Normal form and synchronization of strict-feedback chaotic systems

Chaos, Solitons and Fractals 22 (2004) 927–933

www.elsevier.com/locate/chaos

Normal form and synchronization of strict-feedbackchaotic systems

Feng Wang a, Shihua Chen a,*, Minghai Yu b, Changping Wang c

a College of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR Chinab State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, PR China

c Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada B3H 3J5

Accepted 9 March 2004

Abstract

This study concerns the normal form and synchronization of strict-feedback chaotic systems. We prove that, any

strict-feedback chaotic system can be rendered into a normal form with a invertible transform and then a design

procedure to synchronize the normal form of a non-autonomous strict-feedback chaotic system is presented. This

approach needs only a scalar driving signal to realize synchronization no matter how many dimensions the chaotic

system contains. Furthermore, the R€ossler chaotic system is taken as a concrete example to illustrate the procedure of

designing without transforming a strict-feedback chaotic system into its normal form. Numerical simulations are also

provided to show the effectiveness and feasibility of the developed methods.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

The idea of synchronizing two identical chaotic systems with different initial conditions was first introduced by

Pecora and Carrol [1]. Since then, synchronization in coupled chaotic systems has become a rapidly developing field in

light of its great potential applications in secure communications, chemical reaction, modelling brain activity and so on

[2,3]. A large varieties of approaches have been proposed for chaos synchronization such as the adaptive synchroni-

zation method [4], the sampled-data feedback synchronization method [5], the impulsive synchronization method [6],

and many others [7–9]. Moreover, in the study of synchronizing nonlinear dynamics, several low-dimensional systems

are frequently used as benchmark examples for verification and validation of a proposed theory, method and algorithm.

These examples include Duffing oscillator, Van der Pol oscillator, R€ossler system, Chua’s circuit, Lorenz system and

Chen system [10–14].

It is very interesting to note that, many existing synchronization methods need several controllers to realize syn-

chronization and all those chaotic systems mentioned above can be rewritten into a class of nonlinear systems in a

so-called general strict-feedback form. This motivates the present work, which is to address two important issues

concerning the strict-feedback chaotic system. One issue is to find the normal form of strict-feedback chaotic systems,

the other is to find a scalar driving signal to realize synchronization. We prove that, any strict-feedback chaotic system

can be rendered into a normal form with a invertible transform. Furthermore, a design procedure to synchronize the

normal form of a non-autonomous strict-feedback chaotic system is presented. This approach needs only a scalar

driving signal to realize synchronization no matter how many dimensions the chaotic system contains. In addition, the

R€ossler chaotic system is taken as a concrete example to illustrate the procedure of designing without transforming a

* Corresponding author.

E-mail address: [email protected] (S. Chen).

0960-0779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2004.03.010

mail to: [email protected]

928 F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933

strict-feedback chaotic system into its normal form. Numerical simulations are also provided to show the effectiveness

and feasibility of the developed methods.

2. The normal form of strict-feedback chaotic systems

Many chaotic systems used in the study of nonlinear dynamics can be rewritten into a class of nonlinear systems in

the form

_x1 ¼ f1ðx1; tÞx2 þ g1ðx1; tÞ;_x2 ¼ f2ðx1; x2; tÞx3 þ g2ðx1; x2; tÞ;. . . . . . . . .

_xn�1 ¼ fn�1ðx1; x2; . . . ; xn�1; tÞxn þ gn�1ðx1; x2; . . . ; xn�1; tÞ;_xn ¼ fnðx1; x2; . . . ; xn�1; xn; tÞ;

ð1Þ

where fi, gi ði ¼ 1; 2; . . . ; n� 1Þ are sufficiently smooth nonlinear functions and fn is a continuous nonlinear function. It

is often referred as a general strict-feedback system. If fi 6¼ 0 ði ¼ 1; 2; . . . ; n� 1Þ, it is called a strict-feedback system.

General strict-feedback systems and strict-feedback systems have been thoroughly studied during the past decades.

The main works concentrated on controlling these systems as well as tracking any one-dimensional smooth trajectory

[15,16]. Our aim in this section, however, is to find the normal form of strict-feedback chaotic systems. To this end, we

first define two new variables

u1 ¼ x1;

u2 ¼ f1ðx1; tÞx2 þ g1ðx1; tÞ, F1ðx1; x2; tÞ:ð2Þ

Simple calculation can yields

du2dt

¼ f1ðx1; tÞf2ðx1; x2; tÞx3 þX1

i¼0

o

oxiF1ðx1; x2; tÞ

dxidt

þ f1ðx1; tÞg2ðx1; x2; tÞ, F2ðx1; x2; x3; tÞ; ð3Þ

where x0 stands for t and dx0dt ¼ 1. With the above notation, the first two equations of (1) can be rewritten as

_u1 ¼ u2;

_u2 ¼ F2ðx1; x2; x3; tÞ:ð4Þ

We define the third variable as

u3 ¼ F2ðx1; x2; x3; tÞ: ð5Þ

One can get

du3dt

¼Y3i¼1

fiðx1; . . . ; xi; tÞx4 þX2

i¼0

o

oxiF2ðx1; x2; x3; tÞ

dxidt

þY2i¼1

fiðx1; . . . ; xi; tÞg3ðx1; x2; x3; tÞ, F3ðx1; x2; x3; x4; tÞ: ð6Þ

With this notation, the first three equations of (1) can be rewritten as

_u1 ¼ u2;

_u2 ¼ u3;

_u3 ¼ F3ðx1; x2; x3; x4; tÞ:ð7Þ

Generally, the k-th (36 i6 n� 1) variable is defined as

uk ¼ Fk�1ðx1; x2; . . . ; xk ; tÞ; ð8Þ

its time derivative is

dukdt

¼Yki¼1

fiðx1; . . . ; xi; tÞxkþ1 þXk�1

i¼0

o

oxiFk�1ðx1; . . . ; xk ; tÞ

dxidt

þYk�1

i¼1

fiðx1; . . . ; xi; tÞgkðx1; . . . ; xk ; tÞ, Fkðx1; . . . ; xkþ1; tÞ: ð9Þ

F. Wang et al. / Chaos, Solitons and Fractals 22 (2004) 927–933 929

Then the first k equations of (1) can be transformed into

_u1 ¼ u2;

_u2 ¼ u3;

. . . . . .

_uk�1 ¼ uk ;

_uk ¼ Fkðx1; . . . ; xkþ1; tÞ:

ð10Þ

The last step is to define the n-th variable as

un ¼ Fn�1ðx1; . . . ; xn; tÞ: ð11Þ

Its time derivative is

dundt

¼Yni¼1

fiðx1; . . . ; xi; tÞxn þXn�1

i¼0

o

oxiFn�1ðx1; . . . ; xn; tÞ

dxidt

: ð12Þ

On the other hand, from the definitions of uiði ¼ 1; 2; . . . ; nÞ one can find that xi ði ¼ 1; 2; . . . ; nÞ could be expressed

as functions of t and ui ði ¼ 1; 2; . . . ; nÞ. Thus, the equation (12) could be written in the form

dundt

¼ F ðu1; . . . ; un; tÞ; ð13Þ

from which we have the following theorem:

Theorem 2.1. Any strict-feedback chaotic system can be transformed into the normal form as follows:

_u1 ¼ u2;

_u2 ¼ u3;

. . . . . .

_un�1 ¼ un;

_un ¼ F ðu1; . . . ; un; tÞ:

ð14Þ

3. Synchronizing strict-feedback chaotic systems via a scalar driving signal

In this section, we will propose a design procedure to synchronize the normal form of the non-autonomous strict-

feedback chaotic system. We design such a scalar controller U that the controlled strict-feedback chaotic system in

normal form

_y1 ¼ y2;

_y2 ¼ y3;

. . . . . .

_yn�1 ¼ yn;

_yn ¼ F ðy1; . . . ; yn; tÞ þ U

ð15Þ

is synchronous with (14), the normal form of system (1). To this end, let

U ¼ F ðu1; . . . ; un; tÞ � F ðy1; . . . ; yn; tÞ þ U1; ð16Þ

where U1 is waiting for determination. With this selection, the error system of (15) and (14) can be expressed as

_e ¼ Aeþ BU ; ð17Þ
1


where e ¼ ðe1; e2; . . . ; enÞ ¼ ðy1 � u1; y2 � u2; . . . ; yn � unÞ>, B ¼ ð0; 0; . . . ; 1Þ> and

A ¼

0 1 0 0 . . . 00 0 1 0 . . . 0

. . . . . . . . . . . . . . . . . .0 0 0 0 . . . 1

0 0 0 0 . . . 0

0BBBB@

1CCCCA:

On the other hand, the matrix

½B AB A2B . . . An�1B � ¼

0 0 0 . . . 0 10 0 0 . . . 1 0

. . . . . . . . . . . . . . . . . .0 1 0 . . . 0 0

1 0 0 . . . 0 0

0BBBB@

1CCCCA

is full rank, so the linear control theory [17] confirms that the single-input dynamic system (17) is controllable, i.e., all

the eigenvalues are controllable. Thus, by the pole assigning method [17] we can select an appropriate feedback

gain vector k ¼ ðk1; k2; . . . ; knÞ> such that system (17) is globally asymptotically stable at zero with the state feed-

back control U1 ¼ ke, which implies that dynamical system (15) is synchronous with (14), the normal form of system

(1).

4. Numerical simulation

In order to demonstrate and verify the performance of the proposed method, numerical simulation with the R€osslerchaotic system is presented below. In what follows, the fourth-order Runge–Kutta method is applied to solve the system

of different equations with time step size equal to 0.001 in the numerical simulation.

Suppose the master system is the R€ossler chaotic system which is described by the following dynamical system [10]:

_x ¼ �y � z;

_y ¼ xþ ay;

_z ¼ bþ zðx� cÞ;ð18Þ

where a, b and c are system parameters. Obviously, it is not in the strict-feedback form. However, after a simple state

transformation, i.e. letting u1, u2, u3 replace y, x, z respectively, system (18) can be rendered into the following desired

strict-feedback form:

_u1 ¼ u2 þ au1;

_u2 ¼ �u3 � u1;

_u3 ¼ bþ u3ðu2 � cÞ:ð19Þ

The objective is to design such a scalar driving signal U that the controlled R€ossler chaotic system

_y1 ¼ y2 þ ay1;

_y2 ¼ �y3 � y1;

_y3 ¼ bþ y3ðy2 � cÞ þ U

ð20Þ

is synchronous with R€ossler chaotic system (19). We can transform the system into the normal form following the

procedure as above and then design the controller U by the pole assigning method. However, we can design a controller

directly without transforming R€ossler chaotic system into its normal form. First, let the error variables be e1 ¼ y1 � u1,e2 ¼ y2 � u2, e3 ¼ y3 � u3. Define the first variable

z1 ¼ y1 � u1 ¼ e1:

Its derivative along the solutions of system (19) and (20) is

_z1 ¼ ae1 þ e2:


We select the first partial Lyapunov function as

V1 ¼1

2z21:

By simple calculation one can yield its time derivative along the solutions of system (19) and (20) as follows:

_V1 ¼ �z21 þ z1ððaþ 1Þe1 þ e2Þ: ð21Þ

Now, we define the second variable

z2 ¼ ðaþ 1Þe1 þ e2:

With this notation, Eq. (21) can be rewritten as

_V1 ¼ �z21 þ z1z2:

The second partial Lyapunov function is selected as

V2 ¼ V1 þ1

2z22 ¼

1

2z21 þ

1

2z22:

Its time derivative along the solutions of system (19) and (20) is

_V2 ¼ �z21 � z22 þ z2ððaþ 1Þ2e1 þ ðaþ 2Þe2 � e3Þ: ð22Þ

We define the third variable

z3 ¼ ðaþ 1Þ2e1 þ ðaþ 2Þe2 � e3:

Thus, Eq. (22) can be transformed into

_V2 ¼ �z21 � z22 þ z2z3: ð23Þ

We form the Lyapunov function as

V ðtÞ ¼ V2 þ1

2z23 ¼

1

2ðz21 þ z22 þ z23Þ: ð24Þ

Its time derivative along the solutions of systems (19) and (20) is

dV ðtÞdt

¼ �z21 � z22 � z23 þ z3ðz2 þ z3 þ ðaþ 1Þ2 _e1 þ ðaþ 2Þ _e2 � _e3Þ

¼ �z21 � z22 � z23 þ z3ðða3 þ 3a2 þ 3aÞe1 þ ða2 þ 3aþ 4Þe2 � ðaþ 3Þe3 � y3ðy2 � cÞ þ u3ðu2 � cÞ � UÞ: ð25Þ

Therefore, we have the following theorem:

Theorem 4.1. If we design the scalar driving signal U as

U ¼ ða3 þ 3a2 þ 3aÞe1 þ ða2 þ 3aþ 4Þe2 � ðaþ 3Þe3 � y3ðy2 � cÞ þ u3ðu2 � cÞ; ð26Þ

then the controlled R€ossler chaotic system (20) is globally asymptotically synchronous with R€ossler chaotic system (19).

Proof. Substituting (26) into (25), we have

dV ðtÞdt

¼ �z21 � z22 � z23 ¼ �2V ðtÞ;

from which we can yield V ðtÞ ¼ V ð0Þ expð�2tÞ. This implies limt!þ1 zi ¼ 0 ði ¼ 1; 2; 3Þ, i.e. limt!þ1 ei ¼ 0 ði ¼ 1; 2; 3Þ.Thus, the controlled R€ossler chaotic system (20) is globally asymptotically synchronous with R€ossler chaotic system

(19). h

In the numerical simulation, the system parameters a ¼ b ¼ 15, c ¼ 5:7, with which the R€ossler system behaves

chaotically. The initial conditions are set to be u1ð0Þ ¼ 3:0, u2ð0Þ ¼ 4:0, u3ð0Þ ¼ 5:0 and y1ð0Þ ¼ 6:0, y2ð0Þ ¼ 8:0,y3ð0Þ ¼ 9:0. Fig. 1 presents the results of the numerical simulation. One can see that the synchronization of all variables

is achieved successfully. Fig. 2 illustrates the control signal during the synchronization process.

Fig. 2. The control signal U during the chaos synchronization process.

Fig. 1. Graph of synchronization errors: e1ðtÞ ¼ y1 � u1; e2ðtÞ ¼ y2 � u2; e3ðtÞ ¼ y3 � u3:


5. Conclusion

The strict-feedback chaotic system is studied. We prove that, any strict-feedback chaotic system can be rendered

into a normal form with a invertible transform. A design procedure to synchronize the normal form of the

strict-feedback chaotic system is presented. Furthermore, the R€ossler chaotic system is taken as a concrete example

to illustrate the procedure of designing without transforming the strict-feedback chaotic system into its nor-

mal form. Numerical simulations are also provided to show the effectiveness and feasibility of the developed

methods.

Acknowledgements

The work is supported by 973 Program of PR China (No. 2003CB415205) and the Opening Research Foundation

of the Key Laboratory of Water and Sediment Science (Wuhan University), Ministry of Education, P.R China

(No. 2003A001). The authors would like to thank the referees for their valuable comments.


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Normal form and synchronization of strict-feedback chaotic systems