Chaos, Solitons and Fractals 27 (2006) 1292–1297

www.elsevier.com/locate/chaos

Synchronization of the unified chaotic systemsvia active control

Ahmet Ucar a,*, Karl E. Lonngren b, Er-Wei Bai b

a Department of Electrical and Electronics Engineering, Firat University, Elazıg 23119, Turkeyb Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA

Accepted 21 April 2005

Abstract

This paper investigates the synchronization of coupled unified chaotic systems via active control. The synchroniza-

tion is given in the slave–master scheme and the controller ensures that the states of the controlled chaotic slave system

exponentially synchronize with the state of the master system. Numerical simulations are provided for illustration and

verification of the proposed method.

� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Since the introduction of the Lorenz chaotic attractor [1], there has been great interest in the study and synchroni-

zation of the Lorenz and Lorenz type systems [2–8]. There are several Lorenz type systems as discussed in Chen and

Ueta [9] and Lu et al. [10,11]. Recently a unified chaotic system was introduced in [8], which unified the Lorenz, Chen

and Lu systems by setting a single adjustable parameter instead of three parameters of the individual systems.

In the present paper, we study the synchronization of coupled unified chaotic systems using the active control tech-

nique introduced in [6] and further developed in [12].

In the following section, the dynamics of a unified chaotic system is presented. In Section 3, the synchronization of

the unified chaotic systems is discussed. Numerical simulations are provided in Section 4 to illustrate and verify of the

method. Finally a concluding remark is given.

2. The unified chaotic systems

In 1963, Lorenz developed a system and observed the first chaotic attractor [1]. In 1999, Chen found a similar system

[9] that exhibits chaotic behavior with a topologically different chaotic attractor. More recently, another similar system

was introduced by Lu et al. [10] which produces the chaotic attractor and connects the gap between the Lorenz and

0960-0779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2005.04.104

* Corresponding author.

E-mail addresses: aucar1@firat.edu.tr (A. Ucar), lonngren@engineering.uiowa.edu (K.E. Lonngren), erwei@engineering.uiowa.edu

(E.-W. Bai).

Fig. 1. (a) Lorenz chaotic attractor; (b) Chen chaotic attractor; (c) Lu chaotic attractor; (d) new chaotic attractor.

A. Ucar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297 1293

Chen systems. In [8] Li et al. introduced the system defined in (1) with the parameter a being changed to observe these

three attractors.

_x1 ¼ ð25a þ 10Þðy1 � x1Þ_y1 ¼ ð28 � 25aÞx1 � x1z1 þ ð29a � 1Þy1

_z1 ¼ x1y1 �8 þ a

3z1

ð1Þ

where a 2 [0,1]. It was shown in [8] that the system exhibits chaotic behavior for any a 2 [0,1]. When a 2 [0,0.8), the

system (1) is called a Lorenz system and exhibits chaotic behavior. For a = 0, the chaotic behavior is depicted in

Fig. 1(a) which shows the steady state solution for the initial condition (x(0),y(0),z(0)) = (1,2,3). For a = 0.8, the sys-

tem is known to be the generalized Lu system and its steady state solution is in Fig. 1(b). When a 2 (0.8,1], the system is

called the Chen system as in Fig. 1(c) for a = 1.

In fact, we have found the system defined in (1) also shows chaotic behavior for a = 1.1 as depicted in Fig. 1(d). Our

interest is to design an active control system to synchronize two unified chaotic systems and discuss its validity in the

following section.

3. Chaos synchronization of the unified chaotic systems

Recently chaos synchronization of coupled chaotic system has attracted a great deal of attentions. Many techniques

have been developed to synchronize coupled chaotic systems, such as linear/nonlinear feedback methods, adaptive tech-

niques, time delay feedback approaches and so on. The unified chaotic system defined in (1) has also been studied and

techniques proposed to synchronize coupled unified chaotic systems [8,13,14]. We show in this paper that the technique,

namely the active control method proposed in [6] can be readily designed to synchronize two chaotic systems that also

include systems with time delay if the nonlinearity in the system is known [15–17].

Consider a unified chaotic system of (1) with the subscript �1� as the master system to which the slave system needs to

be synchronized. Since the parameter a has been found to be very important in determining the chaotic attractors

1294 A. Ucar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297

observed from (1), we were led to ask whether it would be possible to synchronize two different chaotic systems with a

different attractor.

Consider the second unified chaotic system with the subscript �2� that contains a different value of the system para-

meter ~a

_x2 ¼ ð25~a þ 10Þðy2 � x2Þ þ la

_y2 ¼ ð28 � 25~aÞx2 � x2z2 þ ð29~a � 1Þy2 þ lb

_z2 ¼ x2y2 �8 þ ~a

3z2 þ lc

ð2Þ

where la(t), lb(t), and lc(t) are active control signals that are yet to be determined. Here the aim of the control signals is

to force the slave system to follow the master system. Thus, one-way synchronization of the two unified chaotic systems

from system 1 to system 2 will be achieved. Without the controls la(t), lb(t), and lc(t), the second system defined by (2)

will have a chaotic behavior depending chosen value of ~a.

In order to obtain the active control signals, we define the error state between the dependent variables of system 2

and system 1 as

ex ¼ x2 � x1

ey ¼ y2 � y1

ez ¼ z2 � z1

ð3Þ

Subtracting the first system (1) from the second system (2) which includes the control signals, we obtain

_ex ¼ 10ðey � exÞ þ 25~aðy2 � x2Þ � 25aðy1 � x1Þ þ laðtÞ_ey ¼ 28ex � 35~ax2 þ 35ax1 � x2z2 þ x1z1 þ 29~ay2 � 29ay1 � ey þ lbðtÞ

_ez ¼ x2y2 � x1y1 �8

3ez �

~a3z2 þ

a3z1 þ lcðtÞ

ð4Þ

Since the nonlinear terms in both systems are known, the control signals can be defined as

laðtÞ ¼ �25~aðy2 � x2Þ þ 25aðy1 � x1Þ þ u1ðtÞlbðtÞ ¼ ~að35x2 � 29y2Þ þ að29y1 � 35x1Þ þ x2z2 � x1z1 þ u2ðtÞ

lcðtÞ ¼ �x2y2 þ x1y1 þ~a3z2 �

a3z1 þ u3ðtÞ

ð5Þ

The substitution of Eq. (5) into (4) leads to

_ex_ey_ez

264

375 ¼

�10 10 0

28 �1 0

0 0 �8=3

264

375

exeyez

264

375þ

1 0 0

0 1 0

0 0 1

264

375

u1

u2

u3

264

375 ð6Þ

Eq. (7) described the error dynamics and can be considered in terms of a control problem where the system to be

controlled is now a linear system with the control input u(t) = [u1,u2,u3]T [18]. Since the error dynamics is full state con-

trollable, the feedback gains can be designed to stabilize the state of the error system, [ex,ey,ez]T so that the error signals

converge to zero as time t goes to infinity. This implies that the two unified chaotic systems with different values the

parameter a are synchronized. There are many possible choices for the controller u(t). We choose

u1

u2

u3

264

375 ¼ �

k11 k12 k12

k21 k22 k23

k31 k32 k33

264

375

exeyez

264

375 ð7Þ

where the constants kij�s are controller gains. For the proper choice of kij�s, the closed loop system characteristic matrix

must have all of the eigenvalues with negative real parts. In this case the control signal defined in (7) yields the error

dynamic function of kij�s as

_ex_ey_ez

264

375 ¼

�10 � k11 10 � k12 0 � k13

28 � k21 �1 � k22 �k23

�k31 �k32 �8=3 � k33

264

375

exeyez

264

375 ð8Þ

The rate of convergence is now determined by the numerical values of the parameters of the feedback gains kij�s. For

the particular choice of feedback gains;

Fig. 2.

master

A. Ucar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297 1295

k11 k12 k12

k21 k22 k23

k31 k32 k33

264

375 ¼

�9 10 0

28 0 0

0 0 �5=3

264

375 ð9Þ

the error system given in (8) is stable and the closed loop system has eigenvalues that are found to be �1, �1 and �1.

The choice of control gains will lead to a stable error system and to synchronization of two unified chaotic systems.

4. Numerical results

Here the numerical results are given to verify the proposed method. In order to demonstrate the efficacy of the pro-

cedure, we keep the master system parameter a = 0 and chose different values of the slave system parameter ~a. First,

consider the case of ~a ¼ 0.8, i.e., the Lu attractor a = 0. In these numerical simulations, the fourth-order Runge-Kutta

is used to solve the master and slave systems defined in (1) and (2), respectively, with time step size 10�3. The initial

values of the master system and slave system are taken as (x1(0),y1(0),z1(0)) = (�1,2,1) and (x2(0),y2(0),z2(0)) =

(1,�2,3), respectively. The controller gains are chosen as in (9) in order to achieve the synchronization within 5 s after

the time of the activation of the controller s.When the control signal is activated at s = 0, the simulation results are illustrated in Fig. 2 for the error system of (8).

Fig. 2(a) shows the error signal ex = x2 � x1; Fig. 2(b) shows the error signal ey = y2 � y1; and Fig. 2(c) shows the error

signal ez = z2 � z1. Fig. 2 shows that the error signals converge to zero and this leads to the synchronization of the Lu

attractor with the Lorenz attractor.

To synchronize the Chen attractor with the Lorenz attractor with a = 0, let the slave system parameter value be~a ¼ 1. The numerical results of the error system are depicted in Fig. 3 when the control signals are applied at the time

s = 50. The results of this calculation clearly indicate that the slave system follows the master system after the control is

applied.

Note that the speed of the convergence time can be adjusted by choosing different values of controller gain defined in

the previous subsection. For example if the desired convergence times of the error dynamics are 5 s for ex, 2.5 s for ey,

1 s for ez, then the controller gains are

k11 k12 k12

k21 k22 k23

k31 k32 k33

264

375 ¼

�9.9 10 0

28 �0.95 0

0 0 �2.6467

264

375 ð10Þ

The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 0. The

Lorenz system (1) parameter was a = 0 and the slave Lu system (2) parameter was ~a ¼ 0.8.

Fig. 3. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 50. The

master Lorenz system (1) parameter was a = 0 and the slave Chen system (2) parameter was ~a ¼ 1.

1296 A. Ucar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297

For those controller gains, the results are shown in Fig. 1(d) for ~a ¼ 1.1 with the Lorenz system with of a = 0. The

error signals are depicted in Fig. 4 when the control signal is activated at s = 0, Fig. 4 shows the convergence times for

ex, ey, and ez, have been achieved.

In the numerical calculations, we have noted that the synchronization speed is very rapid. The evolution of the syn-

chronization can be altered by choosing different control gains. Note that when the convergence time is reduced, the

Fig. 4. The time response of the error signals of coupled unified chaotic systems with active controller applied at a time s = 50. The

master Lorenz system (1) parameter was a = 0 and the slave system (2) parameter was ~a ¼ 1.1.

A. Ucar et al. / Chaos, Solitons and Fractals 27 (2006) 1292–1297 1297

magnitudes of the controller signals are increased. This may lead to a large controller gain and a signal saturation in

practical implementations such as electronically.

5. Conclusion

In this paper, we have obtained a nonlinear active controller that can be used to synchronize two coupled unified

chaotic systems together such that the frequency of oscillation of the slave system will follow the master system.

The desired speed of the convergence time of synchronization can be modified by the linear feedback gain. In practical

applications, some caution should be exercised to balance between the convergence rate and the magnitude of the con-

troller gains. The only limitation of the proposed method is the assumption that the nonlinearity in the systems needs to

be known.

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