Chaos, Solitons and Fractals 42 (2009) 170–179

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

On anti-synchronization of chaotic systems via nonlinear control

M. Mossa Al-sawalha *, M.S.M. NooraniCenter for Modelling & Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

a r t i c l e i n f o

Article history:Accepted 12 November 2008

0960-0779/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.chaos.2008.11.011

* Corresponding author. Tel.: +60 174027145.E-mail address: sawalha_moh@yahoo.com (M. M

a b s t r a c t

Based on the nonlinear control and Lyapunov stability theory, the anti-synchronizationbetween two identical and different chaotic systems is investigated, where the antisyn-chronization can be characterized by the vanishing of the sum of relevant variables in cha-otic systems, such that the Lü system is controlled to be the Lorenz system. Theoreticalanalysis and numerical simulations are shown to verify the results.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Since Pecora and Carroll [1] introduced a method to synchronize two identical chaotic systems with different initial con-ditions, chaos synchronization has attracted a great deal of attention from various fields. It has been intensively studied inthe last few years since it has been widely explored in a variety of fields including physical, chemical and ecological system[2]. Hence various synchronization schemes, such as adaptive control [3–9], linear and nonlinear feedback synchronizationmethods [10–13], active control [14–16], and backstepping design technique [17] have been successfully applied to chaossynchronization. The concept of synchronization has been extended to the scope, such as generalized synchronization[18,19], phase synchronization [18], lag synchronization [20], and even anti phase synchronization (APS) [21,22]. APS canalso be interpreted as anti-synchronization (AS), which is a phenomenon that the state vectors of the synchronized systemshave the same amplitude but opposite signs as those of the driving system. Therefore, the sum of two signals are expected toconverge to zero when either AS or APS appears. Recently, several control method has been applied to anti-synchronize cha-otic systems [23–27].

The objective of this paper is to achieve the anti-synchronization of two identical and different chaotic systems based onnonlinear control and Lyapunov stability theory, where the controllers are designed by using the sum of the relevant vari-ables in chaotic systems. Theoretical analysis and numerical simulations results are presented to justify the theoretical anal-ysis strategy in this paper.

2. Design of controller via nonlinear control method

Consider the following system described by

_x ¼ Axþ Bf ðxÞ; ð1Þ

where x 2 Rn is the state vector, A 2 Rn�n;B 2 Rn are metrices and vectors of system parameters, and f : Rn ! Rn is a nonlinearfunction. Eq. (1) is considered as a drive system.

By introduce an additive control U 2 Rn, then the controlled response system is given by

_y ¼ A1xþ B1gðxÞ þ U; ð2Þ

. All rights reserved.

ossa Al-sawalha).

M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179 171

where y 2 Rn denotes the state vector of the response system, A1 2 Rn�n;B1 2 Rn are metrices and vectors of this controlledresponse system parameters, and g : Rn ! Rn is a nonlinear function. A ¼ A1;B ¼ B1, for two identical chaotic systems,A–A1;B–B1, for two different chaotic systems. The anti-synchronization problem is to design a controller U which anti-syn-chronizes the states of both the drive and response systems.

We add (2) to (1) and get

_e ¼ A1xþ B1gðxÞ þ Axþ Bf ðxÞ þ U; ð3Þ

where e ¼ xþ y, the aim of the anti-synchronization is to make

limt!1keðtÞk ¼ 0:

Then let Lyapunov error function be VðeÞ ¼ 12 eT e, where VðeÞ is a positive definite function. Assuming that the parameters of

the drive and response systems are known and the states of both systems are measurable, we may achieve the anti-synchro-nization by selecting the controller U to make the first derivative of VðeÞ; i:e:; _VðeÞ < 0. Then the states of response system anddrive system are anti-synchronized asymptotically globally.

3. Systems description

The Lorenz system [28] is given by

_x ¼ aðy� xÞ;_y ¼ cx� xz� y;_z ¼ xy� bz;

ð4Þ

where x; y and z are respectively proportional to the convective velocity, the temperature difference between descending andascending flows, and the mean convective heat flow, and a; b and the so-called bifurcation parameter c are real constants.Throughout this paper, we set a ¼ 10; b ¼ 8=3 and c ¼ 28 so the system exhibits chaotic behavior.

Lü attractor is a new chaotic attractor, which connects the Lorenz attractor [28] and Chen attractor [29] and representsthe transition from one to another. This new chaotic attractor was proposed and analyzed by Lü and Chen [30]. The Lü cha-otic system is described by the following system of differential equations:

_x ¼ aðy� xÞ;_y ¼ �xzþ cy;_z ¼ xy� bz;

ð5Þ

where a; b and c are positive system parameters. This system is dissipative for c < aþ b and has a chaotic attractor whena ¼ 36; b ¼ 3 and c ¼ 20 [30].

4. Anti-synchronization of two identical Lorenz systems

Now, we assume that we have two identical Lorenz systems where the master (drive) system with the subscript 1 and theslave (response) system having identical equations denoted by the subscript 2. The master and slave systems are

_x1 ¼ aðy1 � x1Þ;_y1 ¼ cx1 � x1z1 � y1;

_z1 ¼ x1y1 � bz1;

ð6Þ

and

_x2 ¼ aðy2 � x2Þ þ u1;

_y2 ¼ cx2 � x2z2 � y2 þ u2;

_z2 ¼ x2y2 � bz2 þ u3;

ð7Þ

where the systems parameter are chosen such that both system (6) and (7) are in chaotic state when the control functionu ¼ ½u1;u2;u3�T ¼ 0. Our goal is to determine the control function to make the first derivative of VðeÞ, i.e., _VðeÞ < 0. Weadd Eqs. (7) to (6) and get the error equation as follows

_e1 ¼ aðe2 � e1Þ þ u1;

_e2 ¼ ce1 � x2z2 � x1z1 � e2 þ u2;

_e3 ¼ x2y2 þ x1y1 � be3 þ u3;

ð8Þ

where e1 ¼ x2 þ x1, e1 ¼ y2 þ y1, e3 ¼ z2 þ z1. In order to determine the controller, let

u2 ¼ u12 þ u22; where u22 ¼ x2z2 þ x1z1;

u3 ¼ u13 þ u23; where u23 ¼ �x2y2 � x1y1:

172 M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179

Then we rewrite system (8) in the following form

_e1 ¼ aðe2 � e1Þ þ u1;

_e2 ¼ ce1 � e2 þ u12;

_e3 ¼ �be3 þ u13:

ð9Þ

Based on Lyapunov stability theory, when controller satisfies the assumption with VðeÞ ¼ 12 eT e, a positive definite function

and the first derivative of VðeÞ; _VðeÞ < 0, the anti-synchronization of two identical lorenz systems from different initial con-ditions is achieved. Take a Lyapunov function for Eq. (9) into consideration

VðeÞ ¼ 12

eT e

we get the first derivative of VðeÞ:

_V ¼ e1½ae2 � ae1 þ u1� þ e2½ce1 � e2 þ u12� þ e3½�be3 þ u13�: ð10Þ

Therefore, if we choose U as follows,

u1 ¼ �ce2;

u12 ¼ �ae1;

u13 ¼ 0;ð11Þ

then

_V ¼ �ae2

1 � e22 � be2

3: ð12Þ

_VðeÞ < 0 is satisfied. Since _VðeÞ is a negative-definite function, the error states

limt!1keðtÞk ¼ 0:

Therefore, the states of controlled response system and drive system are globally anti-synchronized asymptotically.

4.1. Numerical simulation results

In this section, to verify and demonstrate the effectiveness of the proposed method, we will display the numerical results foranti-synchronizing Lorenz systems under different inial condition, fourth-order Runge–Kutta method is used to solve the sys-tems (6)–(7) with time step size 0.001.We select the parameters of Lorenz system as a ¼ 10; b ¼ 8=3; c ¼ 28, so these systemsexhibits a chaotic behavior. we set x1ð0Þ ¼ �15:8; y1ð0Þ ¼ �17:48; z1ð0Þ ¼ 35:64 and x2ð0Þ ¼ 10:8; y2ð0Þ ¼ 12; z2ð0Þ ¼ 20. Fig. 1displays the time response of states x1; y1; z1 for the drive system (6) and the states x2; y2; z2 for the response system (7). Fig. 2display the anti-synchronization errors of systems (6) and (7). It can be seen from the figures that the anti-synchronization er-rors converge to zero rapidly.

5. Anti-synchronization of two identical Lü systems

In order to observe the anti-synchronization behavior in the Lü system (5), we have two Lü systems where the drive sys-tem with three state variables denoted by the subscript 1 and the response system having identical equations denoted by thesubscript 2. However, the initial condition on the drive system is different from that of the response system. The two Lü sys-tems are described, respectively, by the following equations:

_x1 ¼ aðy1 � x1Þ;_y1 ¼ �x1z1 þ cy1;

_z1 ¼ x1y1 � bz1;

ð13Þ

and

_x2 ¼ aðy2 � x2Þ þ u1;

_y2 ¼ �x2z2 þ cy2 þ u2;

_z2 ¼ x2y2 � bz2 þ u3;

ð14Þ

where the systems parameter are chosen such that both system (13) and (14) are in chaotic state when the control functionu ¼ ½u1;u2;u3�T ¼ 0. Our goal is to determine the control function to make the first derivative of VðeÞ; i:e:; _VðeÞ < 0. We add Eq.(14) to Eq. (13) and get the error equation as follows:

_e1 ¼ aðe2 � e1Þ þ u1;

_e2 ¼ �x1z1 � x2z2 þ ce2 þ u2;

_e3 ¼ x1y1 þ x2y2 � be3 þ u3;

ð15Þ

a

b

c

Fig. 1. The time response of states for drive system (6) and the response system (7) via nonlinear control (a) signals x1 and x2; (b) signals y1 and y2; (c)signals z1 and z2.

M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179 173

where e1 ¼ x2 þ x1, e1 ¼ y2 þ y1, e3 ¼ z2 þ z1. In order to determine the controller, let

u2 ¼ u12 þ u22; where u22 ¼ x2z2 þ x1z1;

u3 ¼ u13 þ u23; where u23 ¼ �x2y2 � x1y1:

Then we rewrite system (15) in the following form

_e1 ¼ aðe2 � e1Þ þ u1;

_e2 ¼ ce2 þ u12;

_e3 ¼ �be3 þ u13:

ð16Þ

Based on Lyapunov stability theory, when controller satisfies the assumption with VðeÞ ¼ 12 eT e, a positive definite function

and the first derivative of VðeÞ; _VðeÞ < 0, the anti-synchronization of two identical Lü systems from different initial condi-tions is achieved. Take a Lyapunov function for Eq. (16) into consideration

VðeÞ ¼ 12

eT e;

a b

c

Fig. 2. Dynamics of anti-synchronization errors for two identical Lorenz systems (drive system (6) and response system (7)). (a) e1 ¼ x2 þ x1; (b)e2 ¼ y2 þ y1; (c) e3 ¼ z2 þ z1.

174 M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179

we get the first derivative of VðeÞ:

_V ¼ e1½ae2 � ae1 þ u1� þ e2½ce2 þ u12� þ e3½�be3 þ u13�: ð17Þ

Therefore, if we choose U as follows,

u1 ¼ 0;u12 ¼ �ae1 � 2ce2;

u13 ¼ 0;ð18Þ

then

_V ¼ �ae2

1 � ce22 � be2

3: ð19Þ

_VðeÞ < 0 is satisfied. Since _VðeÞ is a negative-definite function, the error states

limt!1keðtÞk ¼ 0

Therefore, the states of controlled response system and drive system are globally anti-synchronized asymptotically.

5.1. Numerical simulation results

Fourth order Runge–Kutta integration method is used to solve the systems of differential Eqs. (13) and (14). In addition, atime step size 0.001 is employed. We select the parameters of Lü system as a ¼ 36; b ¼ 3; c ¼ 20, so these systems exhibits a

M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179 175

chaotic behavior. The initial values of the drive and response systems are x1ð0Þ ¼ 0:5; y1ð0Þ ¼ 1; z1ð0Þ ¼ 1 andx2ð0Þ ¼ 5; y2ð0Þ ¼ 2; z2ð0Þ ¼ 2, respectively, Fig. 3 displays the time response of states x1; y1; z1 for the drive system (13)and the states x2; y2; z2 for the response system (14). Fig. 4 display the anti-synchronization errors of systems (13) and(14). It can be seen from the figures that the anti-synchronization errors converge to zero rapidly.

6. Anti-synchronization of two different chaotic systems

In order to achieve the behavior of the anti-synchronization between the Lorenz system and the Lü system by use theproposed method, we assume that the Lorenz system is the drive system whose variables are denoted by subscript 1 andthe Lü system is the response system whose variables are denoted by subscript 2. The drive and the response systemsare described, respectively, by the following equations:

Fig. 3.signals

_x1 ¼ a1ðy1 � x1Þ;_y1 ¼ c1x1 � x1z1 � y1;

_z1 ¼ x1y1 � b1z1;

ð20Þ

a

b

c

The time response of states for drive system (13) and the response system (14) via nonlinear control (a) signals x1 and x2; (b) signals y1 and y2; (c)z1 and z2.

a b

c

Fig. 4. Dynamics of anti-synchronization errors for two identical Lü systems (drive system (13) and response system (14); (a) e1 ¼ x2 þ x1; (b) e2 ¼ y2 þ y1;(c) e3 ¼ z2 þ z1.

176 M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179

and

_x2 ¼ a2ðy2 � x2Þ þ u1;

_y2 ¼ �x2z2 þ c2y2 þ u2;

_z2 ¼ x2y2 � b2z2 þ u3;

ð21Þ

where U ¼ ½u1;u2;u3�T is the controller function. In the following, we design an nonlinear controller to achieve the anti-syn-chronization between the Lorenz system and the Lü system. By adding Eqs. (21) to (20) yields the error dynamical systembetween Eqs. (20) and (21):

_e1 ¼ a2ðe2 � e1Þ þ ða1 � a2Þðy1 � x1Þ þ u1;

_e2 ¼ �x2z2 � x1z1 þ c2e2 � ðc2 þ 1Þy1 þ c1x1 þ u2;

_e3 ¼ x2y2 þ x1y1 � b2e3 þ ðb2 � b1Þz1 þ u3;

ð22Þ

where e1 ¼ x2 þ x1, e1 ¼ y2 þ y1, e3 ¼ z2 þ z1. In order to determine the controller, let

u2 ¼ u12 þ u22 and u3 ¼ u13 þ u23; whereu22 ¼ x2z2 þ x1z1 and u23 ¼ �x2y2 � x1y1

Then we rewrite (22) in the following form:

_e1 ¼ a2ðe2 � e1Þ þ ða1 � a2Þðy1 � x1Þ þ u1;

_e2 ¼ c2e2 � ðc2 þ 1Þy1 þ c1x1 þ u12;

_e3 ¼ �b2e3 þ ðb2 � b1Þz1 þ u13:

ð23Þ

M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179 177

Take a Lyapunov function for Eq. (23) into consideration

Fig. 5.signals

VðeÞ ¼ 12

eT e;

we get the first derivative of VðeÞ:

_V ¼ e1½a2e2 � a2e1 þ ða1 � a2Þðy1 � x1Þ þ u1� þ e2½c2e2 � ðc2 þ 1Þy1 þ c1x1 þ u12� þ e3½�b2e3 þ ðb2 � b1Þz1 þ u13� ð24Þ

Therefore, if we choose U as follows:

u1 ¼ �ða1 � a2Þðy1 � x1Þ;u12 ¼ �2c2e2 � a2e1 þ ðc2 þ 1Þy1 � c1x1;

u13 ¼ �ðb2 � b1Þz1;

ð25Þ

then

_V ¼ �a2e21 � c2e2

2 � b2e23: ð26Þ

a

b

c

The time response of states for drive system (Lorenz system) and the response system (Lü system) via nonlinear control (a) signals x1 and x2; (b)y1 and y2; (c) signals z1 and z2.

a b

c

Fig. 6. Anti-synchronization error in Lorenz system and Lü system via nonlinear control (a) signals x1 and x2; (b) signals y1 and y2; (c) signals z1 and z2.

178 M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179

_VðeÞ < 0 is satisfied. Since _VðeÞ is a negative-definite function, the error states

limt!1keðtÞk ¼ 0

Therefore, this choice will lead the error states e1; e2; e3 to converge to zero as time t tends to infinity and hence the anti-synchronization of two different chaotic systems is achieved.

6.1. Numerical simulation results

Fourth order Runge–Kutta integration method is used to solve the systems of differential Eqs. (20) and (21). In addition, atime step size 0:001 is employed. We select the parameters of the Lü system as a ¼ 36; b ¼ 3; c ¼ 20, and for the Lorenz sys-tem as a ¼ 10; b ¼ 8=3; c ¼ 28, so these systems exhibits a chaotic behavior. The initial values of the drive and response sys-tems are x1ð0Þ ¼ �15:8; y1ð0Þ ¼ �17:48; z1ð0Þ ¼ 35:64 and x2ð0Þ ¼ �10:8; y2ð0Þ ¼ 12; z2ð0Þ ¼ 20, respectively. The diagram ofthe Lü system is controlled to be Lorenz system are shown in Figs. 5 and 6.

7. Conclusion

Based on Lyapunov stability theory, we propose a nonlinear control method to anti-synchronize two identical and twodifferent chaotic systems. The simulation results show that the states of two identical Lorenz and Lü systems are anti-syn-chronized asymptotically globally. For two different chaotic systems, the Lü system is controlled to trace the Lorenz system

M. Mossa Al-sawalha, M.S.M. Noorani / Chaos, Solitons and Fractals 42 (2009) 170–179 179

and the states of two systems become the same finally. This shows that the method proposed has strong robustness. Thesimulation results are presented to show the effectiveness of this approach.

Acknowledgement

This work is financially supported by the Malaysian Ministry of Science, Technology and Environment Sciencefund Grant:06-01-02-SF0177.

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