Commun Nonlinear Sci Numer Simulat 15 (2010) 3067–3072

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat

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

Coexistence of anti-phase and complete synchronization in thegeneralized Lorenz system

Qing Zhang a,c, Jinhu Lü b, Shihua Chen a,*

a School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR Chinab Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, PR Chinac College of Science, Wuhan University of Science and Technology, Wuhan 430065, PR China

a r t i c l e i n f o

Article history:Received 28 June 2009Received in revised form 22 October 2009Accepted 13 November 2009Available online 20 November 2009

Keywords:Generalized Lorenz systemsAdaptive feedback controlCoexistenceAnti-phase synchronizationComplete synchronization

1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.11.020

* Corresponding author.E-mail address: shcheng@whu.edu.cn (S. Chen).

a b s t r a c t

This paper investigates a class of new synchronization phenomenon. Some control strategyis established to guarantee the coexistence of anti-phase and complete synchronization inthe generalized Lorenz system. The efficiency of the control scheme is revealed by someillustrative simulations.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Recently, there has been increasing interest in the study of chaos control and synchronization due to their potentialapplications. Several typical synchronization have been identified such as complete synchronization, phase synchroniza-tion, lag synchronization, generalized synchronization, anti-phase synchronization and so on [1–8,10,11,14]. However,most of the existing works focus on investigating the same kind synchronization in a given system, i.e., all the statesof the response system have the same kind synchronization to the corresponding states of the drive system. For exam-ple, when we say that two systems are complete synchronized (or lag synchronized, or something else) with each other,it means that each pair of the states between the interactive systems is complete synchronous (or lag synchronous, orsomething else). What’s more, the given systems usually are typical benchmark chaotic systems, such as the Lorenz sys-tem, Chen system and Lü system.

In this paper, we consider the generalized Lorenz system, which includes the systems proposed above, and we inves-tigate a class of new synchronization phenomenon – two kinds of synchronization coexist in the same system, i.e., onepart states of the interactive systems is anti-phase synchronous, and the other part is complete synchronous under cer-tain parameter region.

This paper is organized as follows. Section 2 formulate the problem. In section 3, linear feedback control method is pre-sented. In section 4, the adaptive feedback control method with only one controller is designed. Some numerical examplesare given in section 5. We conclude the paper in section 6.

. All rights reserved.

3068 Q. Zhang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3067–3072

2. The problem formulation

In 1963, Lorenz found the first classical chaotic attractor[12]. In 1999, Chen found the Chen attractor which is similar butnot topologically equivalent to Lorenz chaotic attractor [9]. In 2002, Lü found another new critical chaotic system [13]. Thesesystems can be included in the following generalized Lorenz system [15] which is described by

_X ¼ AX þ f ðXÞ; ð1Þ

where

A ¼a11 a12 0a21 a22 00 0 a33

0B@

1CA

is a constant matrix of parameters, X ¼ ðx1; x2; x3Þ> are the state variables and

f ðXÞ ¼ ð0;�x1x3; x1x2Þ>

is the nonlinear part of the system. On the other hand, most of the existing works focused on investigating the same kindsynchronization in a given system, i.e., all the states of the response system are synchronous to the corresponding statesof the drive system. But two kinds of synchronization may coexist in the same system, so we investigate a class of new syn-chronization phenomenon in which anti-phase and complete synchronization coexists in the system.

Throughout the paper, the drive system is chosen as the generalized Lorenz system (1), namely,

_x1 ¼ a11x1 þ a12x2;

_x2 ¼ a21x1 þ a22x2 � x1x3;

_x3 ¼ a33x3 þ x1x2;

8><>:

ð2Þ

and the response system is

_Z ¼ AZ þ f ðZÞ þ U;

namely,

_z1 ¼ a11z1 þ a12z2 þ u1;

_z2 ¼ a21z1 þ a22z2 � z1z3 þ u2;

_z3 ¼ a33z3 þ z1z2 þ u3;

8><>:

ð3Þ

where U ¼ ðu1;u2;u3Þ> are controllers to be designed. We assume the system parameters satisfy the conditiona11 < 0; a33 < 0 and a12–0. Obviously, the system parameters in Lorenz system, Chen system and Lü system satisfy thiscondition.

3. Linear feedback control method

Many methods and techniques have been developed to synchronize the chaotic systems, but the linear feedback controlmethod is one of the most popular methods because it is simplicity for the theoretical analysis and implementation. In thissection, we consider the linear feedback method to make the response system (3) be synchronous with the drive system (2),i.e., the first two pair of state variables between the interactive systems are anti-phase synchronous, while the third pair ofstate variable is complete synchronous. To this end, we define the error variable as

e ¼ ðe1; e2; e3Þ> ¼ ðx1 þ z1; x2 þ z2; x3 � z3Þ>;

and take u1 ¼ u3 ¼ 0;u2 ¼ �ke2 in the response system (3). Thus we get the error dynamical system as follows:

_e1 ¼ a11e1 þ a12e2;

_e2 ¼ a21e1 þ a22e2 � ke2 � x3e1 � x1e3 þ e1e3;

_e3 ¼ a33e3 þ x2e1 þ x1e2 � e1e2:

8><>:

ð4Þ

From the viewpoint of control theory, the synchronization of (2) and (3) is equivalent to the asymptotical stability of theerror system (4) at e ¼ 0. As system (2) is chaotic, x1; x2; x3 are bounded, thus we suppose that M and N are the upper boundsof the absolute values of x2 and x3.

Theorem 1. The anti-phase synchronization and complete synchronization coexistent in the system (3) and system (2) if thecontrollers are selected as u1 ¼ u3 ¼ 0;u2 ¼ �kðx2 þ z2Þ, where k > a22 þmaxf� 1

4aa11ðN þ aja12j þ ja21jÞ2;� a33

4 ðNþaja12j þ ja21jÞ2g and a ¼ M2þ4

4a11a33.

Q. Zhang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3067–3072 3069

Proof. We choose a candidate Lyapunov function as

VðtÞ ¼ 12ðae2

1 þ e22 þ e2

3Þ:

Its time derivative along the trajectories of system (4) is

_VðtÞ ¼ ae1 _e1 þ e2 _e2 þ e3 _e3

¼ ae1ða11e1 þ a12e2Þ þ e2ða21e1 þ a22e2 � ke2 � x3e1 � x1e3 þ e1e3Þ þ e3ða33e3 þ x2e1 þ x1e2 � e1e2Þ¼ aa11e2

1 � ðk� a22Þe22 þ a33e2

3 þ ðaa12 þ a21 � x3Þe1e2 þ x2e1e3

6 aa11e21 � ðk� a22Þe2

2 þ a33e23 þ ðN þ aja12j þ ja21jÞje1jje2j þMje1jje3j ¼ �ðje1j; je2j; je3jÞPðje1j; je2j; je3jÞ>; ð5Þ

where

P ¼�aa11 � 1

2 ðN þ aja12j þ ja21jÞ � 12 M

� 12 ðN þ aja12j þ ja21jÞ k� a22 0

� 12 M 0 �a33

0B@

1CA:

Obviously, from the conditions of the theorem we have �aa11 > 0, �aa11ðk� a22Þ � 14 ðN þ aja12j þ ja21jÞ2 > 0 and

ða11a33a� 14 M2Þðk� a22Þ þ 1

4 ðN þ aja12j þ ja21jÞ2a33 > 0, that is, P is positive definite. Thus, the error system (4) is asymptot-ical stability at e ¼ 0. This complete the proof of the theorem. h

4. Adaptive feedback control method

Although the linear feedback method is easy in the form, it is not precise because the upper bounds M and N in the gen-eralized Lorenz system are not easy to determine and it is very difficult to determine the upper bound of the gain k. Peopleusually take the gain big enough to ensure synchronization and this leads to a waste in practice. In the following, we take theadaptive feedback technique in which we need not to know the upper bound of the generalized Lorenz system. To this end,we take the error variable as

e ¼ ðe1; e2; e3Þ> ¼ ðx1 þ z1; x2 þ z2; x3 � z3Þ>;

and set the controllers as u1 ¼ u3 ¼ 0;u2 ¼ �kðtÞe2; where kðtÞ is a function of t and satisfy the following equation

k0ðtÞ ¼ e22:

Thus, we get the dynamical error system as follows

_e1 ¼ a11e1 þ a12e2;

_e2 ¼ a21e1 þ a22e2 � kðtÞe2 � x3e1 � x1e3 þ e1e3;

_e3 ¼ a33e3 þ x2e1 þ x1e2 � e1e2;

8><>:

ð6Þ

Theorem 2. The anti-phase synchronization and complete synchronization coexistent in the system (3) and system (2) if thecontrollers are selected as u1 ¼ u3 ¼ 0, and u2 ¼ �kðtÞe2, where k0ðtÞ ¼ e2

2.

Proof. We choose a candidate Lyapunov function as

VðtÞ ¼ 12

ae21 þ e2

2 þ e23 þ ðk� k�Þ2

h i;

where k� is a constant satisfying k� > a22 þmaxf� 14aa11ðN þ aja12j þ ja21jÞ2;� a33

4 ðN þ aja12j þ ja21jÞ2g and a > M2þ44a11a33

. Its timederivative along the trajectories of system (6) is

_VðtÞ ¼ ae1 _e1 þ e2 _e2 þ e3 _e3 þ ðkðtÞ � k�Þ _kðtÞ¼ ae1ða11e1 þ a12e2Þ þ e2ða21e1 þ a22e2 � kðtÞe2 � x3e1 � x1e3 þ e1e3Þ þ þe3ða33e3 þ x2e1 þ x1e2 � e1e2Þþ ðkðtÞ � k�Þe2

2

¼ aa11e21 � ðk

� � a22Þe22 þ a33e2

3 þ ðaa12 þ a21 � x3Þe1e2 þ x2e1e3

6 aa11e21 � ðk� a22Þe2

2 þ a33e23 þ ðN þ aja12j þ ja21jÞje1jje2j þMje1jje3j ¼ �ðje1j; je2j; je3jÞPðje1j; je2j; je3jÞ>; ð7Þ

where,

3070 Q. Zhang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3067–3072

P ¼�aa11 � 1

2 ðN þ aja12j þ ja21jÞ � 12 M

� 12 ðN þ aja12j þ ja21jÞ k� � a22 0

� 12 M 0 �a33

0B@

1CA:

Obviously, from the conditions of the theorem we have �aa11 > 0, �aa11ðk� � a22Þ � 14 ðN þ aja12j þ ja21jÞ2 > 0 and

ða11a33a� 14 M2Þðk� � a22Þ þ 1

4 ðN þ aja12j þ ja21jÞ2a33 > 0, that is, P is positive definite. Thus e1; e2; e3; kðtÞ 2 L1. Next, as_V 6 �ðje1j; je2j; je3jÞPðje1j; je2j; je3jÞ>, and P is positive definite, we obtain

R t0 kminðPÞðe2

1 þ e22 þ e2

3Þdt 6R t

0ðje1j; je2j; je3jÞPðje1j; je2j; je3jÞ>dt

6 �R t

0_Vdt ¼ Vð0Þ � VðtÞ

6 Vð0Þ;

where kminðPÞ is the smallest eigenvalue of the symmetrical positive definite matrix P. It follows that e1; e2; e3 2 L2. On theother hand, from Eq. (6), we have e1; e2; e3 2 L1. Thus by Barbalat’s lemma [16], we have e1ðtÞ; e2ðtÞ; e3ðtÞ ! 0 as t approachesto infinity. This implies the result of the theorem.

It should be noted that the variable feedback strength kðtÞ is automatically adapted to a suitable strength, which issignificantly different from the usual linear feedback technique. h

5. Numerical simulation

In the simulations, we choose a12 ¼ �a11 ¼ 36; a21 ¼ 0; a22 ¼ 20; a33 ¼ �3. The initial conditions of the drive and responsesystem are ð2:5;4;6Þ and ð23;14;30Þ respectively. In linear feedback control we take k ¼ 160. In adaptive feedback control,the initial conditions of feedback gain kðtÞ is 0. From Figs. 1and 2. We can see that e1 ¼ x1 þ z1 ! 0; e2 ¼ x2 þ z2 ! 0, ande3 ¼ x3 � z3 ! 0 as time t approaches to infinite. From Fig. 3, we can see that the variable feedback strength k(t) is automat-ically adapted to a suitable strength.

6. Conclusion

In the paper, we investigate a class of new synchronization phenomenon – two kinds of synchronization coexist in thesame system, i.e., one part states of the interactive systems is anti-phase synchronized, and the other part is complete

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

0

10

20

30

x 1(t)+z1(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

0

10

20

x 2(t)+z2(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−30

−20

−10

0

x 3(t)−z3(t)

t

Fig. 1. The convergence dynamics of the error system (4) when k ¼ 160.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

0

10

20

30

x 1(t)+z1(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

0

10

20

x 2(t)+z2(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−30

−20

−10

0

10

x 3(t)−z3(t)

t

Fig. 2. The convergence dynamics of the error system (6).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

k(t)

t

Fig. 3. Feedback gain kðtÞ.

Q. Zhang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3067–3072 3071

synchronized under certain parameter region, which is quite different from conventional synchronization. The efficiency ofthe control scheme is revealed by some illustrative simulations.

Acknowledgements

The work is supported by the National Nature Science Foundation of PR China (Grant No. 70571059 ). The authors arevery grateful to the referees for their valuable comments and suggestions.

References

[1] Pecora LM, Carroll TL. Synchronization in chaotic system. Phys Rev Lett 1990;64:821–4.[2] Rodriguez E, George N. Perceptions shadow: long-distance synchronization of human brain activity. Nature (London) 1999;397:430–43.[3] Robert CE, Selverston AI. Synchronous behavior of two coupled biological neurons. Phys Rev E 1998;81:5692–5.

3072 Q. Zhang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 3067–3072

[4] Grzybowski JMV, Rafikov M, Balthazar JM. Synchronization of the unified chaotic system and application in secure communication. Commun NonlinearSci Numer Simul 2009;14:2793–806.

[5] Luo ACJ. A theory for synchronization of dynamical systems. Commun Nonlinear Sci Numer Simul 2009;14:1901–51.[6] Liu W, Qian X, Yang J, Xiao J. Anti-synchronization in coupled chaotic oscillators. Phys Lett A 2006;354:119–25.[7] Rosenblum MG, Pikovsky AS, Kurths J. Phase synchronization of chaotic oscillators. Phys Rev Lett 1996;76:1804–7.[8] Hu MF, Xu ZY, Zhang R. Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems. Commun Nonlinear Sci Numer

Simul 2008;13(2):456–64.[9] Chen GR, Dong XM. From chaos to order: perspectives, methodologies and application. Singapore: World Scientific; 1999.

[10] Bowong S. Adaptive synchronization between two different chaotic dynamical systems. Commun Nonlinear Sci Numer Simul 2007;12:976–85.[11] Rafikov M, Balthazar J. On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun Nonlinear Sci Numer

Simul 2008;13:1246–55.[12] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20:130–41.[13] Lü J, Chen G, Cheng D, Celikovsky S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12:2917–26.[14] Huang CF, Cheng KH, Yan JJ. Robust chaos synchronization of four-dimensional energy resource systems subject to unmatched uncertainties. Commun

Nonlinear Sci Numer Simul 2009;14(6):2784–92.[15] Celikovsky S, Chen G. On the generalized Lorenz canonical form. Chaos, Solitons Fractals 2005;26:1271–6.[16] Barblart I. Systems dèquations differentielled òscillations nonlinearies. Rev Roumaine Math Pures Appl 1959;4:267–70.