332 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 4, NO. 2, APRIL 2017

Pinning Synchronization Between Two GeneralFractional Complex Dynamical Networks With

External DisturbancesWeiyuan Ma, Yujiang Wu, and Changpin Li

Abstract—In this paper, the pinning synchronization betweentwo fractional complex dynamical networks with nonlinearcoupling, time delays and external disturbances is investigated.A Lyapunov-like theorem for the fractional system with timedelays is obtained. A class of novel controllers is designedfor the pinning synchronization of fractional complex networkswith disturbances. By using this technique, fractional calculustheory and linear matrix inequalities, all nodes of the fractionalcomplex networks reach complete synchronization. In the aboveframework, the coupling-configuration matrix and the inner-coupling matrix are not necessarily symmetric. All involvednumerical simulations verify the effectiveness of the proposedscheme.

Index Terms—External disturbance, fractional complex net-work, nonlinear coupling, pinning control, time delay.

I. INTRODUCTION

FRACTIONAL calculus is as old as the conventionalcalculus. However, fractional calculus has become a hot

topic in recent two decades due to its advantages in appli-cations of physics and engineering. As a generalization ofordinary differential equation, fractional differential equationcan capture non-local relations in space and time. Thus,the fractional-order models are believed to be more accuratethan the integer-order models. Fractional models have beenproven to be excellent instrument to describe the memoryand hereditary properties of various materials and processes,such as dielectric polarization, electrode-electrolyte polariza-tion, electromagnetic waves, viscoelastic systems, quantitativefinance and waves [1]−[4].

It is demonstrated that fractional differential systems alsobehave chaotically or hyperchaotically, such as the fractional

Manuscript received September 11, 2015; accepted January 13, 2016.This work was supported by National Natural Science Foundation of China(11372170, 11471150, 41465002) and Fundamental Research Funds forthe Central Universities (31920130003). Recommended by Associate EditorDingyu Xue.

Citation: W. Y. Ma, Y. J. Wu, and C. P. Li, “Pinning synchronizationbetween two general fractional complex dynamical networks with externaldisturbances,” IEEE/CAA Journal of Automatica Sinica, vol. 4, no. 2, pp. 332−339, Apr. 2017.

W. Y. Ma is with the School of Mathematics and Statistics, LanzhouUniversity, Lanzhou 730000, China, and also with the School of Mathematicsand Computer Science, Northwest University for Nationalities, Lanzhou730000, China (e-mail: mwy2004@126.com).

Y. J. Wu is with the School of Mathematics and Statistics, LanzhouUniversity, Lanzhou 730000, China (e-mail: myjaw@lzu.edu.cn).

C. P. Li is with the Department of Mathematics, Shanghai University,Shanghai 200444, China (e-mail: lcp@shu.edu.cn).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JAS.2016.7510202

Lorenz system [5], the fractional Chua system [6], and thefractional Chen system [7]. Many complex networks usuallyconsist of a large number of highly interconnected fractionaldynamical units. Generally speaking, there are two mainadvantages of the fractional complex dynamical networks: oneis infinite memory; the other is that the derivative order ofa parameter enriches the system performance by increasingone degree of freedom. A fractional neural network is madeup of thousands of neurons and their interactions. On theother hand, fractional differentiation provides neurons witha fundamental and general computation ability which cancontribute to efficient information processing. Time delays areubiquitous in neural networks due to finite switching speedof amplifiers. They usually occur in the signal transmissionamong neurons [8]−[10]. Therefore, it is more valuable andpractical to investigate fractional complex networks with timedelays.

Following these findings, synchronization of fractionalchaotic systems becomes a challenging and interesting realmdue to its potential applications in secure communication andcontrol processing [11]−[16]. Many of complex networksnormally have a large number of nodes, therefore it is usuallyexpensive to control a complex network by designing thecontrollers for all nodes. To reduce the number of controllers,a pinning control method is proposed. Wang and Chen in-vestigated a scale-free dynamical network by controlling afraction of network nodes [17]. Sorrentino et al. explored thepinning controllability of the complex networks [18]. Liangand Wang revealed the relationship between the couplingmatrix and the synchronizability of complex networks viapinning control [19]. Yu et al. studied synchronization viapinning control of general complex dynamical networks [20].Nian and Wang investigated the optimal scheme of pinningsynchronization of directed networks [21]. In addition, Liangand Wang proposed a method to quickly calculating pinningnodes on pinning synchronization in complex networks [22].In [23]−[25], the authors investigated the pinning control ofinteger-order complex networks with time delays. However,pinning synchronization of integer-order complex networkswas well-studied. Due to global dependent property of frac-tional complex networks, as far as we know, the literature onpinning synchronization of fractional complex networks is stillsparse. In [26], based on the eigenvalue analysis and fractional

MA et al.: PINNING SYNCHRONIZATION BETWEEN TWO GENERAL FRACTIONAL COMPLEX DYNAMICAL NETWORKS · · · 333

stability theory, local stability properties of pinned fractionalnetworks were derived. The pinning synchronization of newuncertain fractional unified chaotic systems were discussedin [27]. In [28], Chai et al. proposed a global pinningsynchronization for fractional complex dynamical networks.Xiang et al. investigated the robust synchronization problemfor a class of systems with external disturbances [29]. Wanget al. provided a method to achieve projective synchronizationof two fractional chaotic systems with external disturbances[30]. However, the effects of both time delay and externaldisturbance of the fractional complex network have seldombeen considered.

Lyapunov direct method is important for synchronizationanalysis of complex networks with integer-order, but thismethod is difficult to be directly extended to fractional case[28], [31], [32]. Thus, to find out new ways to cope withthese problems is still challenging. Motivated by the above dis-cussions, pinning synchronization between the drive-responsefractional complex networks with nonlinear coupling and timedelays is studied. A novel modified Lyapunov method is usedto analyze the global asymptotical synchronization criteria offractional systems with time delays. These criteria rely onthe coupling strength and the number of nodes pinned to thenetworks, there is no extra constraint on the two couplingmatrices, such as symmetric or irreducible case.

The rest of this paper is arranged as follows. In Section II,the general drive and response fractional complex dynamicalnetwork models are introduced and some necessary preliminar-ies are given. A Lyapunov-like criteria for delayed fractionalsystem is obtained. In Section III, based on the Lyapunovstability theorem, pinning controllers are designed to ensurethe drive and response systems with external disturbancesachieve synchronization. The illustrative numerical simula-tions are displayed in Section IV. Section V concludes thispaper.

Throughout this paper, let ‖ · ‖ the Euclidean norm, In theidentity matrix. If A is a vector or matrix, its transpose isdenoted by AT . Let λmin(A) and λmax(A) be the smallestand largest eigenvalue of symmetric matrix A, respectively.

II. PRELIMINARIES AND MATHEMATICAL MODELS

A. Fractional Complex Dynamical Networks With NonlinearCoupling and Time Delays

In this section, we introduce some notations, definitions andpreliminaries which will be used later on.

At present, there are several definitions of fractional dif-ferential operators [4], such as Grunwald-Letnikov definition,Riemann-Liouville definition, Caputo definition. Among them,the initial conditions for Caputo derivatives have the sameform as those for integer-order ones. And Caputo derivative notonly has a clearly interpretable physical meaning, but can alsobe properly measured to initializing in the simulation. So itmay be the most appropriate choice for practical applications.Now we give the definition of Caputo fractional derivativeCDα

0,tf(t), of order α with respect to time t as follows

CDα0,tf(t) =

1Γ(m− α)

∫ t

0

(t− τ)m−α−1f (m)(τ)dτ (1)

where m− 1 < α < m ∈ Z+.Consider a general fractional delayed dynamical system

consisting of N nodes, which can be described as follows:

CDα0,txi(t) = Axi(t) + f(xi(t)) + c

N∑

j=1

bijHxj(t)

+ cN∑

j=1

bijHg(xj(t− τ)) + ξi(t) (2)

where i = 1, 2, . . . , N , 0 < α < 1 is the fractional order,xi(t) = (xi1(t), . . . , xin(t))T ∈ Rn is the state variable ofthe ith node. A ∈ Rn×n is a given linear matrix, and f(xi) =[f1(xi), f2(xi), . . . , fn(xi)]T : Rn → Rn is a smooth functiondescribing the nonlinear dynamics of the node. c and c aretwo parameters of the non-delay and delay coupling strengths,respectively. H ∈ Rn×n and H ∈ Rn×n are inner couplingmatrices. B = (bij)N×N and B = (bij)N×N denote thecoupling configuration matrices of the network. If there isa connection from node i to node j (i 6= j), then bij > 0(or bij > 0); otherwise, bij = 0 (or bij = 0). The diagonalelements of matrix B and B are given by bii = −∑N

j=1,j 6=i bij

and bii = −∑Nj=1,j 6=i bij , respectively. g(xi(t − τ)) =

[g1(xi(t− τ)), g2(xi(t− τ)), . . . , gn(xi(t− τ))]T : Rn → Rn

is the nonlinear coupling function. ξi(t) ∈ Rn are externaldisturbance vectors.

If model (2) is referred as the drive system, the responsecomplex network can be chosen as

CDα0,tyi(t) = Ayi(t) + f(yi(t)) + c

N∑

j=1

bijHyj(t)

+ c

N∑

j=1

bijHg(yj(t− τ)) + ηi(t) + ui(t) (3)

where yi(t) = (yi1(t), yi2(t), . . . , yin(t))T ∈ Rn is theresponse state vector of the ith node; ηi(t) ∈ Rn are externaldisturbance vectors. ui(t) ∈ Rn (i = 1, 2, . . . , N) are thecontrollers to be designed; the other parameters have the samemeanings as those in (2).

It is not necessary to assume that the inner couplingmatrix H (or H) and coupling configuration matrix B (or B)are symmetric and irreducible. Meanwhile, the correspondingtopological graph can be directed or undirected. Throughoutthe paper, we always assume that nonlinear functions f(x)and g(x) satisfy the uniform Lipschitz conditions,

‖f(x)− f(y)‖ ≤ L‖x− y‖ (4)

‖g(x)− g(y)‖ ≤ L‖x− y‖. (5)

We also assume the time-varying disturbances ξi(t) andηi(t) are bounded and satisfy the following condition

‖ξi(t)− ηi(t)‖ ≤ Li (6)

where Li > 0.According to systems (2) and (3), the error system is

described by

CDα0,tei(t) = Aei(t) + f(yi(t))− f(xi(t))

334 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 4, NO. 2, APRIL 2017

+ cN∑

j=1

bijHej(t) + [ηi(t)− ξi(t)] + ui(t)

+ cN∑

j=1

bijH[g(yj(t− τ))− g(xj(t− τ))] (7)

where ei(t) = yi(t) − xi(t), i = 1, 2, . . . , N . Thus, ourobjective is to design a suitable controller ui(t) such that errordynamical system (7) is asymptotically stable, i.e.,

limt→∞

‖yi(t; t0, x0)− xi(t; t′0, x′0)‖ = 0, i = 1, 2, . . . , N

which implies the drive system (2) and the response system(3) are synchronized.

B. Some Lemmas

Now we present some lemmas for later use.Lemma 1 [33]: Let x(0) = y(0) and CDα

0,tx(t) ≥CDα

0,ty(t), where α ∈ (0, 1). Then x(t) ≥ y(t).Lemma 2 [9], [10]: Consider the following linear fractional

system with time delays:

CDα0,tX(t) = AX(t) + X(tτ ), α ∈ (0, 1) (8)

where A = (aij)n×n, X(t) = (x1(t), x2(t), . . . , xn(t))T ,X(tτ ) = (

∑nj=1 k1jxj(t − τ1j),

∑nj=1 k2jxk(t− τ2j), . . . ,∑n

j=1 knjxj(t − τnj))T . Let M = (kij + aij)n×n and B =(kij exp(−sτij)+aij)n×n. If all the eigenvalues of M satisfy| arg(λ)| > π/2 and the characteristic equation det(∆(s)) =|sαIn − B| = 0 has no purely imaginary roots for any τij >0, i, j = 1, 2, . . . , n, then the zero solution of system (8) isLyapunov asymptotically stable.

Lemma 3 [34]: Let x(t) = (x1(t), . . . , xn(t))T ∈ Rn be areal continuous and differentiable vector function. Then

CDα0,t[x

T (t)Px(t)] ≤ 2xT (t)P CDα0,tx(t)

where 0 < α < 1, P is a symmetric and positive definitematrix.

Lemma 4 [35]: Let X and Y be arbitrary n-dimensionalreal vectors, K a positive definite matrix, and H ∈ Rn×n.Then, the following matrix inequality holds:

2XT HY ≤ XT HK−1X + Y T KY.

Lemma 5 [36]: Assume that Q = (qij)N×N is symmetric.Let M∗ = diagm∗

1,m∗2, . . . , m

∗l , 0, . . . , 0︸ ︷︷ ︸

N−l

, 1 ≤ l ≤ N , m∗i

> 0 (i = 1, 2, . . . , l), Q − M∗ =(

E − M∗ SST Ql

), M∗

= diagm∗1, . . . , m

∗l , m∗ = min1≤i≤lm∗

i , Ql is the minormatrix of Q by removing its first l (1 ≤ l ≤ N) row-columnpairs, E and S are matrices with appropriate dimensions.When m∗ > λmax(E − SQ−1

l ST ), then Q − M∗ < 0 isequivalent to Ql < 0.

C. Lyapunov-like Method for Fractional Nonlinear SystemWith Time Delays

Consider the Caputo fractional non-autonomous systemwith time delays

CDα0,tx(t) = f(t, x(t), x(t− τ)) (9)

with initial condition x(t) = x0(t), t ∈ [−τ, 0], whereα ∈ (0, 1), f : [0,∞) × Ω → Rn is piecewise continuouson t and locally Lipschitz with respect to x. Ω ∈ Rn is adomain that contains the origin x = 0. We always assumethat (9) has an equilibrium x = 0.

It is well known, by using the Lyapunov direct method,we can get the asymptotic stability of the non-delays systems.Next, we extend the Lyapunov direct method to the time delayscase, which leads to the Lyapunov asymptotic stability. Basedon [9], [10], [33], we could obtain the following theorem.

Theorem 1: Let x = 0 be an equilibrium point of system(9). If there exists a Lyapunov-like function V (t, x(t)) : [−τ,∞] × Ω → R which is continuously differentiable and locallyLipschitz with respect to x such that

α1‖x(t)‖a ≤ V (t, x(t)) ≤ α2‖x(t)‖ab (10)

CDα0,tV (t, x(t)) ≤ −α3‖x(t)‖ab

+ α4‖x(t− τ)‖a (11)

ν < µ sin(απ

2

)(12)

where a, b, α1, α2, α3 are positive constants, µ = α3/α2

and ν = α4/α1. Then x = 0 of system (9) is Lyapunovasymptotically stable.

Proof: It follows from (10) and (11) that

CDα0,tV (t, x(t))

≤ −α3

α2V (t, x(t)) +

α4

α1V (t− τ, x(t− τ)) (13)

where t ≥ 0.Consider the following system:

CDα0,tW (t, x(t))

= −µW (t, x(t)) + νW (t− τ, x(t− τ)) (14)

where W (t, x(t)) has the same initial conditions withV (t, x(t)), µ = α3/α2 and ν = α4/α1. Using Lemma 1,we have

0 ≤ V (t, x(t)) ≤ W (t, x(t)). (15)

By Lemma 2, the characteristic equation of (14) isdet(∆(s)) = sα + µ− ν exp(−sτ) = 0. Suppose that s = ωi= |ω|(cos(π/2)+ i sin(±π/2)). Substituting s into det(∆(s))gives

|ω|α(cos

(απ

2

)+ i sin

(±απ

2

))+ µ

− ν (cos(τω)− i sin(τω)) = 0.

Separating real and imaginary parts gives

|ω|α cos(απ

2

)+ µ = ν cos (τω) (16)

|ω|α sin(±απ

2

)= −ν sin (τω) . (17)

MA et al.: PINNING SYNCHRONIZATION BETWEEN TWO GENERAL FRACTIONAL COMPLEX DYNAMICAL NETWORKS · · · 335

According to (16) and (17), one has

|ω|2α + 2µ cos(απ

2

)|ω|α + µ2 − ν2 = 0. (18)

Obviously, when ν < µ sin(απ/2), no real number ωsatisfies (18). Furthermore, the eigenvalue of M in equationis ν − µ. When ν < µ sin(απ/2), that is ν < µ, implying| arg(λ(M))| > π/2. So, W (t, x(t)) → 0, as t → +∞.

From (15), V (t, x(t)) → 0, as t → +∞, which means allthe solutions of system (9) converge to x = 0. ¥

III. PINNING SYNCHRONIZATION OF FRACTIONALCOMPLEX NETWORKS WITH NONLINEAR COUPLING AND

EXTERNAL DISTURBANCES

In this section, some global asymptotically stable criteriaare presented below.

To realize synchronization between (2) and (3), assume thatfirst l (1 ≤ l ≤ N) nodes are pinned, the pinning controllersare chosen as

ui(t) = −piei(t)− qsgn(ei(t))n∑

j=1

|eij(t)|, 1 ≤ i ≤ l

ui(t) = 0, l + 1 ≤ i ≤ N

(19)

where pi > 0 are feedback gains, sgn(ei(t)) = (sgn(ei1(t)),sgn(ei2(t)), . . . , sgn(ein(t)))T are signum vectors, and q =12l

∑Ni=1 L2

i .Theorem 2: Suppose that the dynamical function f and

nonlinear coupling function g satisfy Lipschitz conditions (4)and (5), respectively. If there exists a matrix P satisfying thefollowing conditions

1) µ =− λmax

[(a + L +

12

+cβ2h(1 + L)

2

)IN

+ chB − P

]> 0 (20)

2) ν < µ sin(απ

2

)(21)

where a = ‖A‖, h = ‖H‖, h = ‖H‖, β1 = maxbij , j 6= i,β2 = max|bii| and ν = cL

2 (Nβ1L + hβ2), then the

fractional response network (3) asymptotically synchronizesto the drive network (2).

Proof: Construct the following Lyapunov-like function:

V (t, e(t)) =12

N∑

i=1

eTi (t)ei(t). (22)

Using (7) and Lemma 3, the fractional derivative ofV (t, e(t)) yields

CDα0,tV (t, e(t)) ≤

N∑

i=1

eTi (t) CDα

0,tei(t)

=N∑

i=1

eTi (t)Aei(t) +

N∑

i=1

eTi (t)[f(yi(t))− f(xi(t))]

+ cN∑

i=1

N∑

j=1

bijeTi (t)Hej(t) +

N∑

i=1

eTi (t)[ηi(t)− ξi(t)]

+ cN∑

i=1

N∑

j=1

bijeTi (t)H[g(yj(t− τ))− g(xj(t− τ))]

−l∑

i=1

pieTi (t)ei(t)−

l∑

i=1

qeTi (t)sgn(ei(t))

n∑j=1

|eij(t)|

≤ (a + L)N∑

i=1

eTi (t)ei(t) + c

N∑

i=1

N∑

j=1

bijeTi (t)Hej(t)

−l∑

i=1

pieTi (t)ei(t)− lq

+ c

N∑

i=1

N∑

j=1

bijeTi (t)H[g(yj(t− τ))− g(xj(t− τ))]

+N∑

i=1

eTi (t)[ηi(t)− ξi(t)]. (23)

From Lemma 4, we have (24), shown at the bottom of thispage, and

N∑

i=1

N∑

j=1

bijeTi (t)H

[g(yj(t− τ))− g(xj(t− τ))

]

≤N∑

i=1

N∑

j=1,j 6=i

bij

[(g(yj(t− τ))− g(xj(t− τ)))T (g(yj(t− τ))− g(xj(t− τ)))

+ eTi (t)Hei(t)

]+ 2hL

N∑

i=1

|bii| · ‖ei(t)‖ · ‖ei(t− τ)‖

≤N∑

i=1

N∑

j=1,j 6=i

bij

[L2eT

j (t− τ)ej(t− τ) + heTi (t)ei(t)

]

+ hLN∑

i=1

|bii|[eTi (t)ei(t) + eT

i (t− τ)ei(t− τ)]

≤ β2h(1 + L)N∑

i=1

eTi (t)ei(t) + L(Nβ1L + β2h)

N∑

i=1

eTi (t− τ)ei(t− τ) (24)

336 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 4, NO. 2, APRIL 2017

N∑

i=1

N∑

j=1

bijeTi (t)Hej(t)

=N∑

i=1

N∑

j=1,j 6=i

bijeTi (t)Hej(t)

+N∑

i=1

biieTi (t)

(H + HT

2

)ei(t)

≤ hN∑

i=1

N∑

j=1,j 6=i

bij‖ei(t)‖ · ‖ej(t)‖

+ ρmin

N∑

i=1

biieTi (t)ei(t) (25)

where ρmin is the minimum eigenvalue of the matrix (H +HT )/2. Using Lemma 4 and (6), we get

N∑

i=1

eTi (t)[ηi(t)− ξi(t)]− lq

≤ 12

N∑

i=1

eTi (t)ei(t)

+12

N∑

i=1

[ηi(t)− ξi(t)]T [ηi(t)− ξi(t)]− lq

≤ 12

N∑

i=1

eTi (t)ei(t) +

12

N∑

i=1

L2i − lq

=12

N∑

i=1

eTi (t)ei(t). (26)

Substituting (24)−(26) into (23), and from Lemma 3, weobtain that

CDα0,tV (t, e(t))

≤ ‖e(t)‖T

[(a + L +

12

+cβ2h(1 + L)

2

)IN

+ chB − P

]‖e(t)‖

+cL

2(Nβ1L + hβ2)‖e(t− τ)‖T · ‖e(t− τ)‖

≤ − µN∑

i=1

eTi ei(t) + ν

N∑

i=1

ei(t− τ)T ei(t− τ) (27)

where ‖e(t)‖ = (‖e1(t)‖, ‖e2(t)‖, . . . , ‖eN (t)‖)T , P =diag(p1, p2, . . . , pl︸ ︷︷ ︸

l

, 0, 0, . . . , 0︸ ︷︷ ︸N−l

), B = (BT + B)/2 and B is

a modifying matrix of B via replacing the diagonal elementsbii by (ρmin/h)bii.

According to Theorem 1, we have ‖ei(t)‖ → 0, that is‖yi(t) − xi(t)‖ → 0 as t → ∞, which means that theasymptotical synchronization between drive system (2) andresponse system (3) is realized. ¥

Furthermore, from (27), let Q = (a + L + 1/2 + cβ2h(1 +

L)/2)IN + chB, and Q − P =(

E − P ∗ SST Ql

), where 1

≤ l ≤ N , P ∗ = diagp1, p2, . . . , pl, Ql is the part matrix

of Q by removing its first l row-column pairs, E and S arematrices with appropriate dimensions. Based on Lemma 5, andsupposing that pi (i = 1, . . . , l) are suitably large, Q−P < 0is equivalent to Ql = [(a+L+1/2+cβ2h(1+L)/2)IN +chB]l< 0. One has λmax[(a+L+1/2+ cβ2h(1+ L)/2)IN +chB]l= (a+L+1/2+ cβ2h(1+ L)/2)+ chλmax(Bl) < 0. So, thefollowing corollary can be immediately obtained.

Corollary 1: Under assumptions (4) and (5), the fractionalresponse network (3) asymptotically synchronizes to the driveone (2) under the controller (28), where pi (i = 1, . . . , l) aresufficiently large, and the following conditions satisfied :

µ = −[(

a + L +12

+cβ2h(1 + L)

2

)

+ chλmax(Bl)]

> 0

ν < µ sin(απ

2

)

in which ν = cL2 (Nβ1L + hβ2).

Corollary 2: If g(xj(t−τ)) = xj(t−τ) and g(yj(t−τ)) =yj(t− τ), the synchronous conditions between (2) and (3) arereduced to:

µ = − λmax

[(a + L +

12

+ cβ2h

)IN

+ chB − P

]> 0

ν < µ sin(απ

2

)

where ν = c2 (Nβ1 + hβ2).

Corollary 3: Assume ξi(t) = ηi(t) = 0, the fractionalcomplex networks (2) and (3) do not contain the disturbances.Under the assumptions (4) and (5), fractional systems (2) and(3) can be asymptotically synchronized under the controllers

ui(t) = −piei(t), 1 ≤ i ≤ l

ui(t) = 0, l + 1 ≤ i ≤ N(28)

and the following control conditions:

µ = − λmax

[(a + L +

cβ2h(1 + L)2

)IN

+ chB − P

]> 0

ν < µ sin(απ

2

)

where ν = cL2 (Nβ1L + hβ2).

IV. NUMERICAL EXAMPLE

In this section, a numerical example is presented. Considera complex network with 10 nodes, the fractional dynamicalequation of each node is described by the following fractionalchaotic Lorenz system

CDα0,txi1 = aL(xi2 − xi1)

CDα0,txi2 = bLxi1 − xi1xi3 − xi2

CDα0,txi3 = xi1xi2 − cLxi3

(29)

MA et al.: PINNING SYNCHRONIZATION BETWEEN TWO GENERAL FRACTIONAL COMPLEX DYNAMICAL NETWORKS · · · 337

where i = 1, 2, . . . , 10. The parameters are chosen as aL = 10,bL = 28, cL = 8/3 and α = 0.995. From (2) and (3), we know

that A =

−a a 0b −1 00 0 −c

, f(xi(t)) =

0−xi1xi3

xi1xi2

,

and that system (29) is chaotic, see Fig. 1.

Fig. 1. Chaotic attractor of fractional Lorenz system with order α = 0.995.

For convenience, let H = H = I , the coupling configura-tion matrices B and B are given as follows,

−2 1 0 1 0 0 0 0 0 00 −2 1 0 1 0 0 0 0 01 0 −3 0 0 1 1 0 0 00 0 1 −1 0 0 0 0 0 00 1 0 1 −3 0 0 0 1 00 0 1 0 1 −2 0 0 0 00 0 0 0 0 1 −3 1 0 10 1 0 1 0 0 0 −3 1 01 0 1 0 0 0 1 0 −4 10 0 1 1 0 0 0 0 0 −2

.

The nonlinear coupling function is chosen as

g(xj(t− τ)) = (xj1(t− τ) + sin(xj1(t− τ)),xj2(t− τ) + sin(xj2(t− τ)),

xj3(t− τ) + sin(xj3(t− τ)))T .

It is known that the Lorenz system is bounded. Actually,‖xi1‖ ≤ 25, ‖xi2‖ ≤ 30, ‖xi3‖ ≤ 60, ‖yi1‖ ≤ 25,‖yi2‖ ≤ 30, ‖yi3‖ ≤ 60, i = 1, 2, . . . , 10, and

‖f(xi)− f(yi)‖≤

√(−xi1xi3 + yi1yi3)2 + (xi1xi2 − yi1yi2)2

≤ 75.83‖ei‖that is L = 75.83. Obviously, L = 2. According to the methodproposed in [15], whose out-degrees are bigger than their in-degrees, it should be selected as pinned candidates. The out-degrees of nodes 2, 3 and 4 are bigger than their in-degrees, sowe choose them as the pinned nodes. Rearrange the networknodes and the new order will be 4, 3, 2, 1, 6, 10, 5, 7, 8, 9.

Under the no disturbance case, that is ξi(t) = ηi(t) = (0,0, 0)T , let pi = 910, i = 1, 2, 3, when α = 0.9, c = 100,c = 0.01, one has µ sin(απ/2) = 0.2808 and ν = 0.2400.From Corollary 3, it is clear that pinning conditions hold.The simulation results are shown in Fig. 2, which showsthe time waveforms of errors ei1, ei2, ei3, i = 1, 2, . . . , 10.

From the figures, fractional complex networks (2) and (3)are synchronized, which demonstrate the effectiveness of theproposed method.

Now, we come to the disturbance case. Let pi = 930,i = 1, 2, 3, ξi(t) = (0, 0, 0)T , ηi(t) = (0.3 sin t cos t, 0.1 sin t,0.5 cos t)T , α = 0.9, c = 100, c = 0.01, one has q = 0.5833,µ sin(απ/2) = 0.2489 and ν = 0.2400. From Theorem 2,it is clear that pinning conditions hold. Fig. 3 illustrates thesynchronization phenomenon in noisy environment. It showsthe error trajectories of drive and response networks, fromwhich we can see that the synchronization between the drivingand response networks is achieved successfully.

Fig. 2. Time evolution of the error states e1i, e2i and e3i with no distur-bance.

V. CONCLUSION

In this paper, we proposed a fractional pinning controllerand presented a synchronization law for the delayed fractionalcomplex networks with nonlinear couplings and disturbances.Some new synchronization criteria are proposed based on theLyapunov-like stability theory. This method can be appliedto many types of fractional complex networks. Furthermore,

338 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 4, NO. 2, APRIL 2017

the coupling configuration matrices and the inner-couplingmatrices are not assumed to be symmetric or irreducible. Itmeans that this method is more general. The numerical resultsshowed the effectiveness of the proposed controllers.

Fig. 3. Time evolution of the error states e1i, e2i and e3i with disturbances.

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Weiyuan Ma received the B. Sc. degree in math-ematics at Northwest Normal University, Lanzhou,China, in 2004, and received the M. Sc. degree inmathematics at Lanzhou University, Lanzhou, China,in 2007. He is working toward the Ph. D. degree incomputational mathematics at Lanzhou University,Lanzhou, China. He is currently an associate profes-sor at the School of Mathematics and Computer Sci-ence, Northwest University for Nationalities, China.His research interests include complex networks,nonlinear systems, and chaos control.

Yujiang Wu received the B. Sc. and M. Sc. degreesin mathematics at Lanzhou University, Lanzhou,China, in 1982 and 1985, respectively, and he re-ceived the Ph. D. degree in mathematics at ShanghaiUniversity, Shanghai, China, in 1997. He is currentlya professor at the School of Mathematics and Statis-tics, Lanzhou University, China. His main researchinterest is scientific computing. Corresponding au-thor of this paper.

Changpin Li received his Ph. D. degree in com-putational mathematics from Shanghai University,China in 1998. After graduation, he worked in thesame university and became a full professor in2007 and director of the Institute of ComputationalMathematics in 2012. Dr. Li is in the broad area ofapplied theory and computation of bifurcation andchaos, dynamics of fractional ordinary differentialequations, dynamics of complex networks, numeri-cal methods and computations of fractional partialdifferential equations, fractional modeling. Dr Li’s

current work is focused on fractional modeling of anomalous diffusion withmathematical analysis and numerical calculations. Dr Li has published nearly100 papers in referred journals with more than 2500 citations. He has co-editedone book published by World Scientific, and has published one book by CRCpress. He is on the editorial boards of several journals such as FractionalCalculus and Applied Analysis, International Journal of Bifurcation andChaos, International Journal of Computer Mathematics, etc. He was alsothe lead guest editor of European Physical Journal-Special Topics (2011),International Journal of Bifurcation and Chaos (2012), and PhilosophicalTransactions of the Royal Society A: Mathematical, Physical and EngineeringSciences (2013). He was awarded the Shanghai Natural Science Prize in 2010,Bao Steel Prize in 2011, and Riemann-Liouville Award: Best FDA Paper(theory) in FDA’12 in 2012.