Chaotification of complex networks with impulsive controlZhiHong Guan, Feng Liu, Juan Li, and Yan-Wu Wang Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 22, 023137 (2012); doi: 10.1063/1.4729136 View online: http://dx.doi.org/10.1063/1.4729136 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/22/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Synchronization in node of complex networks consist of complex chaotic system AIP Advances 4, 077112 (2014); 10.1063/1.4890097 Pinning control of complex networks via edge snapping Chaos 21, 033119 (2011); 10.1063/1.3626024 Engineering synchronization of chaotic oscillators using controller based coupling design Chaos 21, 013110 (2011); 10.1063/1.3548066 Generalized synchronization of complex dynamical networks via impulsive control Chaos 19, 043119 (2009); 10.1063/1.3268587 Stability of piecewise affine systems with application to chaos stabilization Chaos 17, 023123 (2007); 10.1063/1.2734905

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Chaotification of complex networks with impulsive control

Zhi-Hong Guan,a) Feng Liu, Juan Li, and Yan-Wu WangDepartment of Control Science and Engineering, Huazhong University of Science and Technology,Wuhan 430074, People’s Republic of China

(Received 12 December 2011; accepted 21 May 2012; published online 13 June 2012)

This paper investigates the chaotification problem of complex dynamical networks (CDN) with

impulsive control. Both the discrete and continuous cases are studied. The method is presented to

drive all states of every node in CDN to chaos. The proposed impulsive control strategy is effective

for both the originally stable and unstable CDN. The upper bound of the impulse intervals for

originally stable networks is derived. Finally, the effectiveness of the theoretical results is verified by

numerical examples. VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4729136]

Due to some desirable features of chaos, chaotification

(also called anti-control of chaos), which means generat-

ing chaos when it is considered as beneficial has received

a great deal of attention from the nonlinear dynamics

community. Many controllers have been designed to

drive a system from non-chaotic state to chaos; however,

few works have been dedicated to the chaotification prob-

lem in the settings of complex dynamical network (CDN).

In this paper, an impulsive controller is presented to

drive the network chaotic and it is suitable to a relative

wide range of complex networks, which can be stable or

unstable, continuous or discrete.

I. INTRODUCTION

Over the past couple of decades, an increasing interest

has been focused on the theory and applications of complex

networks. What we see everyday are not isolated systems, but

different kinds of interconnected networks of complex sys-

tems, such as the World Wide Web, power grid, transportation

networks, cellular and metabolic networks, etc.1 Because of

the ubiquity of networks, and thanks to the rapid development

of computer technology, many scientists devote themselves to

this research area and have achieved significant progress.

The dynamical behaviors of complex networks have

been a hot topic of research, such as synchronization and

desynchronization, stability, consensus problem, and transi-

tion from non-chaos to chaos. Among all these dynamical

behaviors of complex networks, chaos dynamics has been

one of the most interesting issues and has been intensively

studied in recent years.2–4

Due to some desirable features of chaos, anti-control of

chaos,14,15,18–23 which means generating chaos when it is

considered to be beneficial, has been adopted to provide the

designer with cornucopia of opportunities and rich flexibil-

ity. Examples include secure communication,7,8 human brain

and heart beat regulation,9,10 resonance prevention in me-

chanical systems,11 TCP congestion control.12,13

Recently, Guan and Liu5,14,15,18 proposed an impulsive

controller to drive the several discrete neural networks to

chaos, but it can only guarantee that at least one state of the

networks is chaotic, thus maybe leaving other states unknown,

be it chaotic or stabilized; what’s more, it is only effective for

the discrete cases, leaving the continuous cases yet to be dis-

cussed. Li et al.6 studied the chaotification problem of a linear

model of discrete neural networks, called the Elman networks,

and it is still not effective for those nonlinear or continuous

complex networks. Yuan et al.4 studied the dynamical behav-

iors of the Newman-Watts small-world dynamical networks

using the theoretical analysis and numerical simulation; Li

et al.3 analyzed the required coupling strength to achieve the

transition from non-chaotic to chaotic, and it is found that the

more heterogeneous of a network, the easier of the transition

can be made; Zhang et al.2 studied the emergence of chaos in

several types of networks.

However, most of the existing literatures have focused

on the analysis of the emergence of chaos in complex net-

works without formulating control method to drive the CDN

to chaos when the CDN cannot generate chaos by them-

selves. The design of controller to generate chaos from non-

chaotic states in CDN has remained an unexplored subject.

Due to its theoretical and practical significance, impulsive

control has been a very effective method in chaos control

and synchronization.16,17 In this paper, we present an impul-

sive control implementation aimed to drive the complex net-

works to chaos regardless whether the coupling strength

satisfies the conditions investigated in Refs. 2–4 or not. Both

the discrete and continuous cases are considered. A method

is presented to drive all states of every node in CDN to

chaos. Based on the impulsive equations theory, the upper

bound of the impulse intervals for originally stable networks

is derived. Finally, the effectiveness of the theoretical results

is verified by numerical examples.

The rest of the paper is structured as follows: In Sec. II,

the problem of driving the continuous CDN to chaos is dis-

cussed. In Sec. III, the discrete-time case is considered. In

Sec. IV, simulation results are given to verify the effective-

ness of the proposed method. Finally, some concluding

remarks are stated in Sec. V.

Notations: The notations used in this paper are quite

standard. Rn denotes the n-dimensional Euclidean space; I is

the identity matrix with appropriate dimension. The super-

script T represents transpose of matrix. The notation minð�Þa)Electronic mail: zhguan@mail.hust.edu.cn.

1054-1500/2012/22(2)/023137/5/$30.00 VC 2012 American Institute of Physics22, 023137-1

CHAOS 22, 023137 (2012)

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describes the smallest value in the braces. Furthermore, for a

matrix A 2 Rn�n; kminðAÞ characterizes the smallest eigen-

value of A. Nþ describes the set of all positive integers.

II. CHAOTIFYING COMPLEX DYNAMICAL NETWORKSBY IMPULSIVE CONTROL:THE CONTINUOUS CASE

Consider a CDN consisting of N identical nodes with lin-

early diffusive couplings and each node is an n-dimensional

dynamic system. The system is described as follows:

_xiðtÞ ¼ f ðxiðtÞÞ � cXN

j¼1

aij xjðtÞ; i ¼ 1; 2;…;N; (1)

where xiðtÞ ¼ ðxi1ðtÞ; xi2ðtÞ;…; xinðtÞÞT 2 Rn is the state vari-

able of the ith node, f ð�Þ is a nonlinear vector-valued func-

tion describing the dynamics of the isolated node, and the

constant number c > 0 represents the coupling strength. In

network (1), the coupling matrix A ¼ ðaijÞ 2 RN�N repre-

sents the coupling configuration of the network, if there

exists a connection between node i and node

jði 6¼ jÞ; aij ¼ aji ¼ 1; otherwise, aij ¼ aji ¼ 0 ði 6¼ jÞ and

aii ¼ �XN

j¼1; j 6¼i

aij ¼ �XN

j¼1; j6¼i

aji; i ¼ 1; 2;…;N:

It is evident that 0 is an eigenvalue of matrix A, with all the

other eigenvalues of A being strictly negative. Let

0 ¼ k1 > k2 � k3 � � � � � kN

be the eigenvalues of its coupling matrix A. If the solution of

the isolated node is not located in chaotic regions and the so-

lution s(t) satisfies

_sðtÞ ¼ f ðsðtÞÞ; (2)

where s(t) can be an equilibrium point or a periodic orbit,

then, according to Refs. 2–4, all the Lyapunov exponents of

equation (2) are non-positive, denoted as

0 � hmax ¼ h1 > h2 � � � � � hn:

Next, rewrite the dynamical network (1) in a vector form

_XðtÞ ¼ FðXðtÞÞ � cXðtÞAT ; (3)

where XðtÞ ¼ ðx1ðtÞ; …; xNðtÞÞ 2 Rn�N; FðXðtÞÞ ¼ ðf ðx1Þ;f ðx2Þ; …; f ðxNÞÞ. Similar to Refs. 2–4 and 24, the transver-

sal Lyapunov exponent25–27 in Eq. (3) satisfies

liðkkÞ ¼ hi � ckk; i ¼ 1; 2;…n:

Generally speaking, if the coupled network (3) is chaotic,

then there is at least one positive Lyapunov exponent.2–4

Based on the analysis above, the maximum of liðkkÞ is

l1ðkNÞ ¼ hmax � ckN . Therefore, if the trajectory of system

(3) is boundary and l1ðkNÞ > 0, the dynamical network (3)

is chaotic. That is, the inequality

c >jhmaxjjkN j

(4)

is very important for the emergence of chaos in complex net-

work (3). In some practical cases, the coupling strength ccannot be changed. If the coupling strength fails to satisfy

the requirement of (4), a controller must be designed to drive

the network to chaos. In this paper, an impulsive control

strategy is presented to drive the network to chaos. As A is

an irreducible and symmetric matrix, there exists Usatisfying

U�1AU ¼ ~K;

where ~K is a diagonal matrix whose elements are

the eigenvalues of A, and U is a unitary matrix. By linea-

rizing (3) near s(t), the following equation can be

derived:

_XðtÞ ¼ Df ðsðtÞÞXðtÞ � cXðtÞAT ;

where Df(s(t)) is the Jacobian matrix of f ð�Þ on s(t). Let

yðtÞ ¼ XðtÞU;

where yðtÞ ¼ ðy1ðtÞ;…; yNðtÞÞ 2 Rn�N , then

_yiðtÞ ¼ Df ðsðtÞÞyiðtÞ � ckiyiðtÞ; i ¼ 1; 2;…N:

Assumption 1. Gi :¼ Df ðsðtÞÞ � ckiI can be diagonal-

ized, where i ¼ 1; 2;…;N.

According to assumption 1, there exists a nonsingular

matrix Q 2 Rn�n satisfying QGiQ�1 ¼ Ki, denote ziðtÞ

¼ QyiðtÞ, the following equation is derived:

_ziðtÞ ¼ KiziðtÞ; i ¼ 1; 2;…;N: (5)

Lemma 1. The following map is chaotic:20

qkþ1 ¼ ecqk mod ðrÞ; qk 2 Rm: (6)

All the Lyapunov exponents of the map (6) are strictly posi-

tive, and the orbit of the map is uniformly bounded.

Lemma 2. The continuous system is chaotic if it is

observed being chaotic at discrete time sequences.20

Next, we seek a control method to drive the Eq. (5) cha-

otic like the map in Lemma 1.

At first, a limited bound M is chosen, M is an arbitrary

positive scalar which can represent the limitation of a sys-

tem, such as word length in a computer. When the state

signal is larger than M, it will cause overflow and be

“wrapped around,” which corresponds to the modulus

operation that “wraps around” at M. When all the signals

are within the bound of M, we adopt impulsive control

to drive every state of every node to increase

“exponentially,” when it gets out M at time tðnÞ, modulus

operation is active immediately to make the state signal

small again. Denote the “wrap around” time sequences

with a superscript on it like this ftð1Þ; tð2Þ;…g, the con-

trolled CDN can be written as

023137-2 Guan et al. Chaos 22, 023137 (2012)

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ðaÞ_ziðtÞ ¼ KiziðtÞ; t 6¼ tk

Dzi ¼ ziðtþk Þ � ziðtkÞ ¼ Bkzi; t ¼ tk;

ziðtþ0 Þ ¼ zi0; k ¼ 1; 2;…; i ¼ 1; 2;…;N;

8><>:

ðbÞ zijðtþÞ ¼ zijðtÞ mod ðMÞ; t ¼ tð1Þ; tð2Þ;…; (7)

where zij is the jth state of zi, when jzijj > M at time tðlÞ,(b) is active, otherwise (a) is active. For simplicity, let

Bk ¼ bkI.Theorem 1. Assume that assumption 1 holds,

denote kij ¼ kjðDf ðsðtÞÞ � ckiÞ; a ¼ minfkijg for all

i ¼ 1; 2;… N; j ¼ 1; 2;…n, and the bounded impulse inter-

val Dk ¼ tk � tk�1. If there exists a constant l > 1 such that

jð1þ bmÞeaDm j � l; m 2 Nþ; (8)

then the impulsive controlled systems in Eq. (7) is chaotic.

Proof. For any t 2 ðtk; tkþ1�, without loss of generality,

suppose tkþ1 < tð1Þ, which implies tð0Þ < t < tð1Þ. From _ziðtÞ¼ KiziðtÞ, it can be derived that

ziðtÞ ¼ eKiðt�tkÞziðtþk Þ ¼ eKiðt�tkÞðI þ bkIÞziðtkÞ ¼ � � �

¼ eKiðt�t0Þð1þ bkÞð1þ bk�1Þ � � � ð1þ b1Þziðt0Þ

¼ eKiðt�tkÞð1þ bkÞeKiDkð1þ bk�1ÞeKiDk�1 � � �

� ð1þ b1ÞeKiD1 ziðt0Þ;

therefore,

zijðtÞ ¼ ekijðt�tkÞð1þ bkÞekijDkð1þ bk�1ÞekijDk�1 � � �

� ð1þ b1ÞekijD1 zijðt0Þ;

where zijðtÞ is the jth state of ziðtÞ; kij is the jth diagonal ele-

ment of the matrix Ki, as Ki ¼ QGiQ�1.

From Eq. (8), there exists a constant l > 1 such that

jð1þ bmÞekijDm j � jð1þ bmÞeaDm j � l; m ¼ 1; 2;…; k;

then,

jzijðtÞj � lkqjzijðt0Þj;

where q is a constant number satisfying

q � ekijðt�tkÞ; t 2 ðtk; tkþ1�:

When jzijðtÞj get larger enough, the modulus operation

is active to make jzijðtÞj small at tð1Þ. As for

tðlÞ � t � tðlþ1Þ; ðl 2 Nþ; l � 1Þ, the analysis is quite simi-

lar to the case when tð0Þ < t < tð1Þ. In summary, jzijðtÞjincrease exponentially, when it tries to exceed M, it is

“wrap around,” so jzijðtÞj is chaotic according to Lemmas

1 and 2.

It is obvious that jzijðtÞj is chaotic for any

i ¼ 1; 2;…;N; j ¼ 1; 2;…; n, which means all the states of

zðtÞ ¼ ðz1ðtÞ; z2ðtÞ;…; zNðtÞÞ is chaotic. Thus, the proof is

completed.

Remark 1. From Eq. (8), the following can be derived:

lnj1þ bmj þ aDm > 0; m 2 Nþ: (9)

If the isolated node is stable, a < 0, then from Eq. (9), we

have

Dm < �1

alnj1þ bmj; m 2 Nþ:

The impulsive interval has an upper bound � 1a lnj1þ bmj.

Equation (8) can always be satisfied by choosing the suitable

impulse interval Dm and the control gain bm.

Remark 2. As yðtÞ ¼ ðy1ðtÞ; y2ðtÞ;…; yNðtÞÞ ¼ Q�1zðtÞ,and XðtÞ ¼ ðx1ðtÞ;…; xNðtÞÞ ¼ yðtÞU�1, X(t) is chaotic.

Remark 3. We did not assume the original states to be

stable, the presented method can also be used for unstable

cases.

III. CHAOTIFYING COMPLEX DYNAMICALNETWORKS BY IMPULSIVE CONTROL:THE DISCRETE CASE

For the discrete CDN which is described as

xiðnþ 1Þ ¼ f ðxiðnÞÞ � cXN

j¼1

aijxjðnÞ; i ¼ 1; 2;…;N; (10)

similar to the procedure in continuous case, the impulsive

controlled system can be described as

ðaÞziðnþ 1Þ ¼ KiziðnÞ; n 6¼ nk

ziðnþ 1Þ ¼ ðBk þ IÞKiziðnÞ; n ¼ nk;

zið0Þ ¼ zi0; k ¼ 1; 2;…; i ¼ 1; 2;…;N:

8><>:

ðbÞ zijðnþ 1Þ ¼ zijðnÞmod ðMÞ; n ¼ nð1Þ; nð2Þ;…: (11)

Theorem 2. Assume that the assumption 1 holds, denote

the bounded impulse interval Dk ¼ nk � nk�1. If there exists

a constant l > 1 such that

jð1þ bmÞaDm j � l; m 2 Nþ (12)

then the impulsive controlled systems in Eq. (11) is chaotic,

where a and bm are given in Theorem 1.

The proof is very similar to Theorem 1 and the details

are omitted here.

Remark 4. From Eq. (12), the following can be derived:

lnj1þ bmj þ Dmln a > 0; m 2 Nþ: (13)

If the isolated node is stable, a < 1, then from Eq. (13), we

have Dm < � lnj1þbmjln a ;m 2 Nþ. The impulsive interval has an

upper bound � lnj1þbmjln a .

IV. SIMULATION EXAMPLE

In this section, simulation examples are given to verify

the effectiveness of the proposed chaotification schemes for

continuous and discrete complex networks, respectively.

Choose a dynamical network with five nodes for the

continuous case, and its coupling matrix

023137-3 Guan et al. Chaos 22, 023137 (2012)

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A1 ¼

�2 1 0 1 0

1 �3 1 1 0

0 1 �2 0 1

0 0 1 1 �2

0BB@

1CCA:

The f ðxiðtÞÞ in Eq. (1) is

f ðxiÞ ¼�3xi1ðtÞ � xi1ðtÞx2

i3ðtÞ0:2xi2ðtÞ þ x3

i2ðtÞ�5xi3ðtÞ � xi3ðtÞx2

i1ðtÞ

0@

1A; i ¼ 1; 2;…; 5:

The equilibrium �x ¼ ð0; 0; 0Þ is unstable. The states xi1

and xi3 are stable, while the state xi2 increases to infinity.

According to the theoretical results in Sec. II, let

Dm ¼ 1s; bm ¼ 160; M ¼ 300, random initial states are cho-

sen and the simulation result is illustrated in Figs. 1 and 2.

For the discrete case, choose a dynamical network with

four nodes, and its coupling matrix

A2 ¼

�3 1 1 1

1 �2 1 0

1 1 �3 1

1 0 1 �2

0BB@

1CCA

its eigenvalues are k1 ¼ 0; k2 ¼ �2; k3 ¼ k4 ¼ �4: The

f ðxiðnÞÞ in Eq. (10) is

f ðxiðnÞÞ ¼0:3xi1ðnÞ þ 0:2xi2ðnÞ þ x2

i1ðnÞxi3ðnÞ�xi1ðnÞ þ 1:2xi2ðnÞ þ x2

i3ðnÞ�0:5xi3ðnÞ

0@

1A;

where i¼ 1,2,3,4. The equilibrium �x ¼ ð0; 0; 0Þ is stable.

Df ð�xÞ ¼0:3 0:2 0

�1 1:2 0

0 0 0:5

0@

1A. Let Dm ¼ 3; bm ¼ 11. Ran-

dom initial states are chosen and the simulation result is

illustrated in Figs. 3 and 4.

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

time

x1(1

)

FIG. 1. Evolution of x11 with time t(s).

0

200

400

600

0

200

400

600−100

0

100

200

300

400

500

x2(1)x2(2)

x2(3

)

FIG. 2. Phase portrait of node 2.

0 200 400 600 800 1000−5

−4

−3

−2

−1

0

1

2

3

4

5

n times iterations

the

first

sta

te o

f x1

FIG. 3. Evolution of x11 with iteration times

FIG. 4. Phase portrait of node 1.

023137-4 Guan et al. Chaos 22, 023137 (2012)

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V. CONCLUSIONS AND POTENTIAL APPLICATION

In this paper, an impulsive control method is proposed

to chaotify complex dynamical networks both for continuous

case and discrete case. Some sufficient conditions are given

to guarantee that all the states of every node in the networks

are chaotic. We give the impulsive interval upper bound for

originally stable complex networks. Finally, both the contin-

uous and discrete nonlinear networks are given as examples

to verify the effectiveness of the proposed method.

The chaotification problem in complex networks has great

potential application in parallel/distributed computation, espe-

cially in the area of secure communication. Due to information

explosion as well as information security problem brought

about by information eavesdropping or the defects in the net-

work protocol itself, great amount of information need encryp-

tion and decryption. The proposed method can be used to

process data with very high efficiency in parallel/distributed

processors which can be treated as the nodes in the complex

networks. In order to meet the demand of real-time encryption,

generating more pseudo-random numbers is of pivotal impor-

tance to mix the plain message. In some hardware devices,

such as Field-programmable gate array (FPGA), where the

processor performs parallel operations, it can hardly engender

enough pseudo-random numbers when only one low-

dimensional chaotic system is implemented. Taking the parallel

computation capability of FPGA into consideration, a network

of chaotic systems can be implemented on its processor and

each chaotic node can be calculated in parallel, thus generating

more pseudo-random numbers to meet the real-time demand.

ACKNOWLEDGMENTS

This work was supported in part by the National Natural

Science Foundation of China under Grant Nos. 60973012,

61073025, 61073026, 61074124, 61170031 and the Graduate

Student Innovation Fund of Huazhong University of Science

and Technology under Grant No. 0109184979.

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