij eTi (t)ΓH(ej (t))

i=1 j=1

ei (t + s)ds)

−τ

sij [ψj (cj ei (t)) − wj cj ei (t)][ψj (cj ei (t)) − δj cj ei (t)]. 9

(14)

ACCEPTED MANUSCRIPT

By applying Kronecker product to rewrite the above inequality (14), we get that V˙ (t) + S ∗ ≤eT (t)IN ⊗ (A + AT + 2Q + 2τ R)e(t) + 2eT (t)(IN ⊗ B)e(t − τ (t)) ˜ − 2eT (t − τ (t))Qe(t − τ (t)) + 2eT (t)(IN ⊗ D)Ψ(Ce(t)) εij eTi (t)ΓH(ej (t)) + 2c

i=1 j=1

− 2τ

N X

(

i=1

Z

N X N X

(2)

gij ΓH(ej (t))

i=1 j=1

0

−τ

Z ei (t + s)ds)T R(

0

ei (t + s)ds)

−τ

CR IP T

N X N X

+ 2c

T T ˜ ˜ ˜ − 2Φ(Ce(t)) SΦ(Ce(t)) + 2Φ(Ce(t)) S(IN ⊗ ∆)(IN ⊗ C)e(t) ˜ + 2eT (t)(IN ⊗ C)T (IN ⊗ W )T SΦ(Ce(t))

− 2eT (t)(IN ⊗ C)T (IN ⊗ W )T S(IN ⊗ ∆)(IN ⊗ C)e(t).

(15)

2c

n X

AN US

Since G(2) is a zero-row-sum matrix, it follows from Lemma 2 for θ1 and Definition 2 for z with pk (x) = hk (x) − βx that γk e˜k (t)T G(2) (˜ pk (xk (t)) − p˜k (sk (t)))

k=1

k=1

+c

θ1

n X

k=1

≤c

n X γk

k=1

−c

n X

γk θ1 [˜ pk (xk (t)) − p˜k (sk (t))]T M [˜ pk (xk (t)) − p˜k (sk (t))] e˜k (t)T G(2) G(2)T e˜k (t) γk θ 1

k=1 n X γk k

θ1

X j>i

k=1

n X

k=1

CE

− 2c

AC

≤c

=c =c

n X

k=1 n X

k=1 n X

k=1

mij [(pk (xkj (t)) − pk (skµj (t))) − (pk (xki (t)) − pk (skµi (t)))]2

e˜ (t)T G(2) G(2)T e˜k (t)

PT

≤c

θ1

e˜k (t)T G(2) G(2)T e˜k (t)

M

n X γk

ED

≤c

γk θ1

X j>i

mij [(pk (xkj (t)) − pk (skµj (t)))2 + (pk (xki (t)) − pk (skµi (t)))2 ]

n X X γk k T (2) (2)T k e˜ (t) G G e˜ (t) − 2cz 2 γk θ 1 mij (ekj (t)2 − eki (t)2 ) θ1 j>i

γk k T (2) (2)T k e˜ (t) G G e˜ (t) − 2cz 2 θ1

k=1 n X k=1

γk θ1 (1 −

1 k T k )˜ e (t) e˜ (t) N

1 1 γk e˜k (t)T [ (G(2) )2 + 2z 2 θ1 (1 − )IN ]˜ ek (t) θ1 N

=ceT (t)(

1 1 ((G(2) )2 ⊗ Γ) + 2z 2 θ1 (1 − )IN ⊗ Γ)e(t). θ1 N

10

(16)

ACCEPTED MANUSCRIPT

Based on inequality (16), it gives (2) gij ΓH(ej (t))

= 2c

i=1 j=1

N X N X i=1 j=1

= 2c

n X

(2) ˜ j (t)) − H(s ˜ µ (t))) gij Γ(H(x j

˜ k (xk (t)) − h ˜ k (sk (t))) γk e˜k (t)T G(2) (h

k=1 n X

= 2cβ

k

T

(2) k

γk e˜ (t) G

e˜ (t) + 2c

k=1

n X

k=1

γk e˜k (t)T G(2) (˜ pk (xk (t)) − p˜k (sk (t)))

CR IP T

2c

N X N X

1 1 ≤ ce (t)[2β(G ⊗ Γ) + ((G(2) )2 ⊗ Γ) + 2z 2 θ1 (1 − )(IN ⊗ Γ)]e(t). θ1 N PN Similarly, we define the matrix Σ = [εij ]N ×N with εii = − j=1,j6=i εij for θ2 in Lemma 2, we have T

N X N X i=1 j=1

εij eTi (t)ΓH(ej (t)) ≤ ceT (t)[2β(Σ ⊗ Γ) +

1 1 2 (Σ ⊗ Γ) + 2z 2 θ2 (1 − )(IN ⊗ Γ)]e(t). θ2 N

(17)

(18)

AN US

2c

(2)

Also, we could rewrite the matrix Σ in block form as Σ = [Σkl ]N ×N , where Σkl ∈ R(qk −qk−1 )×(ql −ql−1 ) . Correspondingly, we have Σkl = 0 if k 6= l for k, l = 1, 2, · · · , d. R0 T ˜ Define η(t) = (eT (t), eT (t − τ (t)), ΨT (Ce(t))) and ξi (t) = −τ ei (t + s)ds. From analysis in (15)-(18), based on Lemma 4, it derives that

M

V˙ (t) + S ∗ ≤η T (t)Πη(t) − τ T

≤η (t)Πη(t),

N X

ξiT (t)Rξi (t)

i=1

(19)

ED

in view of the selection of Lyapunov function in (11) and condition (10) in Theorem 1, we have V˙ (t) ≤ −S < 0 for suitable εij (i, j = 1, 2, · · · , N ) such that condition (10) in Theorem 1 to be satisfied when (i, j) ∈ Ek∗ . Based (1) on Lyapunov stability theorem and the adaptive laws (8), ei (t) and gij (t) are bounded and ei (t) → 0(t → ∞), (1)

PT

gij (t) → εij (t → ∞) for i, j = 1, 2, · · · , N . Therefore, the cluster synchronization of the Lur’e network (4) is achieved under the distributed adaptive pinning control scheme (5). That completes the proof.

CE

Remark 2. Since the spanning tree denotes a connected undirected graph with no cycles for a network, therefore, based on the selected edges in the networks, the spanning tree could be different. Normally, the spanning tree with largest algebraic connectivity should be controlled ([23]). This proposition could be verified in numerical simulation section.

AC

Remark 3. From the network model (4), one can find that the time-varying coupling strength for the first 1 1 coupling term could be decided by cgij (t) where c is coupling gains and gij (t) is the adaptive updating coupling 1 weights. For a given c, a suitable coupling matrix G (t) could be obtained according to the nonlinear adaptive law 1 1 (5). And it can be found in simulation section that if c is a very small constant, then limt→∞ gij (t) = gij would be correspondingly larger, and vice versa. Remark 4. In view of the nonlinearly couplings among the systems, the asymmetrical distributed adaptive 1 pinning control strategy (5) is designed in order to obtain the suitable coupling weights gij . Almost all previous works utilized the symmetrical adaptive pinning control for complex systems focused on the linearly adaptive updating law, such as [23], [18], [19] and [21]. However, in practical applications, most of couplings among different systems are in nonlinear ways, therefore, the nonlinearly distributed adaptive updating law (5) should be applied to the networks.

11

ACCEPTED MANUSCRIPT

3.2 The Lur’e network with nonidentical nodes In this subsection, we will discuss the cluster synchronization of the Lur’e network with nonidentical nodes. That is, the system could be described as (1). With considering the distinct d clusters, the model from (1) could be rewritten as x˙ i (t) =Aµi xi (t) + Bµi xi (t − τ (t)) + Dµi fµi (Cxi (t)) + c

d X X

˜ j (t)), gij ΓH(x

k=1 j∈Vk

CR IP T

the functions fµi j (cj xi (t)), i = 1, 2, · · · , N, j = 1, 2, · · · , m are assumed to satisfy the sector section (2). Let sµi (t) = [s1µi (t), s2µi (t), · · · , snµi (t)]T ∈ Rn (i = 1, 2, · · · , N ) be the solution of an isolated node in the i−th cluster satisfying s˙ µi (t) = Aµi sµi (t) + Bµi sµi (t − τ (t)) + Dµi fµi (Csµi (t))

where limt→+∞ ksi (t) − sj (t)k = 6 0, for i 6= j, i, j = 1, · · · , d. Then, we have the following error system by defining error vector as ei (t) = xi (t) − sµi (t):

+c

N X j=1

(2)

N X

(1)

gij (t)ΓH(ej (t))

j=1

AN US

e˙i (t) = Aµi ei (t)+Bµi ei (t − τ (t)) + Dµi ψµi (Cei (t)) + c

gij ΓH(ej (t)), i ∈ Vk , k = 1, 2, · · · , d,

(20)

where functions ψµi (Cei (t)) = fµi (Cxi (t)) − fµi (Csµi (t)) for i, j = 1, 2, · · · , N. Corresponding to (9), for the nonidentical situation, we have [ψµi j (cj ei (t) − wµi j cj ei (t)][ψµi j (cj ei (t) − δµi j cj ei (t)] ≤ 0

(21)

for all cj ei (t) 6= 0, i = 1, · · · , N, j = 1, · · · , m and t ∈ R , where

M

+

ψµi (Cei (t)) =[ψµi 1 (c1 ei (t)), ψµi 2 (c2 ei (t)), · · · , ψµi m (cm ei (t))]T .

ED

In the following, we will give the main results of the cluster synchronization for nonidentical Lur’e networks. In this situation, the dynamics of nodes in different clusters are distinct.

CE

PT

Theorem 2. Suppose that Assumption 1 holds and all of the nonlinear coupling functions hi (·) ∈ N CF (β, z) with the positive constants β, z for i = 1, 2, · · · , n. If there exist positive scalars θ1 and θ2 , diagonal positive definite matrices Sµi = diag{sµi 1 , sµi 2 , · · · , sµi m } and L = diag{l1 , l2 , · · · , ln }, and positive definite matrices Q and R such that the following two conditions hold: (i) Π∗ is a negative definite matrix:  ∗  Π11 Bµi Dµi + C T (Wµi + ∆µi )Sµi  −2Q 0 Π∗ =  ∗ (22) ∗ ∗ −2Sµi

AC

(ii) There exists a symmetrical matrix Σ such that Λ is a negative definite matrix:   Λ11 Σ Λ= ∗ −θ2 IN where

Π∗11 =Aµi + ATµi + Q + QT + τ R + τ RT − 2C T Wµi Sµi ∆µi C + 2L,

Wµi =diag{wµi 1 , wµi 2 , · · · , wµi m }, ∆µi = diag{δµi 1 , δµi 2 , · · · , δµi m }, 2 1 2lk 1 IN ≤ 0, Λ11 =2β(G(2) + Σ) + G(2) + 2z 2 (θ1 + θ2 )(1 − )IN − θ1 N γk 12

(23)

ACCEPTED MANUSCRIPT

for i = 1, 2, · · · , N, and k = 1, 2, · · · , n, then cluster synchronization of the network (1) with distinct communities could be realized under the distributed adaptive pinning method (5). Proof. Construct the same Lyapunov function as (11). Then, calculate the derivative of V (t) in (11) along the trajectories of the error system (20) with considering the distributed adaptive pinning controller (8), we have N X i=1

+ 2c

eTi (t)[Aµi ei (t) + Bµi ei (t − τ (t)) + Dµi ψµi (Cei (t))] N X N X

(1)

gij (t)ΓH(ej (t)) + 2c

i=1 j=1

N X N X

(2)

gij ΓH(ej (t))

CR IP T

V˙ 11 (t) =2

(24)

i=1 j=1

V˙ 2 (t), V˙ 3 (t) and V˙ 4 (t) are similar in equation (12). Then, according to sector condition (21) and S-procedure in Definition 4, we derive V˙ (t) ≤ 0 for all cj ei (t) 6= 0 if there exist scalars sµi j ≥ 0 such that V˙ (t) + S 0 < 0 for i = 1, 2, · · · , N, j = 1, 2, · · · , m, where we select S = −2

N X m X i=1 j=1

sµi j [ψµi j (cj ei (t)) − wµi j cj ei (t)][ψµi j (cj ei (t)) − δµi j cj ei (t)].

AN US

0

Therefore, we could get V˙ (t) + S 0 ≤2

N X

eTi (t)(Aµi + Q + τ R)ei (t) + 2

i=1

+2

i=1

N X

eTi (t)Dψµi (Cei (t)) + 2c

N X

− 2τ

N X i=1

Z (

0

ei (t + s)ds)T R(

−τ

N X m X

PT −2

N X N X

eTi (t − τ (t))Qei (t − τ (t)) + 2c

ED

i=1

i=1 j=1

eTi (t)Bµi ei (t − τ (t)) (2)

gij ΓH(ej (t))

i=1 j=1

M

i=1

−2

N X

Z

0

N X N X

εij eTi (t)ΓH(ej (t))

i=1 j=1

ei (t + s)ds)

−τ

sµi j [ψµi j (cj ei (t)) − wµi j cj ei (t)][ψµi j (cj ei (t)) − δµi j cj ei (t)].

(25)

CE

It follows from the similar proof process in Theorem 1, we could verify that the cluster synchronization of Lur’e network (1) with distinct communities could be achieved based on the conditions in Theorem 2.

AC

Remark 5. Theorem 2 gives conditions for the cluster synchronization of the time-varying delayed networks consist of nonidentical Lur’e systems. From the inequality (21), we could know that in this situation, Kronecker product can’t be used because that the dynamical behaviors of each cluster are distinct, therefore, we obtain dispersed conditions (22) and (23) which not only for each cluster µi (i = 1, 2, · · · , N ), but also for each component k(k = 1, 2, · · · , n). Remark 6. For a given network with fixed structure, we could determine the spanning tree of each cluster according to the connectivity of each node. And for the fixed spanning tree, the edge-based distributed adaptive control scheme (5) could be imposed to the edges, or so-called coupling weights. However, since the adaptive control scheme (5) is asymmetrically designed, for certain time points, it is possible that some coupling weights could get to zero when the coupling weights update, that denotes the original connections in spanning tree will be vanished. This possible situation even doesn’t affect the results derived in this paper.

13

ACCEPTED MANUSCRIPT

Chua circuit 2

CR IP T

Chua circuit 1

1 1

0.5

0.5

0

s22

s12

0 −0.5

−0.5

−1

5 0 −5 −10

s13

−4

0

−2

2

4

2 0

−5

s11

−10

−2 −4

s21

(b)

M

Chua circuit 3

40

ED

20 s32

4

0

6

s23

(a)

0

PT

−20 −40 200

AN US

−1 5

−1.5 10

100

100 50

0

−100

CE

s33

−200

−100

0 −50 s31

(c)

(d)

AC

Figure 1: The dynamics are described by three Chua’s circuits with different parameters (a)-(c). The network consists of three clusters (d) where solid line means gij = 1, the dashed line means gij = −1, and otherwise gij = 0. The red line in (d) gives the spanning tree of every cluster.

14

ACCEPTED MANUSCRIPT

4

Numerical simulation

Consider the following Chua’s circuits with time-varying delay [30]    x˙ 1 (t) = ai (x2 (t) − hi (x1 (t))) x˙ 2 (t) = x1 (t) − x2 (t) + x3 (t)   x˙ 3 (t) = −bi x2 (t) + ci x3 (t) − bi ζi %i x1 (t − τ (t))

(26)



 −ai (pi − qi ) , 0 Di =  0

Ci =



1

0

CR IP T

where nonlinear function hi (x1 (t) = qi x1 (t) + 21 (pi − qi )(|x1 (t) + 1| − |x1 (t) − 1|) for i = 1, 2, 3. Rewrite the network equation (26) into Lur’e form by Yalcin et al [31], we get     −ai qi ai 0 0 0 0 −1 1  , B =  0 0 0 , Ai =  1 0 −bi ci −bi ζi %i 0 0 0



,

CE

PT

ED

M

AN US

and nonlinear function fi (x1 (t)) = 21 (|x1 (t) + 1| − |x1 (t) − 1|) belonging to the sector [ω, δ] = [−1, 1]. For the above Chua’s circuit, we select the following parameters to describe the dynamical behaviors of the nodes in different clusters: the first cluster a1 = 9.78, b1 = 14.97, c1 = 0, p1 = 1.31, q1 = 0.75, ζ1 = 0.02, %1 = 0.5, the second cluster a2 = 10, b2 = 14.87, c2 = 0, p2 = 1.27, q2 = 0.68, ζ2 = 0, %2 = 0 and the third cluster a3 = 9, b3 = 14.5, c3 = 0, p3 = 8/7, q3 = 5/7, ζ3 = 0, %3 = 0, and the time-varying delay choose τ (t) = et 0.001 1+e t ∈ [0, 0.001]. In the simulation, we select 15 nodes which are separated into three clusters, cluster 1= {1, 2, 3, 4, 5}, cluster 2= {6, 7, 8, 9, 10} and cluster 3= {11, 12, 13, 14, 15}, the structure of the network can be found in (a)-(c) of Fig.1. Then set the coupling gain c = 0.1, the inner coupling matrix Γ = I3 . The nonlinear coupling function ˜ i (t)) = (h1 (x1 (t), h2 (x2 (t), h3 (x3 (t))T with hj (xj (t)) = (xj + 2sinxj ) ∈ N CF (β, z), β = 1, z = 2 for H(x i i i i i i i = 1, 2, · · · , 15, j = 1, 2, 3. Select the related parameters as follows: θ1 = 2, θ2 = 3 and Γ = 3I3 . Construct a network consists of 15 nodes which are separated into three clusters as (d) in Fig.1, where the solid line denotes gij = 1, the dashed line means gij = −1 and otherwise, gij = 0. Besides, the red line in (d) gives the spanning three of each cluster. We execute the simulation by using SIMULINK in MATLAB. For the above given parameters, we plot the evolution of the coupling gains in Fig.2 according to the adaptive updating laws (5). By verifying the conditions in Theorem 2, we obtain S = 420.3108 and lk = 844.2769(k = 1, 2, 3), which ensure the maximum eigenvalue of matrix Λ is negative. And we plot the evolution of state curves in each clusters in Fig.3. Therefore, we verified the cluster synchronization can be achieved by applying the proposed distributed adaptive pinning control strategy.

AC

Remark 7. In this numerical simulation, we focus on Chua’s circuits with different parameters in order to reflect the nonidentical clusters. Fifteen Lue’s systems were separated into three clusters. Simulation results showed the validity of the main results. In this regard, the main results could also be applied to other systems which could be written as Lue’e form ([10]- [14] ) x˙ i (t) = Aµi xi (t) + Dµi fµi (Cxi (t)),

such as Chua’s circuits, Lorenz system, Chen system, the Goodwin model, the repressilator, the toggle switch and the swarm model.

5

Conclusion

This paper investigated cluster synchronization for dynamical networks consist of delayed nonlinearly coupled Lur’e systems with identical and distinct communities. In view of the nonlinearly coupled Lur’e networks, for the 15

ACCEPTED MANUSCRIPT

The adaptive coupling gains gij(t).

14

g12 g

13

12

g

14

g15

10

g

67

g68

8

g

CR IP T

69

gij

g610

g1112

6

g1113 g1114

4

g1115

0

0

AN US

2

5

10 t

15

20

Figure 2: The adaptive coupling gains of coupling matrix G(t).

ED

M

first time, an asymmetrical edge-based distributed adaptive pinning control protocol was proposed for obtaining some appropriate coupling weights by using the local information of the nodes. In each cluster, the couplings belong to the given spanning tree were fully pinned. Sufficient conditions of cluster synchronization for identical and nonidentical Lur’e network are derived respectively based on S-procedure, Kronecker product and Lyapunov stability theory. Finally, we presented some illustrative examples to demonstrate the effectiveness of the theoretical results.

PT

Acknowledgements

CE

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201).

References

AC

[1] Alex Arenas, Albert D´ıaz-Guilera, Jurgen Kurths, Yamir Moreno, and Changsong Zhou. Synchronization in complex networks. Physics Reports, 469(3):93–153, 2008.

[2] Yong Xu, Hua Wang, Yongge Li, and Bin Pei. Image encryption based on synchronization of fractional chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 19(10):3735–3744, 2014. [3] Sergio Vazquez, Juan Antonio Sanchez, Manuel R Reyes, Jose Leon, Juan M Carrasco, et al. Adaptive vectorial filter for grid synchronization of power converters under unbalanced and/or distorted grid conditions. IEEE Transactions on Industrial Electronics, 61(3):1355–1367, 2014. [4] Meng Zhan, Xingang Wang, Xiaofeng Gong, GW Wei, and C-H Lai. Complete synchronization and generalized synchronization of one-way coupled time-delay systems. Physical Review E, 68(3):036208, 2003. 16

The state curves of the 5 nodes in the first cluster. 20 0 −20

0

2 4 6 8 t The state curves of the 5 nodes in the first cluster.

10 0 −10

0

2

4

6 8 t The state curves of the 5 nodes in the first cluster.

2

4

50 0 −50

0

6

8

AN US

0 −5

10

0

6 8 t The state curves of the 5 nodes in the second cluster.

10

2

0

2 4 6 8 t The state curves of the 5 nodes in the second cluster.

10

2

0 −10

4

M

0 −2

10

0

2

4

6

8

10

t

The state curves of the 5 nodes in the third cluster.

5

PT

x3i (t),i=11,12,13,14,15x2i (t),i=11,12,13,14,15x1i (t),i=11,12,13,14,15

CE

10

The state curves of the 5 nodes in the second cluster. 5

ED

x3i (t),i=6,7,8,9,10

x2i (t),i=6,7,8,9,10

x1i (t),i=6,7,8,9,10

t

AC

10

CR IP T

x3i (t),i=1,2,3,4,5

x2i (t),i=1,2,3,4,5

x1i (t),i=1,2,3,4,5

ACCEPTED MANUSCRIPT

0

−5

0

2 4 6 8 t The state curves of the 5 nodes in the third cluster.

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2 4 6 8 t The state curves of the 5 nodes in the third cluster.

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5 0 −5

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Figure 3: The state curves of the nodes in different clusters.

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