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Published in IET Control Theory and ApplicationsReceived on 29th January 2013Revised on 16th April 2013Accepted on 7th May 2013doi: 10.1049/iet-cta.2013.0090

ISSN 1751-8644

Pinning synchronisation in fixed and switchingdirected networks of Lorenz-type nodesGuanghui Wen1, WenwuYu1, Yu Zhao2, Jinde Cao1

1Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of China2College of Engineering, Peking University, Beijing 100871, People’s Republic of ChinaE-mail: wenguanghui@gmail.com

Abstract: This study addresses the global pinning synchronisation of directed networks with Lorenz-type node dynamics.By using tools from M -matrix theory and Lyapunov stability theory, the interesting issues of what kind of nodes should bepinned and how large the control strength between neighbouring nodes should be selected for achieving global synchronisationin both fixed and switching networks are clearly addressed. It is theoretically shown that global pinning synchronisation ofan arbitrarily given fixed network can be achieved if the network topology contains a directed spanning tree and the couplingstrength is larger than a derived critical value depending both on the node dynamics and the network topology. By suitablyconstructing multiple Lyapunov functions, it is further proved that global pinning synchronisation in switching networks witha suitable coupling strength can be guaranteed if each possible network topology contains a directed spanning tree and thedwell time of switching is less than a positive threshold. Numerical examples are finally given for illustration.

1 Introduction

Many real large-scale dynamical systems can be modelledas complex networks where the nodes are the elementsand the links represent the interaction among the neigh-bouring elements. Recently, the study of complex networkshas received much attention from researchers ranging fromphysics to mathematics and to control theory [1]. In thenumerous research topics on complex networks, synchroni-sation control and its related problems such as the distributedconsensus of multi-agent systems have received particularattention because of its broad applications in engineering[2–9].

Roughly speaking, synchronisation in complex dynam-ical networks can be divided into local synchronisation[10] and global synchronisation [11]. Analytical methodsfor these two kinds of problems are generally different.By using the transverse stability analysis approach, localsynchronisation has been studied in [12–15]. In contrast,global synchronisation is more interesting since it meansthat synchronisation in complex networks can be achievedunder arbitrarily given initial conditions. In the last decade,many researchers have contributed to the fruitful results ofglobal synchronisation of complex networks under differentenvironments [16–20]. Within the context of global syn-chronisation control, global pining synchronisation controlin complex networks has been recognised as one of themost important topics, which can be described as how todesign some pinned controllers for a small fraction of nodesto make the states of the whole network converge to adesired trajectory [11, 21]. Concerning the pinning synchro-nisation of complex networks, it does not need to assume

the network topology satisfy any connectivity conditions.By using the augmented network approach, it is proved thatpinning synchronisation in a network can be achieved if thecorresponding augmented network meets some mild connec-tivity conditions [7, 22, 23]. However, it is still unclear whatkinds of and how many nodes should be pinned such thatthe augmented network meets these connectivity conditions.Another key factor affecting the achievement of pinning syn-chronisation is the node dynamics. In this work, the globalpinning synchronisation in networks of Lorenz-type systemsthat include Chen and Lü systems as special cases will beaddressed. As it is well-known, the Lorenz system as oneof the paradigms of chaos has been taking as a focal sub-ject in non-linear control since the year of 1963 [24]. As adual Lorenz system, Chen system was found in 1999 [25].The work in [25] is really a breakthrough in the study ofLorenz family systems. Then, the Lü system was introducedin [26] as a transition system between Lorenz and Chensystems. Since it is hard or impossible to verify whetherthe Lü system satisfies the Lipschitz condition, the syn-chronisation criteria derived in the some exiting works withthe Lipschitz condition may not be valid for coupled net-works with Lorenz-type nodes [27]. Note also that even forthe non-linear systems that do satisfy the Lipschitz con-dition, some conservatism will be involved in calculatingthe Lipschitz constants. Thus, the synchronisation conditionbased on the derived Lipschitz constants will be conser-vative. Furthermore, another common assumption alwaysmade in existing literature on pinning synchronisation isthat the network topology is fixed. It is more interestingto study pinning synchronisation under a switching networktopology.

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Based on the above analysis, this paper investigatesthe global pinning synchronisation of coupled Lorenz-typenodes under fixed and switching directed topologies. Byusing tools from M -matrix theory and Lyapunov stabilitytheory, some sufficient conditions for realising global pin-ning synchronisation in networks of Lorenz-type nodes areprovided where the Lipschitz condition is removed. Par-ticularly, the problems of what kinds of and how manynodes should be pinned for both fixed and switching directedtopologies are addressed in a detail. It is proved that theglobal pinning synchronisation in switching networks withan appropriately selected coupling strength can be guaran-teed if each possible network topology contains a directedspanning tree and the dwell time of the switching is lessthan a derived positive threshold. At last, some numericalsimulations are provided to verify the theoretical analysis.

The rest of the paper is organised as follows. In Section 2,some preliminaries on graph theory and the problem for-mulation are provided. The theoretical results are presentedand discussed in Section 3. Numerical examples are givenin Section 4. At last, Section 5 concludes the paper.

Throughout the paper, let Rn×n and N be the sets ofn × n real matrix space and positive natural numbers,respectively. The superscript T means the transpose forreal matrices. Let diag{x1, x2, . . . , xn} be a diagonal matrixwith xi, i = 1, 2, . . . , n, being its ith diagonal element.Take 1n(0n) be the n-dimensional column vector with eachentry being 1(0). Matrices, if not explicitly stated, areassumed to have compatible dimensions. Notation ‖ · ‖ rep-resents the Euclidean norm. For a real symmetric matrixS, λmax(S) and λmin(S) represent its largest and small-est eigenvalues, respectively. Furthermore, a column vectorx = (x1, x2, . . . , xn)

T ∈ Rn is said to be positive if and onlyif xi > 0, for all i = 1, 2, . . . , n.

2 Preliminaries and problem formulation

2.1 Preliminaries

Let G(A) be a directed graph with the set of nodesV = {v1, v2, . . . , vN }, the set of directed links E ⊆ V × V ,and a weighted adjacency matrix A = [aij]N×N with non-negative elements aij, i, j = 1, 2, . . . , N . An link eij in G isdenoted by the ordered pair of nodes (υj, υi), where υj and υi

are called the parent and child nodes, respectively, and eij ∈E if and only if aij > 0. Note that only simple graphs areconsidered here, that is, self-loops and multiple links are for-bidden. A ‘path’ from nodes υr0 to υrk in G(A) is a sequenceυr0 , υr1 , . . . , υrk of distinct nodes such that each (υri , υri+1),i = 0, 1, . . . , k − 1, is a link in E . A directed graph G(A) iscalled ‘strongly connected’ if there exits a directed betweenany two distinct nodes. A ‘strongly connected component’is an induced subgraph that is maximal, subject to beingstrongly connected. An isolated node, that is, a node withzero in-degree and out-degree, is also regarded as a stronglyconnected component. A directed tree is a directed graphwhere every node υ, except the root node r has no par-ent, has exactly one parent, and there exists a unique pathfrom r to υ. A directed spanning tree of a network G is adirected tree, which contains all the nodes of G. The Lapla-cian matrix L = [lij]N×N of G is defined as: lij = −aij, i �= jand lij = ∑N

k=1,k �=i aik , i = j, where i, j = 1, 2, . . . , N .

Lemma 1 [28]: Suppose that the directed graph G(A) con-tains a directed spanning tree. Then, 0 is a simple eigenvalue

of its Laplacian matrix L, and all the other eigenvalues of Lhave positive real parts.

Lemma 2 [29]: For matrices A, B, C and D with appropriatedimensions, one has

(1) (A ⊗ B)T = AT ⊗ BT;(2) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD);(3) A ⊗ (B + C) = A ⊗ B + A ⊗ C;

Definition 1 [29]: A non-singular real matrix A is called anM -matrix if all of its off-diagonal elements are non-positive,and each of its eigenvalues has positive real part.

2.2 Problem formulation

Consider a directed network of Lorenz-type nodes, thedynamics of ith node are given by

xi(t) = Axi(t) + βxi(t)Bxi(t) + α

N∑j=1

aij(t)H (xj(t) − xi(t))

(1)

where

A =⎛⎜⎝−(25γ + 10) (25γ + 10) 0

(28 − 35γ ) (29γ − 1) 0

0 0 − (γ + 8)

3

⎞⎟⎠B =

(0 0 00 0 −10 1 0

), β = (1, 0, 0) (2)

γ ∈ [0, 1] is a parameter, α > 0 represents the couplingstrength among the nodes, A(t) = [aij(t)]N×N is the adja-cency matrix of the communication topology at time t,and H ∈ R3×3 is the positive-definite inner linking matrix,i = 1, 2, . . . , N . Note that systems (1) will become the cou-pled Lorenz, Chen and Lü systems if γ = 0, 1, and 0.8,respectively. By the definition of the Laplacian matrix for agraph, it follows from (1) that

xi(t) = Axi(t) + βxi(t)Bxi(t) − α

N∑j=1

lij(t)Hxj(t) (3)

where L(t) = [lij(t)]N×N is the Laplacian matrix of commu-nication topology G(A(t)), i = 1, 2, . . . , N .

The control goal here is to design some pinned controllersto some designed nodes such that the states of all the nodesin (1) to converge to a common target trajectory s(t) in thesense of limt→∞‖xi(t) − s(t)‖ = 0, for all i = 1, 2, . . . , N ,with

s(t) = As(t) + βs(t)Bs(t) (4)

with arbitrarily given s(0) ∈ R3. Motivated by the works in[11, 21, 22, 30], pinning network (1) by using some linearcontrollers −αci(t)H (xi(t) − s(t)) to its ith node leads to

xi(t) = Axi(t) + βxi(t)Bxi(t) − α

N∑j=1

lij(t)Hxj(t)

− αci(t)H (xi(t) − s(t)) (5)

where ci(t) ∈ {0, 1} and ci(t) = 1 if the ith node of (1) ispinned at time t.

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Let ei(t) = xi(t) − s(t), i = 1, 2, . . . , N , it thus followsfrom (5) that

ei(t) = Aei(t) + βxi(t)Bxi(t) − βs(t)Bs(t)

− α

N∑j=1

lij(t)Hej(t) − αci(t)Hei(t) (6)

where i = 1, 2, . . . , N . By the definition of B, it follows from(6) that

ei(t) = Aei(t) + βxi(t)Bei(t) + βei(t)Bs(t)

− α

N∑j=1

lij(t)Hej(t) − αci(t)Hei(t), i = 1, 2, . . . , N

(7)

Obviously, ei(t) = 03 is a fixed point of system (7), for eachi = 1, 2, . . . , N . Furthermore, it is not hard to verify thatglobal pinning synchronisation in networks (3) with targettrajectory given in (4) is achieved if and only if the zeroequilibrium point of (7) is globally attractive.

3 Theoretical results

The theoretical results of this paper are provided anddiscussed in this section.

3.1 Fixed directed network topology

In this subsection, global pinning synchronisation of net-work (1) with target trajectory given in (4) under a fixednetwork topology is studied. Without loss of generality, letG(A(t)) = G(A), for all t ≥ 0.

By taking the target system (4) as a virtual leader node ofthe complex networks under consideration, one obtains theaugmented network topology G(A) that consists of N + 1nodes. Labelling the index of the virtual node as N + 1,the Laplacian matrix L of the augmented network topologyG(A) can be thus partitioned as

L =(

L ζ

0TN 0

)∈ R(N+1)×(N+1) (8)

where ζ ∈ RN and L ∈ RN×N . It is easy to verify thatL = L + diag{c1(t), c2(t), . . . , cN (t)}, where L is the Lapla-cian matrix of G(A). Before moving on, the followingassumption is introduced.

Assumption 1: There exists at least one directed spanningtree in the augmented network topology G(A).

Remark 1: Since the target system (4) evolves without beingaffected by any other nodes in the considered complexnetwork, Assumption 1 holds if there exist some directedpaths from the virtual node labelled as N + 1 to each nodei, i = 1, 2, . . . , N .

According to Remark 1, one obtains that Assumption 1will hold if all of the nodes 1, 2, . . . , N are pinned, that is,ci(t) = 1, for all i = 1, 2, . . . , N and t ≥ 0. However, it ismore interesting to study how to make Assumption 1 hold ifonly a small fraction of the nodes in G(A) could be selectedand pinned. To do this, the following algorithm is proposed

to determine at least how many and what kinds of nodesshould be pinned such that Assumption 1 holds.

Algorithm 1: Find the strongly connected components ofG(A) by using the Tarjan’s algorithm [31]. Note that thetime complexity of this operation is O(N + E), where Nand E are respectively the numbers of nodes and links ofG(A). Suppose that there are ω strongly connected compo-nents for G(A), labelled as W1, W2, . . ., Wω. Set mi = 0,i = 1, 2, . . . , ω and h = 1.

(1) Check whether there exists at least one node nk belong-ing to Wh, which is reachable from a node ng belonging toWj, j = 1, 2, . . . , ω, j �= h. If it holds, go to step (2); if itdose not hold, go to step (3).(2) Check whether the following condition holds: h < ω.If it holds, let h = h + 1 and reperform step (1); else stop.(3) Arbitrarily selected one node in Wh and pinned, let mh =1; Check whether the following condition holds: h < ω. Ifit holds, let h = h + 1 and reperform step (1); else stop.

Remark 2: Note that one may use the graphconncompfunction of Matlab [32] to search the strongly connectedcomponents of G(A). Furthermore, it is not hard to ver-ify that there exists a directed path from the leader toeach node in G(A), if the δ = ∑ω

i=1 mi selected nodes inG(A) are pinned. And, the δ is equal to the number of thestrongly connected components without incoming links inG(A). Furthermore, the number δ the set of the selectednodes are, respectively, called the ‘forest dimension’ andthe ‘node basis’ of directed graph G(A) [33]. Note alsothat δ = 1 if the network topology G(A) contains a directedspanning tree, that is, there exists a directed path from theleader, labelled as N + 1, to each node in G(A) if the rootnode of a directed spanning tree within this graph is selectedand pinned. Alternatively, when the network topology G(A)of the followers consists of τ separate components with eachof them having a directed spanning tree, it follows fromAlgorithm 1 that δ = τ and the root node of the spanningtree in each strongly connected component should be pinned.For an arbitrarily given G(A), it can be verified that δ deter-mined in Algorithm 1 corresponds to the minimum numberof nodes should be pinned such that Assumption 1 holds.Obviously, if there are some additional nodes besides the δselected nodes are also pinned, the Assumption 1 still holds;however, Assumption 1 can never be ensured if there only� nodes are selected and pinned, where � < δ.

Remark 3: For the case of G(A) is undirected graph, it canbe seen from Algorithm 1 that δ is equal to the number ofdisjoined connected components in G(A).

With Assumption 1 and according to Lemma 1, one hasthat zero is a simple eigenvalue of the Laplacian matrix Ldefined in (8) and all the other eigenvalues of L have positivereal parts. It thus follows from Definition 1 that L given in(8) is an M -matrix. Then, from Theorem 2.3 in [34], oneobtains the following lemma.

Lemma 3: Under Assumption 1, there exist positive vectorsξ = (ξ1, ξ2, . . . , ξN )T ∈ RN , such that Lξ = 1N and L +LT > 0, where = diag{1/ξ1, 1/ξ2, . . . , 1/ξN }.

Based on the above analysis, one may obtain the followingtheorem that summarises the main result of this section.

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Theorem 1: Suppose that Assumption 1 holds and the targettrajectory s(t) satisfies ‖s(t)‖ ≤ κ , for all t ≥ 0 where κ is agiven positive scalar. Then, pinning synchronisation in com-plex network (5) with fixed network topology G(A) canbe achieved exponentially if there exists a positive scalarc0 such that the following conditions hold: α > (λmax(AT +A) + 2κ + c0)/(ξminλmin( L + LT )λmin(H )), where ξmin =mini∈{1,2,...,N } ξi, ξ = (ξ1, ξ2, . . . , ξN )T is defined in Lemma 3.

Proof: Note that the global stability of the error systems (7)for its zero equilibrium point implies that the global pinningsynchronisation in complex network (5) will be achieved.Construct the following Lyapunov function

V (t) =N∑

i=1

(1/ξi)ei(t)Tei(t) (9)

where ξi, i = 1, 2, . . . , N , are defined in Lemma 3. Takingthe time derivative of V (t) along the trajectories of (7) andusing the facts eT

i (t)βxi(t)Bei(t) = 0, for all i = 1, 2, . . . , N ,yields

V (t) =N∑

i=1

(1/ξi)ei(t)T(AT + A)ei(t)

+ 2N∑

i=1

(1/ξi)ei(t)T(βei(t)Bs(t))

− 2α

N∑i=1

(1/ξi)ei(t)T

N∑j=1

lijHej(t)

− 2α

N∑i=1

(1/ξi)ci(t)ei(t)THei(t) (10)

Furthermore, according to (2), one has

ei(t)T (βei(t)Bs(t))

= ei1(t)ei3(t)s2(t) − ei1(t)ei2(t)s3(t)

≤ κ (|ei1(t)ei3(t)| + |ei1(t)ei2(t)|)≤ κei(t)

Tei(t) (11)

where ei(t) = (ei1(t), ei2(t), ei3(t))T and ‖s(t)‖ ≤ κ , for allt ≥ 0. According to Lemma 2 and by the definitions of Land , it follows from (10) and (11) that

V (t) ≤ e(t)T{ ⊗ (AT + A) + 2(κ/ξmin)I3N

− α[(

L + LT ) ⊗ H

]}e(t) (12)

where e(t) = (e1(t)T, e2(t)T, . . . , eN (t)T)T and ξmin =mini∈{1,2,...,N } ξi. From the condition

α ≥ (1/ξmin)[λmax(AT + A) + 2κ + c0

]λmin( L + LT )λmin(H )

(13)

and (12), one obtains

V (t) ≤ −c0V (t), for all t ≥ 0 (14)

Thus, one may obtain that pinning synchronisation in com-plex network (5) with fixed directed network topology canbe achieved exponentially. �

Remark 4: Suppose that the global pinning synchronisationin network (5) with network topology G(A) can be achievedfor some given coupling strength α. It can be seen from (14)the convergence rate of synchronisation is characterised bythe positive scalar c0.

From Theorem 1, one may directly obtain the followingcorollary where the proof is omitted for brevity.

Corollary 1: Suppose that Assumption 1 holds and the tar-get trajectory s(t) satisfies ‖s(t)‖ ≤ κ , for all t ≥ 0, whereκ is a given positive scalar. Then, pinning synchronisationin complex network (5) with fixed network topology G(A)can be achieved asymptotically if the coupling strength αsatisfies the following condition

α > λmax(�) (15)

where � = [( L + LT ) ⊗ H ]−1[ ⊗ (AT + A + 2κI3N )].

3.2 Switching directed network topology

The underlying topology of the complex network consid-ered in this section is modelled by a switching directedgraph. Let G = {G(A1), G(A2), . . . , G(Ap)}, p ≥ 2, indicatethe set of all possible directed network topologies. Supposethat there exists an infinite sequence of uniformly boundednon-overlapping time intervals [tk , tk+1), k ∈ N, with t1 = 0,tk+1 − tk ≥ τ0, and the dwell time τ0 > 0, over which theinteraction graph is fixed. The time sequence tk , k ∈ N isthen called the switching sequence, at which the interactiongraph changes. Furthermore, introduce a switching signalσ(t) : [0, +∞) → {1, 2, . . . , p}. Then, let G(Aσ(t)) be thecommunication topology of the complex network at timet. Note that G(Aσ(t)) ∈ G, for all t ≥ 0. The error dynamicalsystem (7) can be rewritten as

ei(t) = Aei(t) + βxi(t)Bei(t) + βei(t)Bs(t)

− α

N∑j=1

lσ(t)ij Hej(t) − αci(t)Hei(t), i = 1, 2, . . . , N .

(16)

where Lσ(t) = [lσ(t)ij ]N×N is the Laplacian matrix of network

topology G(Aσ(t)). Throughout this section, the time deriva-tives of functions ei(t) and xi(t) at any switching instantrepresents its right derivative.

By taking the target system (4) as a virtual node ofthe considered complex networks, one obtains the aug-mented network topology G(Aσ(t)) which consists of N + 1nodes. Labelling the index of the virtual node as N + 1, theLaplacian matrix Lσ(t) of the augmented network topologyG(Aσ(t)) can be thus partitioned as

Lσ(t) =(

Lσ(t) ζ σ(t)

0TN 0

)∈ R(N+1)×(N+1), (17)

where ζ σ(t) ∈ RN and Lσ(t) ∈ RN×N . It is easy to verify thatLσ(t) = Lσ(t) + diag{c1(t), c2(t), . . . , cN (t)}, where Lσ(t) is theLaplacian matrix of G(Aσ(t)) at time t. To derive the mainresults of this section, the following assumption is needed.

Assumption 2: There exists at least one directed spanningtree in the augmented network topology G(Aσ(t)).

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Remark 5: Applying Algorithm 1 to each possible networktopology G(Ai), i = 1, 2, . . . , p, one obtains that Assump-tion 2 will hold if the selected nodes are pinned.

Similar to Lemma 3, one could obtain the followinglemma.

Lemma 4: Under Assumption 2, there exist positive vectorsξσ(t) = (ξ

σ(t)1 , ξσ(t)

2 , . . . , ξσ(t)N )T ∈ RN , such that Lσ(t)ξ σ(t) =

1N and σ(t)Lσ(t) + (Lσ(t))T σ(t) > 0, where σ(t) = diag{1/ξ

σ(t)1 , 1/ξ

σ(t)2 , . . . , 1/ξ

σ(t)N } and t ≥ 0.

Based on the above analysis, one may obtain the followingtheorem that is the main result of this section.

Theorem 2: Suppose that Assumption 2 holds and the tar-get trajectory s(t) satisfies ‖s(t)‖ ≤ κ , for all t ≥ 0 whereκ is a given positive scalar. Then, pinning synchronisationin complex network (5) with switching directed topologiescan be achieved if there exists a positive scalar c0 such thatthe following conditions hold: α > (λmax(AT + A) + 2κ +c0)/(ξminλmin(

σ(t)Lσ(t) + (Lσ(t))T σ(t))λmin(H )), and

τ0 > lnρ/c0

where ρ = ξmax/ξmin, ξmax = maxi,j ξij , ξmin = mini, j ξ

ij , i ∈

{1, 2, . . . , p}, and j ∈ {1, 2, . . . , N }.

Proof: Construct the following multiple Lyapunov functions

V (t) =N∑

i=1

(1/ξσ(t)i )ei(t)

Tei(t) (18)

where ξσ(t)i , i = 1, 2, . . . , N , are defined in Lemma 4. Taking

the time derivative of V (t) along the trajectories of (16) andusing the facts eT

i (t)βxi(t)Bei(t) = 0, for all i = 1, 2, . . . , N ,yields

V (t) =N∑

i=1

(1/ξσ(t)i )ei(t)

T(AT + A)ei(t)

+ 2N∑

i=1

(1/ξσ(t)i )ei(t)

T(βei(t)Bs(t))

− 2α

N∑i=1

(1/ξσ(t)i )ei(t)

TN∑

j=1

lσ(t)ij Hej(t)

− 2α

N∑i=1

(1/ξσ(t)i )ci(t)ei(t)

THei(t) (19)

Similar to the steps in the proof of Theorem 1, one has

V (t) ≤ e(t)T{ σ(t) ⊗ (AT + A) + 2

(κ/ξmin

)I3N

− α[(

σ(t)Lσ(t) + (Lσ(t))T σ(t)) ⊗ H

]}e(t) (20)

where e(t) = (e1(t)T, e2(t)T, . . . , eN (t)T)T, σ(t) = diag{1/ξσ(t)1 ,

1/ξσ(t)2 , . . . , 1/ξ

σ(t)N } and ξmin = mini, j ξ

ij , i ∈ {1, 2, . . . , p},

j ∈ {1, 2, . . . , N }. From the condition

α >(1/ξmin)[λmax(AT + A) + 2κ + c0]

λmin( σ(t)Lσ(t) + (Lσ(t))T σ(t))λmin(H )

and (20), one obtains

V (t) ≤ −c0V (t), for all t ≥ 0 (21)

As the pinned complex network (5) switches when t = tk ,k = 2, 3, . . .. It then follows from (21) that

V (t−2 ) < V (0)e−c0(t2−t1)

≤ V (0)e−c0τ0 (22)

where τ0 is the dwell time such that tk+1 − tk ≥ τ0, for allk ∈ N. Furthermore, it is easy to check that

V (t2) ≤ ρV (t−2 )

where ρ = ξmax/ξmin, ξmax = maxi,j ξij , ξmin = mini, j ξ

ij , i ∈

{1, 2, . . . , p}, j ∈ {1, 2, . . . , N }. Based on the above analysis,one has

V (t2) < ρe−c0τ0 V (0)

that is

V (t2) < e(−c0τ0+lnρ)V (0) (23)

Furthermore, one obtains that −c0τ0 + lnρ < 0 by notingthat τ0 > lnρ/c0. By recursion, one may obtain that pin-ning synchronisation in network (5) with switching directedtopologies can be achieved exponentially. �

Under Assumption 2, Theorem 2 indicates that global pin-ning synchronisation in network (5) with switching directedtopologies can be achieved if both the coupling strengthbetween neighbouring nodes and the dwell time for eachpossible network topology are respectively larger than theirthresholds. In some practical applications, one may only tunethe coupling strength while cannot design the dwell time forswitching since the switching operation may be triggeredby link breakdown, actuator failures and so forth. It thusinteresting to study under what conditions global pinningsynchronisation can be ensured with an arbitrary switching.Before moving forward, the following assumption is given.

Assumption 3: Each possible topology G(Ai), i = 1, 2, . . . , p,is undirected.

Then, one may obtain the following corollary directlyfrom Theorem 2.

Corollary 2: Suppose that Assumptions 2, 3 hold and thetarget trajectory s(t) satisfies ‖s(t)‖ ≤ κ , for all t ≥ 0, whereκ is a given positive scalar. Then, pinning synchronisationin complex network (5) with switching topologies can beachieved for an arbitrarily given dwell time τ0 if there existsa positive scalar c0 such that the following condition holds

α > maxi=1,2,...,p

λmax(�i) (24)

where �i = {[ Li + (Li)T ] ⊗ H }−1[ ⊗ (AT + A + 2κI3N )],i = 1, 2, . . . , p.

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Proof: From Assumptions 2 and 3, one obtains that thematrix Lσ(t) given in (17) is positive-definite. Thus, onemay construct the following common Lyapunov for the errordynamical system (16)

V (t) =N∑

i=1

ei(t)Tei(t) (25)

Following the steps given in the proof of Theorem 2, thiscorollary can be proved. �

Remark 6: Note that whether it is possible to construct acommon Lyapunov for error dynamical system (16) withoutAssumption 3 still is a challenging issue up to date.

Remark 7: Since the threshold value of coupling strength forachieving synchronisation in fixed or switching networks ofLorenz-type nodes is derived by using a Lyapunov-basedapproach, it is thus sometimes conservative (see the numer-ical simulations in Example 1). Note also that it is very hardor even impossible to obtain some necessary and sufficientconditions on selecting the coupling strength for synchro-nisation of the considered networks as the node dynamicsare indeed non-linear. Nevertheless, the topic of how tofind a less conservative coupling strength for synchronis-ing the states of coupled Lorenz-type nodes is interestingand derives future research.

4 Numerical simulations

In this section, two numerical simulations are given forillustration.

Example 1: Consider a fixed network (1) that consists of 31identical Chen systems, described as follows

xi(t) = Axi(t) + βxi(t)Bxi(t) + α

31∑j=1

aijH (xj(t) − xi(t))

where

A =(−35 35 0

−7 28 00 0 −3

), B =

(0 0 00 0 −10 1 0

), β = (1, 0, 0)

H = diag{1630, 95.06, 1546}, i = 1, . . . , 31. The networktopology G(A) is given in Fig. 1, where the weights on eachlink are assumed to be 5. Next, Algorithm 1 is employedto determine at least how many and what kinds of nodesshould be pinned. From Algorithm 1, one obtains that G(A)contains six strongly connected components and each ofthem has no incoming link from other strongly connectedcomponents. It thus follows from Algorithm 1 that at leastsix nodes should be pinned, that is, at least one node ineach strongly connected component without incoming linksshould be pinned. The six strongly connected componentswithout incoming links are respectively labelled as Wi, i =1, 2, . . . , 6, and highlighted by turquoise shading in Fig. 2.In Fig. 2, the node labelled as 32 is the virtual leader whosedynamics are given as

s(t) = As(t) + βs(t)Bs(t) (26)

where s(t) is the state of the leader at time t ≥ 0. Tomake the augmented network topology contains a directedspanning tree, the nodes labelled as 6, 11, 18, 19, 26, 29are selected and pinned. The weights on each pinning link(indicated by the dashed lines in Fig. 2) are set as 6.85.In simulations, the initial states of the leader is given ass(0) = (1, 0.5, −0.35)T. It is theoretically shown in [35] thatall the solutions of system (26) are globally bounded. LetD1 = 31, D2 = 36.5, and D3 = 61, one obtains that the solu-tion s(t) = [s1(t), s2(t), s3(t)]T of Chen system (26) satisfiess1(t) ≤ D1, s2(t) ≤ D2, and s3(t) ≤ D3. Thus, the parameterκ can be set as κ = 77.56. The state trajectory of system(26) is given in Fig. 3 by using the fourth-order Runge–Kutta method. From Corollary 1, one has that the globalpinning synchronisation in the network can be realised ifthe coupling strength α ≥ 7.69595. For notational conve-nience, one may use x32(t) to denote the states of the leader.

13

14

15 16

18

17

41

3 62 5

11

10

7

9 128

29

28

30 31

20

23

21 22

19

25

24

Fig. 1 Network topology G(A)

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30 31

2726

5W

13

14

15 16

18

1720

41

3 62 5

11

10

7

9 128

23

29

28

21 22

19

25

32

24

1W

2W

3W

4W

6W

Fig. 2 Augmented network topology G(A)

−200

20−20

020

0

10

20

30

40

50

60

s2(t)

s1(t)

s 3(t)

Fig. 3 State trajectory of Chen system

Choose α = 7.696, the state trajectories of the nodes are,respectively, given in Figs. 4–6. Use

E(t) =√√√√ 1

31

31∑j=1

‖xj(t) − x32(t)‖2 (27)

to denote the synchronisation errors. Fig. 7 demonstrates thatthe global pinning synchronisation is indeed achieved. Next,the conservativeness of estimation on the coupling strengthfor achieving global pinning synchronisation is numericallystudied. It is shown by simulations that pinning synchro-nisation may be still guaranteed if the coupling strengthα ∈ (0.5, 7.69595]. The states of the networks with α = 0.5are plotted in Figs. 8–10, respectively, and the synchronisa-tion errors E(t) are provided in Fig. 11. Note that how toreduce the conservativeness of estimation on the coupling

0 0.02 0.04 0.06 0.08 0.1−100

−80

−60

−40

−20

0

20

40

60

80

100

Time

x i1(t

),i=

1,2,

...,3

2

Fig. 4 State trajectories of xi1(t), i = 1, 2, . . . , 32, with couplingstrength α = 7.696

strength for guaranteeing global synchronisation still is achallenging issue, leaving an interesting yet non-trivial issuefor future study.

Example 2: Consider a switching network (1) that consistsof 7 identical Chen systems, described as follows

xi(t) = Axi(t) + βxi(t)Bxi(t) + α

4∑j=1

aσ(t)ij H (xj(t) − xi(t))

where H = diag{30, 30, 30} and the other parameters are setthe same as those given in Example 1. The communicationtopologies G(A1) and G(A2) are given in Fig. 12a and b,respectively, where the weights on each link are set as 4.By using Algorithm 1, one obtains that G(A1) contains twodisjoined connected components, labelled as W 1

1 and W 12 ,

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0 0.02 0.04 0.06 0.08 0.1

−1000

−500

0

500

1000

Time

x i2(t

),i=

1,2,

...,3

2

Fig. 5 State trajectories of xi2(t), i = 1, 2, . . . , 32, with couplingstrength α = 7.696

0 0.02 0.04 0.06 0.08 0.1

−600

−400

−200

0

200

400

600

800

Time

x i3(t

),i=

1,2,

...,3

2

Fig. 6 State trajectories of xi3(t), i = 1, 2, . . . , 32, with couplingstrength α = 7.696

0 0.02 0.04 0.06 0.08 0.1

500

1000

1500

2000

2500

3000

3500

4000

E(t

)

Time

0.015 0.02 0.025 0.03

50

100

150

200

Fig. 7 Evolution of E(t) in Example 1, with coupling strengthα = 7.696

respectively; and G(A2) contains three disjoined connectedcomponents labelled, respectively, as W 2

1 , W 22 and W 2

3 . Notethat the disjoined connected components are highlighted by

0 0.2 0.4 0.6 0.8 1−200

−150

−100

−50

0

50

100

150

200

250

300

Time

x i1(t

),i=

1,2,

...,3

2

Fig. 8 State trajectories of xi1(t), i = 1, 2, . . . , 32, with couplingstrength α = 2.50

0 0.2 0.4 0.6 0.8 1

−2000

−1500

−1000

−500

0

500

1000

1500

2000

Time

x i2(t

),i=

1,2,

...,3

2

Fig. 9 State trajectories of xi2(t), i = 1, 2, . . . , 32, with couplingstrength α = 2.50

0 0.2 0.4 0.6 0.8 1

−150

−100

−50

0

50

100

150

200

250

300

Time

x i3(t

),i=

1,2,

...,3

2

Fig. 10 State trajectories of xi3(t), i = 1, 2, . . . , 32, with couplingstrength α = 2.50

turquoise shading in Fig. 13. To make each possible aug-mented network topology contains a directed spanning tree,the nodes labelled as 1, 4 in G(A1) are pinned, and the

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0 0.2 0.4 0.6 0.8 1

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

x 104

E(t

)

Time

Fig. 11 Evolution of E(t) in Example 1, with coupling strengthα = 2.50

3

1

2

4 3

4

1

25 6

7

7

5

6a b

Fig. 12 Switching topologies G(A1) (a) and G(A2) (b)

nodes labelled as 1, 3 and 5 in G(A2) are pinned. Theweights on each directed link originated from the leader areset as 4.85. It is further assumed that the topology switchesbetween G(A1) and G(A2) periodically. In Fig. 12, the nodelabelled as 5 is the target system whose dynamics are thesame as those defined in Example 1. It can be seen thatAssumptions 2 and 3 hold.

It follows from Corollary 2 that the global pinningsynchronisation in the network can be realised withan arbitrarily given dwell time τ0 > 0, if the couplingstrength α ≥ 17.495. Choose α = 17.50 and τ0 = 0.002 s,the state trajectories of the nodes are respectively given inFigs. 14–16. Use

E(t) =√√√√1

7

7∑j=1

‖xj(t) − x8(t)‖2 (28)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−100

−80

−60

−40

−20

0

20

Time

x i1(t

),i=

1,...

,8

Fig. 14 State trajectories of xi1(t) with τ0 = 0.02s, i = 1, 2, . . . , 5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−10

−8

−6

−4

−2

0

2

4

6

8

10

12

Time

x i2(t

),i=

1,...

,8

Fig. 15 State trajectories of xi2(t) with τ0 = 0.02s, i = 1, 2, . . . , 5

to denote the synchronisation errors. Fig. 17 demonstratesthat global synchronisation with switching topologies isindeed achieved. Furthermore, the synchronisation errorsE(t) defined with dwell time τ0 = 0.001 s are also numeri-cally studied. Fig. 18 indicates that global pinning synchro-nisation with dwell time τ0 = 0.001 s is also achieved, whichverify the Corollary 2 very well.

3

1

2

8

4 3

4

1

25 6

7

7

5

6

8

11W

12W 2

2W2

1W2

3W

a b

Fig. 13 Augmented topologies G(A1) (a) and G(A2) (b)

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0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−6

−4

−2

0

2

4

6

8

10

Time

x i3(t

),i=

1,...

,8

Fig. 16 State trajectories of xi3(t) with τ0 = 0.02s, i = 1, 2, . . . , 5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

5

10

15

20

25

30

35

40

45

50

55

E(t

)

Time

Fig. 17 Evolution of E(t) in Example 2 with τ0 = 0.02s andcoupling strength α = 17.50

0 0.005 0.01 0.015 0.02

5

10

15

20

25

30

35

40

45

50

55

E(t

)

Time

Fig. 18 Evolution of E(t) in Example 2 with τ0 = 0.001s andcoupling strength α = 17.50

5 Concluding remarks

By using tools from M -matrix theory and Lyapunov sta-bility theory, global pinning synchronisation of a class ofcomplex networks with Lorenz-type nodes has been inves-tigated in the present paper. Both the fixed and switchingtopologies have been considered. The challenging issues ofwhat kind of nodes and at least how many nodes should beselected and pinned for achieving global synchronisation inthe closed-loop networks have been clearly addressed. Par-ticularly, the search algorithm presented here can be directlyimplemented by using the Matlab Toolbox with a lineartime complexity. It has been theoretically shown that theglobal pinning synchronisation can be ensured if the cou-pling strength among the neighbouring nodes is larger thana derived positive quantity and the switching among possibledirected network topologies is not too fast. Future work willfocus on solving an H∞ pinning synchronisation problemover switching directed complex networks.

6 Acknowledgements

The authors thank Professor Guanrong(Ron) Chen for send-ing us the useful references and the helpful discussions. Theauthors also thank the anonymous reviewers for their use-ful suggestions regarding the numerical simulations as wellas presentation of this paper. This work was supported bythe National Natural Science Foundation of China underGrant no. 61104145, the Natural Science Foundation ofJiangsu Province of China under Grant no. BK2011581,the Research Fund for the Doctoral Program of HigherEducation of China under Grant no. 20110092120024, theFundamental Research Funds for the Central Universitiesof China, and by the Information Processing and Automa-tion Technology Prior Discipline of Zhejiang Province OpenResearch Foundation under Grant numbers 20120801 and20120802.

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