Topology identification of uncertain nonlinearly coupled complex networkswith delays based on anticipatory synchronizationYanqiu Che, Ruixue Li, Chunxiao Han, Shigang Cui, Jiang Wang et al. Citation: Chaos 23, 013127 (2013); doi: 10.1063/1.4793541 View online: http://dx.doi.org/10.1063/1.4793541 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v23/i1 Published by the AIP Publishing LLC. Additional information on ChaosJournal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors

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Topology identification of uncertain nonlinearly coupled complex networkswith delays based on anticipatory synchronization

Yanqiu Che,1,2,a) Ruixue Li,1 Chunxiao Han,1 Shigang Cui,1 Jiang Wang,3 Xile Wei,3

and Bin Deng3

1Tianjin Key Laboratory of Information Sensing & Intelligent Control, Tianjin University of Technologyand Education, Tianjin 300222, People’s Republic of China2Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China3School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China

(Received 20 September 2012; accepted 5 February 2013; published online 21 February 2013)

This paper presents an adaptive anticipatory synchronization based method for simultaneous

identification of topology and parameters of uncertain nonlinearly coupled complex dynamical

networks with time delays. An adaptive controller is proposed, based on Lyapunov stability theorem

and Barb�alat’s Lemma, to guarantee the stability of the anticipatory synchronization manifold

between drive and response networks. Meanwhile, not only the identification criteria of network

topology and system parameters are obtained but also the anticipatory time is identified. Numerical

simulation results illustrate the effectiveness of the proposed method. VC 2013 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4793541]

The research on complex networks, from natural to man-

made ones, pervades almost all scientific and technological

fields. Mathematically, complex networks can be described

as graphs composed of nodes representing individual units

that are linked by edges denoting physical or functional

connections. Among various problems in the study of com-

plex networks, topology identification is a significant one.

This paper investigates the topology identification of the

uncertain complex networks with unknown parameters,

nonlinear coupling and delays in states and coupling,

which are not fully considered in the existing research.

Moreover, in the previous widely used auto-synchroniza-

tion-based method, the stability of the complete synchroni-

zation (CS) manifold will be destroyed because of the

existence of time delays in real systems. As a result, the

estimation of the unknown topology and parameters will

be practically impossible. In this paper, the proposed novel

adaptive identification approach is based on anticipatory

synchronization (AS), that is, the states of the response sys-

tem synchronize with the future states of the drive system

with an adaptive determined anticipatory time, which is

much more practical. Numerical simulations are given to

illustrate the effectiveness of the proposed approach.

I. INTRODUCTION

Complex networks, which successfully describe many

natural and man-made systems including gene network, neu-

ral networks, World Wide Web, and communication net-

work, to name but a few, have gained growing attention in

the past decades.1–6 Various issues in complex networks

have been investigated, such as analysis of network dynam-

ics,2–4 small-world and scale-free characteristics,5,6 and syn-

chronization of large-scale complex networks with certain

types of topology.7–10 Recently, topology identification for

complex networks is emerging as an attractive topic of

research because of its essential role in lots of complex net-

works.11–20 Since a complex network is composed of a large

number of interacting nodes, it is clear that small perturba-

tions in system parameters of nodes may dramatically

change the cooperative behavior and network features.21,22

Moreover, due to the finite speed of signal transmission over

a distance, time delays ubiquitously exist in complex net-

works,23,24 such as neural networks and communication net-

works. Therefore, simultaneous identification of topology

and parameter for uncertain complex networks with time

delays is of theoretical and practical importance.

Thus far, a few methods have been proposed for topol-

ogy identification of complex networks. These methods can

basically be divided into two categories, i.e., perturbation-

based12,13,17,19 and synchronization-based11,14–16,18,20 ones.

In the former category, researchers design various controller

to drive the complex dynamical network with unknown topol-

ogy to a steady-state, and then, by means of small perturba-

tion, transform the identification problem to solving a large

linear equation. In the latter category, researchers consider

the complex dynamical network with unknown topology as a

drive system, and construct a response system and design a

controller to make the states of the response system synchron-

ize with that of the drive one. During the process of synchro-

nization (CS,11,14–16,18 lag synchronization (LS),20 or AS),

the unknown topology can be identified. Most of the afore-

mentioned methods assume that (i) parameters of nodes are

known a priori, (ii) the nodes of complex networks are line-

arly coupled, or (iii) there exists only instant (at time t) ordelayed (at time t� s) information communication between

nodes. However, all these assumptions are not the usual cases

in real-world networks. Moreover, the CS-based method

requires complete synchronization between current states of

the drive and response system, which will be practically

impossible because of the presence of time delays during the

signal transmission. On the contrary, in many applications,a)Electronic mail: yqche@tju.edu.cn.

1054-1500/2013/23(1)/013127/7/$30.00 VC 2013 American Institute of Physics23, 013127-1

CHAOS 23, 013127 (2013)

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essential drive-response synchronization occurs regardless of

the natural coupling delays between drive and response sys-

tems,1,2 such as chemical transport delay between cells in a

living organism, optical delay in the synchronized movement

of swarms of birds, communication delay between multiple

robots in the execution of shared tasks, and transmission

delay in secure communication. In this regard, LS- or AS-

based methods, which require the response system to syn-

chronize with the past or future states of the drive system,

have important technological implications. Particularly, AS is

a state that the response system may anticipate the drive sys-

tem in real time, which is more reasonable and has been

observed in many practical systems such as lasers,25,26 elec-

tronic circuits,27 and neuronal systems.28 Thus, AS has

attracted a great deal of attention.29–36 However, to the best

of our knowledge, AS-based topology identification technol-

ogy has not been fully studied.

Motivated by the existing works, in this paper, using

adaptive control techniques, we propose an AS-based

approach to identify unknown network topological structure

and system parameters for general uncertain delayed nonli-

nearly coupled complex dynamical networks. Based on

Lyapunov stability theorem and Barb�alat’s Lemma,37

adaptive identification rules are proposed for the nonli-

nearly coupled networks with node and coupling delays.

During the AS process, not only the topology and parame-

ters but also the anticipatory time are identified, which

makes our approach much more practical.

The rest of this paper is organized as follows. Section II

introduces some useful mathematical preliminaries. Section III

presents the main theory for the identification of network topo-

logical structure and system parameters for nonlinearly coupled

uncertain complex dynamical networks with delays. After that,

in Sec. IV, a network of Hindmarsh-Rose (HR) neuronal

models38 is used to illustrate the effectiveness and reliability of

the suggested technique. Finally, Sec. V draws the conclusions.

II. PRELIMINARIES

Consider the following uncertain dynamical complex

networks consisting of N different nodes which are n-dimen-

sional dynamical units:

_xiðtÞ ¼ fiðxiðtÞ; xiðt� siÞ; aiÞ þXN

j¼1

aijgijðxjðtÞÞ

þXN

j¼1

bijhijðxjðt� sijÞÞ; i ¼ 1; 2;…;N (1)

where xi ¼ ðxi1; xi2;…; xinÞT 2Rn�1 and ai 2Rmi�1

for i¼ 1;…;N are the state vectors and unknown system

parameter vectors of the i-th node, respectively. fiðxiðtÞ;xiðt� siÞ;aiÞ 2Rn�1 is a known function of the i-th node dy-

namics. gijðxjÞ ¼ ðgij1ðxjÞ;gij2ðxjÞ;…;gijnðxjÞÞT and hijðxjÞ¼ ðhij1ðxjÞ;hij2ðxjÞ;…;hijnðxjÞÞT 2Rn�1 are known nonlinear

inner-coupling functions. A¼ ðaijÞN�N and B¼ ðbijÞN�N 2RN�N are unknown or uncertain outer-coupling configuration

matrices. si and sij are time-varying node delay of i-th node

and coupling delay from node j to node i, respectively.

In the following, we denote xs ¼ xðt� sÞ and omit the

time variable t whenever no confusion may arise.

Suppose that uncertain quantities in the i-th node

dynamic systems can be expressed linearly with respect to

unknown constant parameters ai, then rewrite system Eq. (1)

as

_xi ¼ Fiðx; xsii Þ þ �Fiðxi; x

sii Þai þ

XN

j¼1

aijgijðxjÞ

þXN

j¼1

bijhijðxsi

j

j Þ; i ¼ 1; 2;…;N (2)

where Fiðxi; xsii Þ 2 Rn�1 and �Fiðxi; x

sii Þ 2 Rn�mi are known

functions of the i-th node dynamics.

We now introduce some assumptions and lemmas that

will be required throughout this study.

Assumption 1: The functions gijð�Þ and hijð�Þ are

Lipschitzian, that is, for any vectors x, y 2 Rn�1, there exist

nonnegative constants Lgij and Lhij for i ¼ 1; 2;…;N such that

jjgijðxÞ � gijðyÞjj � Lgijjjx� yjjjjhijðxÞ � hiðyÞjj � Lhijjjx� yjj:

(3)

Hereafter, the norm jjxjj of vector x is defined as jjxjj ¼ xTx.

Assumption 2: There exists a nonnegative constant Msuch that

jjfiðxi;xsi ;aiÞ � fiðyi;y

si ;aiÞjj2 �Mðjjxi� yijj

2þ jjxsi � ys

i jj2Þ:(4)

Remark 1. Note that the assumptions 1 and 2 hold as long

as @F@x

(here, F denotes gij, hij, or fi) are uniformly bounded.39

Almost all well-known finite-dimensional chaotic and hyper-

chaotic systems, such as Chua’s circuit, hyperchaotic L€u sys-

tem, generalized Lorenz system, and neuronal systems such

as FitzHugh-Nagumo (FHN) and HR models have the form of

Eq. (2), which meet the conditions of assumptions 1 and 2.

Assumption 3: Time-varying delays siðtÞ and sijðtÞ are dif-

ferentiable and satisfy 0� _siðtÞ�l<1 and 0� _sijðtÞ�g<1,

respectively, where l and g are constants.

Clearly, assumption 3 is certainly valid for constant sðtÞ.Lemma 1. For any vectors x; y 2 Rn�1, the matrix in-

equality 2xTy � xTxþ yTy holds.

Definition 1. Two systems of complex networks are said

to be AS if limt!1 jjyðt� sdðtÞÞ � xðtÞjj ¼ 0 for a positive

anticipatory time sdðtÞ, where x and y are the system states

of the drive and response complex networks, respectively.

The main goal of this paper is to identify the unknown

or uncertain coupling strengths aij and bij (for i; j ¼ 1;…;N),

namely, its network topological structure, and all unknown

system parameter vectors ai (for i ¼ 1;…;N) of its node dy-

namical systems.

III. GENERAL THEORY

In this paper, we consider system Eq. (2) as a drive net-

work, and construct another complex dynamical network

called the response network with actual control as follows:

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_xsd

i ¼ Fiðxsdi ; x

sdþsii Þ þ �Fiðxsd

i ; xsdþsii Þasd

i

þXN

j¼1

asdij gijðxsd

j Þ þXN

j¼1

bsd

ij hijðxsdþsi

j

j Þ þ ui;

ui ¼ �sd~xi; _sd ¼ cXN

i¼1

jj~xijj2;

_asd

i ¼ �b �FTi ðx

sdi ; x

sdþsii Þ~xi;

_asd

ij ¼ �da~xTi gijðxsd

j Þ;_bsd

ij ¼ �db~xTi hijðx

sdþsij

j Þ;

(5)

where xi 2 Rn�1 is the response state vector of the i-th node,

ui 2 Rn is its controller, ~xi ¼ xsdi � xi is the synchronization

error between drive and response networks. ai 2 Rmi and

aij; bij 2 R are the estimated values of parameter vector ai

and unknown coupling strengths aij and bij, respectively, for

i; j ¼ 1;…N. c, b, da and db are any positive constants.

Theorem 1. Suppose that assumptions 1, 2, and 3 hold.

Then, the uncertain parameter vectors faigNi¼1 and coupling

configuration matrices A ¼ ðaijÞN�N and B ¼ ðbijÞN�N of the

uncertain general complex dynamical network Eq. (2) can be

identified by the estimated values faigNi¼1; A ¼ ðaijÞN�N and

B ¼ ðbijÞN�N , respectively, via the response network Eq. (5).

See proof of Theorem 1 in Appendix.

IV. EXAMPLE

To verify the effectiveness of the proposed method in to-

pology and parameter identification of complex networks,

we consider a network consists of 5 non-identical HR neuro-

nal systems38 with instant electrical coupling and delayed

chemical synaptic coupling

_xi1 ¼ xi2 � aix3i1 þ bix

2i1 � xi3 þ Iext þ ge

XN

j¼1

aijxj1

þ gsðxi1 � VsÞXN

j¼1

bijCðxsi

j

j Þ;

_xi2 ¼ ci � dix2i1 � xi2;

_xi3 ¼ ri½siðxi1 � x0Þ � xi3�;

(6)

where xi1 are the membrane potentials, xi2 are associated with

the fast current Naþ or Kþ, and xi3 with the slow current, for

example, Ca2þ. The nominal values of the parameters

are ai¼1:0;bi¼3:0;ci¼1:0, di¼5:0;si¼4:0;ri¼0:006, and

x0¼�1:60. Iext is the external current input. In the range of

2:92< Iext<3:40, individual HR systems show chaotic spike-

bursting behaviors.40 ge and gs are electrical and chemical

synaptic coupling strengths, respectively. The reversal poten-

tial Vs>xi1ðtÞ, i.e., the synapse is excitatory. The sigmoidal

function CðxjÞ¼ 1=f1þexp½�10ðxj�HsÞ�g describes the

synaptic coupling function. The matrix (aij) describes the elec-

trical coupling, while the matrix (bij) describes the synaptic

coupling.

In the following simulations, we fix Iext ¼ 3:2;Hs ¼ �0:25;Vs ¼ 2; ge ¼ 0:1, and gs ¼ 0:2 and assume that

the parameters di (the extension to more than one unknown

parameters is immediate), aij, and bij are unknown and

sij ¼ 0:2 for i; j ¼ 1;…; 5.

Comparing Eq. (6) with Eq. (2), we have xi ¼ ðxi1; xi2;xi3ÞT ;FiðxÞ¼ ðxi2 � ax3

i1 þ bx2i1; c� xi2; r½sðxi1 � x0Þ� xi3�ÞT ;

�FiðxÞ ¼ ð0;�x2i1; 0Þ

T, ai ¼ di; gijðxjÞ ¼ ðgexj1; 0; 0ÞT ; hijðx

sij

j Þ¼ ðgsðxi1 � VsÞCðx

sij

j Þ; 0; 0ÞT, n¼ 3 and N¼ 5.

As is well-known, failures in components of a dynamical

system can be represented by abrupt changes in the system

parameters. Similarly, repairs of the failed components and

system reconfigurations also cause abrupt changes in the sys-

tem parameters.41 Taking the neuronal system as an example,

in the pathological condition, the coupling between different

neurons may become stronger or weaker than that in normal

states.42 Motivation for applicability of our method to a large

class of realistic problems, we consider system models with

abrupt (but infrequent) parameters changes in the simulations.

That is, when t < 1500, for i; j ¼ 1;…; 5, we set the system

parameter di ¼ ð4:9; 4:95; 5; 5:05; 5:1ÞT and the coupling ma-

trix (aij) and (bij)

A ¼

�3 0 1 1 1

0 �2 1 1 0

1 1 �4 1 1

1 1 1 �3 0

1 0 1 0 �2

26666664

37777775;

B ¼

0 1 0 1 1

1 0 1 1 0

1 1 0 0 1

0 1 1 0 1

1 0 1 1 0

26666664

37777775:

When t � 1500, we switch di to ð4:8; 4:9; 5; 5:1; 5:2ÞT ,

and switch (aij) and (bij) to

A ¼

�2 1 0 0 1

1 �3 1 0 1

0 1 �2 1 0

0 0 1 �2 1

1 1 0 1 �3

26666664

37777775;

B ¼

1 0 1 0 0

0 1 0 0 1

0 0 1 1 0

1 0 0 1 0

0 1 0 0 1

26666664

37777775:

The parameters and topology are estimated by means of

Theorem 1. The control gains are chosen as c ¼ b ¼ 1 and

da ¼ db ¼ 10. The initial values estimated for the unknown

parameters and coupling strengths are set to be zero. The ini-

tial states of the drive and response systems are randomly

given.

The simulation results are shown in Figs. 1–4. In Fig. 1,

waveforms of the states x11ðtÞ and x11ðtÞ, and the state errors

~xij ¼ xijðt� sdÞ � xijðtÞ (i ¼ 1;…; 5; j ¼ 1;…; 3) versus

time are shown in (a) and (b), respectively. The state errors

013127-3 Che et al. Chaos 23, 013127 (2013)

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tend to zero asymptotically as time evolves, which indicates

anticipatory synchronization between the drive and response

systems. Figs. 2 and 3 show the identification of unknown

system parameters di and network topology, respectively.

Figs. 2(a) and 2(b) describe the estimation values of di and

the corresponding estimation errors, respectively. In Fig. 3,

the examples of estimation values of a2j and b2j and the

corresponding estimation errors ~aij ¼ aijðt� sdÞ � aij and~bij ¼ bijðt� sdÞ � bij for i; j ¼ 1;…; 5 are shown in (a, c)

and (b, d), respectively. Clearly, the estimated values of pa-

rameters and coupling topology finally evolve to the real

values of the unknown parameters and topology. Note that

the dynamic evolution of the response system in Figs. 1–3 at

t¼ 1500 when the drive system undergoes abrupt parame-

ters changes anticipates the dynamics and parameters

changes of the drive system. This implies that the proposed

approach can be applied for online “anticipating” of the net-

work topology and system parameters. Fig. 4 shows the

time series of the anticipatory time sd which tends to a con-

stant as time evolves. The identification is very successful.

Thus, the numerical simulations illustrate the effectiveness

of Theorem 1.

FIG. 1. Anticipatory synchronization of

Hindmarsh-Rose networks. (a) Time series of

states of drive and response network, x11ðtÞ(solid line) and x11ðtÞ (dotted-dashed line). (b)

Time series of anticipatory synchronization

errors ~xij ¼ x ijðt� sdÞ � xijðtÞ; i ¼ 1;…; 5;j ¼ 1;…; 3.

FIG. 2. Identification of system parameters

di (i ¼ 1;…; 5) in a Hindmarsh-Rose net-

work. (a) Time series of the estimation val-

ues d i and (b) time series of the estimation

errors ~di ¼ d iðt� sdÞ � di.

FIG. 3. Identification of network topology

aij and bij, i; j ¼ 1;…; 5 in a Hindmarsh-

Rose network. (a) and (c) are the time series

of the estimation values a2j and b2j, respec-

tively. (b) and (d) are the time series of the

estimation errors ~aij ¼ aijðt� sdÞ � aij and~bij ¼ bijðt� sdÞ � bij, respectively.

013127-4 Che et al. Chaos 23, 013127 (2013)

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V. CONCLUSIONS

We have proposed a new method based on adaptive an-

ticipatory synchronization to simultaneously identify the

unknown network topological structure and system parame-

ters of uncertain general complex dynamical networks with

nonlinear delayed coupling. According to the Lyapunov sta-

bility theorem and Barb�alat’s Lemma, the proposed method

guarantees the stable anticipatory synchronization between

drive and response systems, and thus ensures effective identi-

fication of the unknown network topological structure and

system parameters. This method also has the ability to dynam-

ically anticipate the parameters and topological structure.

Moreover, it operates in anticipatory synchronization, and the

anticipatory time is simultaneously identified, which makes it

superior than the previous methods that always require a com-

plete synchronization.11,14–16,18 The simulation results have

demonstrated the validity of the proposed method.

As in many previous studies,11,13–18,20 to keep things

simple, we also assume that all the states of each node are

completely available, in principle, which is not practical for

the case in which only partial state variables of partial nodes

are accessible. For example, in neuronal networks, usually

only the membrane potentials can be measurable. The pertur-

bation method12 works only for the network that automati-

cally reaches a stable stationary state, which fails for network

systems with complex dynamical behaviors (e.g., chaos).

Although the delayed feedback based steady state control

method19 accomplishes topology detection using only one

state variables of each node, its main disadvantages, including

careful calculation of control parameters, influence of dynam-

ical behavior of systems, and inability to time-varying topol-

ogy, limit its applications. On the contrary, our method does

not have these limitations and can be extended to practical

applications by combination with nonlinear state-observation

techniques. As verified by many theoretical and experimental

results, the unscented Kalman filter (UKF) is powerful in state

estimation of nonlinear system.43–45 An UKF aided AS-based

method for topology identification is under our investigation.

ACKNOWLEDGMENTS

This work was supported by The National Natural

Science Foundation of China (Grants Nos. 50907044,

60901035, 61072012, 61172009, and 61104032) and The

Science and Technology Development Program of Tianjin

Higher Education (Grants Nos. 20100819 and 20110834).

We are grateful to the editor and reviewers for their valuable

comments and suggestions. We would also acknowledge the

support of Tianjin University of Technology and Education

(Grants Nos. KYQD10009 and KJ11-04).

APPENDIX: PROOF OF THEOREM 1

Proof. Denote ~xi ¼ xsdi � xi; ~ai ¼ asd

i � ai; ~aij ¼ asdij

�aij; ~bij ¼ bsd

ij � bij. Then, the error dynamics is described as

_~x i ¼ Fiðxsdi ; x

sdþsii Þ þ �Fiðxsd

i ; xsdþsii Þasd

i � Fiðxi; xsii Þ � �Fiðxi; x

sii Þai

þXN

j¼1

asdij gijðxsd

j Þ þXN

j¼1

bsd

ij hijðxsdþsi

j

j Þ �XN

j¼1

aijgijðxjÞ �XN

j¼1

bijhijðxsi

j

j Þ þ ui

¼ fiðxsdi ; x

sdþsii ; aiÞ � fiðxi; x

sii ; aiÞ þ �Fiðxsd

i ; xsdþsii Þ~ai þ

XN

j¼1

~aijgijðxsdj Þ þ

XN

j¼1

aijðgijðxsdj Þ � gijðxjÞÞ

þXN

j¼1

~bijhijðxsdþsi

j

j Þ þXN

j¼1

bijðhijðxsdþsi

j

j Þ � hijðxsi

j

j ÞÞ þ ui: (A1)

In order to guarantee ~xiðtÞ ! 0; ~aiðtÞ ! 0; ~aijðtÞ ! 0, ~bijðtÞ ! 0, and sdðtÞ ! s�d as t!1, consider the following

Lyapunov function candidate:

2V ¼XN

i¼1

~xTi ~xi þ

1

b

XN

i¼1

~aTi ~ai þ

1

da

XN

i¼1

XN

j¼1

~a2ij þ

1

db

XN

i¼1

XN

j¼1

~b2

ij þ1

cðsd � s�dÞ

2

þ M

1� l

XN

i¼1

ðt

t�si

~xTi ðsÞ~xiðsÞdsþ j

XN

i¼1

XN

j¼1

ðt

t�sij

~xTj ðsÞ~xjðsÞds; (A2)

FIG. 4. Time series of anticipatory time sdðtÞ.

013127-5 Che et al. Chaos 23, 013127 (2013)

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where s�d and j are sufficiently large positive constants to be determined.

Differentiating Eq. (A2) about time and noting Eqs. (5) and (A1), one has

_V jð5ÞðA1Þ ¼XN

i¼1

~xTi

_~x i þ1

b

XN

i¼1

~aTi

_asd

i þ1

da

XN

i¼1

XN

j¼1

~aij_asd

ij þ1

db

XN

i¼1

XN

j¼1

~bij_bsd

ij þ1

cðsd � s�dÞ _sd

þ M

2ð1� lÞXN

i¼1

½jj~xijj2 � ð1� _siÞjj~xsii jj

2� þ j2

XN

i¼1

XN

j¼1

½jj~xjjj2 � ð1� _sijÞjj~x

sij

j jj2�

¼XN

i¼1

~xTi ½fiðx

sdi ; x

sdþsii ; aiÞ � fiðxi; x

sii ; ai� � s�d

XN

i¼1

jj~xijj2

þXN

i¼1

XN

j¼1

aij~xTi ðgijðxsd

j Þ � gijðxjÞÞ þXN

i¼1

XN

j¼1

bij~xTi ðhijðx

sdþsij

j Þ � hijðxsi

j

j ÞÞ

þ M

2ð1� lÞXN

i¼1

½jj~xijj2 � ð1� _siÞjj~xsii jj

2� þ j2

XN

i¼1

XN

j¼1

½jj~xjjj2 � ð1� _sijÞjj~x

sij

j jj2�: (A3)

From assumptions 1, 2, and Lemma 1, one gets

~xTi ½fiðxsd

i ; xsdþsii ; aiÞ � fiðxi; x

sii ; ai� �

M þ 1

2jj~xijj2 þ

M

2jjxsi

i jj2; (A4)

aij~xTi ðgijðxsd

j Þ � gijðxjÞÞ � jaijj � jj~xjj � jjðgijðxsdj Þ � gijðxjÞjj � Lgijjaijj � jj~xjj � jj~xjjj �

Lgijjaijj2ðjj~xijj2 þ jj~xjjj2Þ; (A5)

bij~xTi ðhijðx

sdþsij

j Þ � hijðxsi

j

j ÞÞ � jbijj � jj~xijj � jjhijðxsdþsi

j

j Þ � hijðxsi

j

j Þjj � Lhijjbijj � jj~xijj � jj~xsi

j

j jj �Lhijjbijj

2ðjj~xijj2 þ jj~x

sij

j jj2Þ: (A6)

Denote aM ¼ maxi;j¼1;2;…;Nfjaijjg; bM ¼ maxi;j¼1;2;…;Nfjbijjg; LgM ¼ maxi¼1;2;…;NfLgijg, and LhM ¼ maxi¼1;2;…;NfLhijg.According to Eqs. (A4)–(A6), one obtains

_V jð5ÞðA1Þ �M þ 1

2þ M

2ð1� lÞ þ N LgMaM þLhMbM

2þ j

2

� �� s�d

� �XN

i¼1

jj~xijj2

þMð _si � lÞ2ð1� lÞ

XN

i¼1

jj~xsii jj

2 þXN

i¼1

XN

j¼1

LhMbM þ jð_sij � 1Þ

2jj~xsi

j

j jj2: (A7)

Letting

j ¼ LhMbM

1� g;

s�d ¼M þ 1

2þ M

2ð1� lÞ þ N LgMaM þLhMbM

2þ j

2

� �þ 1;

(A8)

noting assumption 3 and denoting eðtÞ ¼ ð~xT1 ðtÞ; ~xT

2 ðtÞ;…;~xT

NðtÞÞT 2 RnN�1, one has

_V jð5Þð7Þ � �eTe � 0: (A9)

From the above inequality, one has

0 � limt!1

ðt

0

eTðsÞeðsÞds � Vð0Þ � limt!1

VðtÞ: (A10)

Obviously, the right part of the above inequality is

bounded since both V(0) and V(t) are bounded according to

Eqs. (A2) and (A9). Therefore, eðtÞ 2L2. Moreover, eðtÞ is

bounded, so eðtÞ 2L1. According to Eq. (A1), _eðtÞ exists

and is bounded for t � 0. Thus, it follows from Barb�alat’s

Lemma37 that limt!1 eðtÞ ¼ 0. Then, one has limt!1_~x iðtÞ ¼ 0 and x

sd

i converges to

�i ¼�

xsdi : �Fiðxsd

i ; xsdþsii Þ~ai þ

XN

j¼1

~aijgijðxsdj Þ

þXN

j¼1

~bijhijðxsdþsi

j

j Þ ¼ 0

�as t!1; (A11)

where i ¼ 1; 2;…;N. Thus, one has ~aiðtÞ ! 0; ~aijðtÞ ! 0,

and ~bijðtÞ ! 0 as t!1.39 It implies that the unknown or

uncertain system parameter vectors ai and coupling matrices

013127-6 Che et al. Chaos 23, 013127 (2013)

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A and B can be successfully identified using updating law

Eq. (5) in the process of anticipatory synchronization

between the drive and response systems.

This completes the proof.

1A. Pikovsky, M. rosenblum, and J. Kurths, Synchronization: A UniversalConcept in Nonlinear Science (Cambridge University Press, 2001).

2G. V. Osipov, J. Kurths, and C. Zhou, Synchronization in OscillatoryNetworks (Springer, 2007).

3S. Boccaletti, V. Latora, Y. Moreno, M. Ghavez, and D. U. Hwang,

“Complex networks: Structure and dynamics,” Phys. Rep. 424, 175–308

(2006).4H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A.-L. Barab�asi, “The

large-scale organization of metabolic networks,” Nature 407, 651–654

(2000).5D. J. Watts and S. H. Strogatz, “Collective dynamics of “small-world”

networks,” Nature 393, 440 (1998).6A. L. Barab�asi and R. Albert, “Emergence of scaling in random networks,”

Science 286, 509 (1999).7M. Barahona and L. M. Pecora, “Synchronization in small-world systems,”

Phys. Rev. Lett. 89, 054101 (2002).8L. Kocarev and P. Amato, “Synchronization in power-law networks,”

Chaos 15, 024101 (2005).9C. S. Zhou and J. Kurths, “Dynamical weights and enhanced synchroniza-

tion in adaptive complex networks,” Phys. Rev. Lett. 96, 164102 (2006).10A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. S. Zhou,

“Synchronization in complex networks,” Phys. Rep. 469, 93 (2008).11D. Yu, M. Righero, and L. Kocarev, “Estimating topology of network,”

Phys. Rev. Lett. 97, 188701 (2006).12M. Timme, “Revealing network connectivity from response dynamics,”

Phys. Rev. Lett. 98, 224101 (2007).13D. Yu and U. Parlitz, “Driving a network to steady states reveals its coop-

erative architecture,” Europhys. Lett. 81, 48007 (2008).14X. Wu, “Synchronization-based topology identification of weighted gen-

eral complex dynamical networks with time-varying coupling delay,”

Physica A 387, 997–1008 (2008).15H. Liu, J. A. Lu, J. L€u, and D. J. Hill, “Structure identification of uncertain

general complex dynamical networks with time delay,” Automatica 45,

1799–1807 (2009).16L. Chen, J. Lu, and C. K. Tse, “Synchronization: an obstacle to identifica-

tion of network topology,” IEEE Trans. Circuits Syst., II: Express Briefs

56, 310–314 (2009).17D. Yu, “Estimating the topology of complex dynamical networks by

steady state control: Generality and limitation,” Automatica 46, 2035–

2040 (2010).18J. Zhao, Q. Li, J. A. Lu, and Z.-P. Jiang, “Topology identification of com-

plex dynamical networks,” Chaos 20, 023119 (2010).19D. Yu and U. Parlitz, “Inferring network connectivity by delayed feedback

control,” PLos ONE 6, e24333 (2011).20Y. Q. Che, R. X. Li, C. X. Han, J. Wang, S. G. Cui, B. Deng, and X. L.

Wei, “Adaptive lag synchronizationbased topology identification scheme

of uncertain general complex dynamical networks,” Eur. Phys. J. B 85,

265 (2012).21A. Roxin, H. Riecke, and S. A. Solla, “Self-sustained activity in a small-

world network of excitable neurons,” Phys. Rev. Lett. 92, 198101 (2004).

22X. Wang, Y. Lu, M. Jiang, and Q. Ouyang, “Attraction of spiral waves by

localized inhomogeneities with small-world connections in excitable

media,” Phys. Rev. E 69, 056223 (2004).23K. Pyragas, “Synchronization of coupled time-delay systems: analytical

estimations,” Phys. Rev. E 58, 3067–3071 (1998).24Q. Zhang, J. Lu, J. L€u, and C. Tse, “Adaptive feedback synchronization of

a general complex dynamical network with delayed nodes,” IEEE Trans

Circuits and Syst., II: Express Briefs 55, 183–187 (2008).25T. Heil, I. Fischer, W. Els€asser, J. Mulet, and C. R. Mirasso, “Chaos syn-

chronization and spontaneous symmetry-breaking in symmetrically delay-

coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).26S. Tang and J. M. Liu, “Experimental verification of anticipated and re-

tarded synchronization in chaotic semiconductor lasers,” Phys. Rev. Lett.

90, 194101 (2003).27N. J. Corron, J. N. Blakely, and S. D. Pethel, “Lag and anticipating syn-

chronization without time-delay coupling,” Chaos 15, 023110 (2005).28M. Ciszak, F. Marino, R. Toral, and S. Balle, “Dynamical mechanism of

anticipating synchronization in excitable systems,” Phys. Rev. Lett. 93,

114102 (2004).29H. U. Voss, “Anticipating chaotic synchronization,” Phys. Rev. E 61, 5115

(2000).30H. Huijberts, H. Nijmeijer, and T. Oguchi, “Anticipating synchronization

of chaotic lur’e systems,” Chaos 18, 019901 (2008).31G. Ambika and R. E. Amiritkar, “Anticipatory synchronization with vari-

able time delay and reset,” Phys. Rev. E 79, 056206 (2009).32Q. Han, C. Li, and T. Huang, “Anticipating synchronization of a class of

chaotic systems,” Chaos 19, 023105 (2009).33Q. Han, C. Li, and J. Huang, “Anticipating synchronization of chaotic sys-

tems with time delay and parameter mismatch,” Chaos 19, 013104 (2009).34H. Wei and L. Li, “Estimating parameters by anticipating chaotic syn-

chronization,” Chaos 20, 023112 (2010).35F. S. Matias, P. V. Carelli, and C. R. Mirasso, “Anticipated synchroniza-

tion in a biologically plausible model of neuronal motifs,” Phys. Rev. E

84, 021922 (2011).36C. Mayol, C. R. Mirasso, and R. Toral, “Anticipated synchronization and

the predict-prevent control method in the fitzhugh-nagumo model system,”

Phys. Rev. E 85, 056216 (2012).37J.-J. E. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, NJ,

1991).38J. L. Hindmarsh and R. M. Rose, “A model of neuronal bursting using 3

coupled 1st order differential-equations,” Proc. R. Soc. Lond. B 221,

87–102 (1984).39H. K. Khalil, Nonlinear Systems, 3rd ed. (Prentice Hall, NJ, 2002).40Y. Q. Che, J. Wang, K. M. Tsang, and W. L. Chan, “Unidirectional syn-

chronization for hindmarsh-rose neurons via robust adaptive sliding mode

control,” Nonlinear Anal. Real World Appl. 11, 1096–1104 (2010).41A. S. Willsky, “A survey of design methods for failure detection in

dynamic systems,” Automatica 12, 601–611 (1976).42T. I. Netoff, R. Clewley, S. Arno, T. Keck, and J. A. White, “Epilepsy in

small-world networks,” J. Neurosci. 24, 8075–8083 (2004).43S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear

estimation,” Proc. IEEE 92, 401–422 (2004).44G. Ullah and S. J. Schiff, “Tracking and control of neuronal hodgkin-

huxley dynamics,” Phys. Rev. E 79, 040901(R) (2009).45G. Ullah and S. J. Schiff, “Assimilating seizure dynamics,” PLOS

Comput. Biol. 6, e1000776 (2010).

013127-7 Che et al. Chaos 23, 013127 (2013)

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