Anti-Synchronization of Liu-Chen-Liu and Liu-Su-Liu Chaotic Systems by Active Nonlinear Control

Home , Su (surname)

ISSN No. 0976-5697 Volume 1, No. 4, Nov-Dec 2010 International Journal of Advanced Research in Computer Science RESEARCH PAPER Available Online at www.ijarcs.info Anti-synchronization of Liu-Chen-Liu and Liu-Su-Liu Chaotic Systems by Active Nonlinear Control

Dr. V. Sundarapandian Professor, Research and Development Centre Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600 062, INDIA [email protected]

Abstract : The purpose of this paper is to study chaos anti-synchronization of identical Liu-Chen-Liu chaotic systems (Liu, Chen and Liu, 2007), identical Liu-Su-Liu chaotic systems (Liu, Su and Liu, 2006), and non-identical Liu-Chen-Liu and Liu-Su-Liu chaotic systems using active nonlinear control. Sufficient conditions for achieving anti-synchronization of the identical and different Liu-Chen-Liu and Liu-Su-Lu chaotic sytsems using active nonlinear control are derived based on Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the nonlinear feedback control method is effective and convenient to anti-synchronize identical and different Liu-Chen- Liu and Liu-Su-Liu chaotic systems. Numerical simulations are also given to illustrate and validate the anti-synchronization results for Liu- Chen-Liu and Liu-Su-Liu chaotic systems.

Keywords: Chaos Anti-synchronization, Nonlinear Control, Liu-Chen-Liu System, Liu-Su-Liu System, Active Control.

anti-synchronization of Liu-Chen-Liu and Liu-Su-Liu chaotic I. INTRODUCTION systems. In Section VI, we present the conclusions of this paper. Chaotic systems are dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly II. PROBLEM STATEMENT AND OUR METHODOLOGY referred to as the butterfly effect [1]. Since the pioneering work of Pecora and Carroll [2], chaos Consider the chaotic system described by the dynamics synchronization has attracted a great deal of attention from x= Ax + f( x ) (1) various fields and it has been extensively studied in the last two # n decades [2-17]. Chaos theory has been explored in a variety of where x ∈ R is the state of the system, A is the n× n matrix fields including physical [3], chemical [4], ecological [5] of the system parameters and f : Rn→ R n is the nonlinear systems, secure communications [6-8] etc. In the recent years, part of the system. We consider the system (1) as the master or various schemes such as PC method [2], OGY method [9], drive system. active control [10-12], adaptive control [13-14], time-delay As the slave or response system, we consider the following feedback approach [15], backstepping design method [16], chaotic system described by the dynamics sampled-data feedback synchronization method [17], sliding mode control [18], etc. have been successfully applied for y# = By + g( y ) + u (2) chaos synchronization. Recently, active control has been n applied to anti-synchronize identical chaotic systems [19-20] where y ∈ R is the state vector of the response system, B is and different hyperchaotic systems [21]. the n× n matrix of the system parameters, g : Rn→ R n is In most of the chaos anti-synchronization approaches, the n the nonlinear part of the response system and u ∈ R is the master-slave or drive-response formalism is used. If a controller of the response system. particular chaotic system is called the master or drive system and another chaotic system is called the slave or response If A= B and f= g , then x and y are the states of two system, then the idea of the anti-synchronization is to use the identical chaotic systems. If A≠ B and f≠ g , then x and output of the master system to control the slave system so that y are the states of two different chaotic systems. the states of the slave system have the same amplitude but For the anti-synchronization of the chaotic systems (1) and opposite signs as the states of the master system (2) using active control, we design a feedback controller asymptotically. In other words, the sum of the states of the u master and slave systems are expected to converge to zero which anti-synchronizes the states of the master system (1) and asymptotically when anti-synchronization appears. the slave system (2) for all initial conditions of the systems. This paper has been organized as follows. In Section II, we If we define the anti-synchronization error as give the problem statement and our methodology. In Section e= y + x , (3) III, we discuss the chaos anti-synchronization of two identical then the anti-synchronization error dynamics is obtained as Liu-Chen-Liu systems ([22], 2007). In Section IV, we discuss the chaos anti-synchronization of two identical Liu-Su-Liu e# =++ By Ax gy() + fx () + u (4) chaotic systems ([23], 2006). In Section V, we discuss the

© 2010, IJARCS All Rights Reserved 408 Dr. V. Sundarapandian et al , International Journal of Advanced Research in Computer Science, 1 (4), Nov. –Dec, 2010,408-413 Thus, the global anti-synchronization problem is essentially Figure 1 illustrates the chaotic portrait of the Liu-Chen-Liu to find a feedback controller u so as to stabilize the error system, which has a reversed butterfly-shaped chaotic attractor. dynamics (4) for all initial conditions e(0)∈ Rn , i.e. lime ( t )= 0 (5) t→∞ for all initial conditions e(0)∈ Rn . We use Lyapunov stability theory as our methodology. We take as a candidate Lyapunov function V( e )= eT Pe , (6) where P is a positive definite matrix. Note that V : Rn→ R n is a positive definite function by construction. We assume that the parameters of the master and slave systems are known and that the states of both systems (1) and (2) are measurable. If we find a feedback controller u so that

T (7) V#( e )= − e Qe , Figure 1. Chaotic Portrait of the Liu-Chen-Liu System (8) where Q is a positive definite matrix, then V# : Rn→ R n is a negative definite function. The anti-synchronization error e is defined by Thus, by Lyapunov stability theory [24], the error dynamics eyxi= i + i , ( i = 1,2,3) (10) (4) is globally exponentially stable and hence the condition (5) The error dynamics is obtained as will be satisfied for all initial conditions e(0)∈ R n . Then the e= ae( − e ) + u states of the master system (1) and slave system (2) will be #1 21 1 globally exponentially anti-synchronized. e#2=+ be 1 kyy( 1313 + xx ) + u 2 (11)

III. ANTI -SYNCHRONIZATION OF IDENTICAL LIU -CHEN -LIU e#3=−− ce 3 hyy( 1212 + xx ) + u 3 CHAOTIC SYSTEMS To find an anti-synchronizing controller, we first let

u2= u 2a + u 2 b In this section, we apply the active nonlinear control (12) technique for the anti-synchronization of two identical Liu- u3= u 3a + u 3 b Chen-Liu chaotic systems ([22], 2007). where Thus, the master system is described by the Liu-Chen-Liu dynamics u2b = − kyy( 13 + xx 13 ) (13) x= ax( − x ) #1 2 1 u3b = hyy( 12 + xx 12 )

x#2= bx 1 + kx 13 x (8) Substituting (12) and (13) into (11), we obtain

x#3= − cx 3 − hx 12 x e#1= ae( 21 − e ) + u 1 e= be + u (14) where x1, x 2 , x 3 are the states of the system and #2 1 2 a

abc, ,, h , k are positive parameters of the system. e#3= − ce 3 + u 3 a The slave system is also described by the Liu-Chen-Liu Next, we consider the candidate Lyapunov function dynamics as 1 1 T 2 2 2 (15) y ay( y ) u Ve( ) = ee =() eee1 ++ 2 3 #1= 21 − + 1 2 2 (9) 3 y#2= by 1 + ky 13 y + u 2 which is a positive definite function on R .

y#3=− cy 3 − hy 123 y + u A simple calculation gives 2 2 where y, y , y are the states of the system and Ve#()=− ae − ce + () abe + e + eu 1 2 3 131 2 11 (16) T +eu22a + eu 33 a u= [ u1 u 2 u 3 ] Therefore, we choose is the nonlinear controller to be designed. The Liu-Chen-Liu system (8) is a new 3-D chaotic system u1= −(a + b ) e 2 proposed by Liu, Chen and Liu ([22], 2007). u2a = − e 2 (17) The Liu-Chen-Liu system (8) is chaotic when u = 0 a=10, b = 40, c = 2.5, h = 1 and k =16. 3a

Dr. V. Sundarapandian et al , International Journal of Advanced Research in Computer Science, 1 (4), Nov. –Dec, 2010,408-413

Substituting (17) into (14), the error dynamics simplifies to IV. ANTI -SYNCHRONIZATION OF IDENTICAL LIU -SU-LIU CHAOTIC SYSTEMS e#1= − ae 1 − be 2 In this section, we apply the active nonlinear control e#2= be 1 − e 2 (18) technique for the anti-synchronization of two identical Liu-Su- e#3= − ce 3 Liu chaotic systems ([23], 2006). Substituting (17) into (16), we obtain Thus, the master system is described by the Liu-Su-Liu 2 2 2 dynamics V#( e ) = − ae 1−e 2 − ce 3 (19) x#1=α( x 2 − x 1 ) which is a negative definite function on R3 since a and c are (21) positive constants. x#2=β x 1 + xx 13

Hence, by Lyapunov stability theory [24], the error x#3= −γ x 3 − xx 12 dynamics (18) is globally exponentially stable. where x x x are the states of the system and are Combining (12), (13) and (17), the anti-synchronizing 1, 2 , 3 α, β , γ nonlinear controller u is obtained as positive parameters of the system. u= −( abe + ) The slave system is also described by the Liu-Su-Liu 1 2 dynamics as

u2=−− e 2 kyy( 13 + xx 13 ) (20) y#1=α( yyu 21 − ) + 1 u3=h( yy 12 + xx 12 ) y#2=β y 1 + yy 13 + u 2 (22) Thus, we have proved the following result. y=−γ y − yyu + Theorem 1. The identical Liu-Chen-Liu systems (8) and (9) #3 3 12 3 are exponentially and globally anti-synchronized for any initial where y1, y 2 , y 3 are the states of the system and conditions with the nonlinear controller u defined by (20). T Numerical Results u= [ u1 u 2 u 3 ] For the numerical simulations, the fourth-order Runge- is the nonlinear controller to be designed. −6 Kutta method with time-step 10 is used to solve the systems The Liu-Su-Liu system (21) is a new 3-D chaotic system using MATLAB. derived by Liu, Su and Liu ([23], 2007). The Liu-Su-Liu For the Liu-Chen-Liu system (8), the parameter values are system (21) is chaotic when taken as those which result in the chaotic behaviour of the α=10, β = 35 and γ =1.4. system, viz. a=10, b = 40, c = 2.5, h = 1 and k =16. Figure 3 illustrates the chaotic portrait of the Liu-Su-Liu The initial values of the master system (8) are taken as chaotic system (21).

x1(0)= 7, x 2 (0) = 3, x 3 (0) = 12 while the initial values of the slave system (9) are taken as

y1(0)= 2, y 2 (0) = 10, y 3 (0) = 9 Figure 2 shows the anti-synchronization between the states of the master system (8) and the slave system (9).

Figure 3. Chaotic Portrait of the Liu-Su-Liu System (21)

Figure 2. Anti-synchronization of Identical Liu-Chen-Liu Chaotic Systems The anti-synchronization error e is defined by

eyxi= i + i , ( i = 1,2,3) (23)

Dr. V. Sundarapandian et al , International Journal of Advanced Research in Computer Science, 1 (4), Nov. –Dec, 2010,408-413

The error dynamics is obtained as Numerical Results For the numerical simulations, the fourth-order Runge- e#1=α( ee 21 − ) + u 1 Kutta method with time-step 10 −6 is used to solve the systems e=β e + yy + xx + u (24) #2 1 13 13 2 using MATLAB. e#3=−γ e 3 − yy 12 − xx 12 + u 3 For the Liu-Su-Liu system (20), the parameter values are To find an anti- synchronizing controller, we first let taken as those which result in the chaotic behaviour of the system, viz. α=10, β = 35 and γ =1.4. u2= u 2a + u 2 b (25) The initial values of the master system (21) are taken as u3= u 3a + u 3 b x1(0)= 7, x 2 (0) = 9, x 3 (0) = 15 where while the initial values of the slave system (22) are taken as u2b = − yy 13 − xx 13 (26) y(0)= 14, y (0) = 10, y (0) = 4 u= yy + xx 1 2 3 3b 12 12 Figure 4 shows the anti-synchronization between the states Substituting (25) and (26) into (24), we obtain of the master system (21) and the slave system (22).

e#1=α( ee 21 − ) + u 1

e#2=β e 1 + u 2 a (27)

e#3= −γ e 3 + u 3 a Next, we consider the candidate Lyapunov function

1T 1 2 2 2 Ve( ) = ee =() eee1 ++ 2 3 (28) 2 2 A simple calculation gives V#( e ) = −−++αe2 γ e 2 ( αβ ) eeeu + 1 3 1211 (29) +eu22a + eu 33 a Therefore, we choose

u1= −(α + β ) e 2

u2a = − e 2 (30) Figure 4. Anti-Synchronization of Identical Liu-Su-Liu Chaotic Systems

u3a = 0 Substituting (30) into (27), the error dynamics simplifies to V. ANTI -SYNCHRONIZATION OF LIU -CHEN -LIU AND LIU -SU-LIU CHAOTIC SYSTEMS e#1= −α e 1 − β e 2 In this section, we apply the active nonlinear control e#2=β e 1 − e 2 (31) technique for the anti-synchronization of non-identical Liu- e#3= − γ e 3 Chen-Liu and Liu-Su-Liu chaotic systems. As the master Substituting (30) into (29), we also obtain system, we consider the Liu-Chen-Liu chaotic system ([22], 2 2 2 2007) described by V#( e ) = − α e 1−e 2 − γ e 3 (32) x#1= ax( 2 − x 1 ) which is a negative definite function on R3 since α and γ are x#2= bx 1 + kx 13 x (34) positive constants. Hence, by Lyapunov stability theory [24], the error x#3= − cx 3 − hx 12 x dynamics (31) is globally exponentially stable. As the slave system, we consider the Liu-Su-Liu chaotic Combining (25), (26) and (30), the anti-synchronizing system ([23], 2006) described by nonlinear controller u is obtained as y#1=α( yyu 21 − ) + 1 u1= −(α + β ) e 2 y#2=β y 1 + yy 13 + u 2 (35) u2=− e 2 − yy 13 − xx 13 (33) y#3=−γ y 3 − yyu 12 + 3 u3=y 12 y + xx 12 where all the parameters are positive real constants and Thus, we have proved the following result. T u= [ u1 u 2 u 3 ] is the nonlinear controller to be designed. Theorem 2. The identical Liu-Su-Liu systems (21) and (22) are exponentially and globally anti-synchronized for any initial The anti-synchronization error e is defined by conditions with the nonlinear controller u defined by (33). eyxi= i + i , ( i = 1,2,3) (36)

Dr. V. Sundarapandian et al , International Journal of Advanced Research in Computer Science, 1 (4), Nov. –Dec, 2010,408-413

The error dynamics is obtained as synchronized for any initial conditions with the nonlinear controller u defined by (46). e=α( ee −+− ) ( a α )( xxu −+ ) #1 21 211 Numerical Results e#2=+−β e 1( b β ) x 113 + yy + kxx 132 + u (37) For the numerical simulations, the fourth-order Runge- −6 e#3=−+−γ e 3( γ cx ) 312 − yy − hxxu 123 + Kutta method with time-step 10 is used to solve the systems To find an anti-synchronizing controller, we first let using MATLAB. For the Liu-Chen-Liu system (34), the parameter values are u1= u 1a + u 1 b taken as those which result in the chaotic behaviour of the

u2= u 2a + u 2 b (38) system, viz. a=10, b = 40, c = 2.5, h = 1 and k = 16. u= u + u For the Liu-Su-Liu system (35), the parameter values are 3 3a 3 b taken as those which result in the chaotic behaviour of the where system, viz. α=10, β = 35 and γ =1.4. u1b =−( a −α )( xx 2 − 1 ) The initial values of the Liu-Chen-Liu system (34) are taken as u2b =−−( bβ ) x 11313 − yy − kxx (39) x1(0)= 14, x 2 (0) = 5, x 3 (0) = 8 u3b =−−(γ cx ) 312 + yy + hxx 12 while the initial values of the Liu-Su-Liu system (35) are taken Substituting (38) and (39) into (37), we get as e#1=α( ee 21 − ) + u 1 a y1(0)= 5, y 2 (0) = 12, y 3 (0) = 4 e e u (40) #2=β 1 + 2 a Figure 5 shows the anti-synchronization between the states of the Liu-Chen-Liu system (34) and Liu-Su-Liu system (35). e#3= −γ e 3 + u 3 a Next, we consider the candidate Lyapunov function

1T 1 2 2 2 Ve( ) = ee =() eee1 ++ 2 3 (41) 2 2 A simple calculation gives V#( e ) = − αe2− γ e 2 ++( αβ ) eeeu + 1 3 1211 a (42) +eu22a + eu 33 a Therefore, we choose

u1a = −(α + β ) e 2

u2a = − e 2 (43)

u3a = 0 Substituting (43) into (40), the error dynamics simplifies to

e#1= −α e 1 − β e 2 Figure 5. Anti-synchronization of Liu-Chen-Liu and Liu-Su-Liu Chaotic Systems e#2=β e 1 − e 2 (44) e= − γ e #3 3 VI. CONCLUSIONS Substituting (43) into (42), we obtain 2 2 2 In this paper, we have used active control method based V#( e ) = − α e −e − γ e (45) 1 2 3 on Lyapunov stability theory to achieve global chaos anti- which is a negative definite function on R3 since α and γ synchronization for the following 3-D chaotic systems. are positive constants. (A) Identical Liu-Chen-Liu chaotic systems (2007) Hence, by Lyapunov stability theory [24], the error (B) Identical Liu-Su-Liu chaotic systems (2006) dynamics (44) is globally exponentially stable. (C) Non-Identical Liu-Chen-Liu and Liu-Su-Liu systems. Combining (38), (39) and (43), the anti-synchronizing Since the Lyapunov exponents are not required for these nonlinear controller u is obtained as calculations, the active control method is very effective and convenient to achieve global chaos anti-synchronization for u1=−+(α β ) eaxx 2 −− ( α )( 21 − ) identical and different Liu-Chen-Liu and Liu-Su-Liu chaotic u=−e−( b − β ) x −− y y kx x (46) systems. Numerical simulations are also given to illustrate and 2 2 113 13 validate the proposed active control method for the global u3=−(γ − c) x 312 + y y + hx 12 x chaos anti-synchronization of the chaotic systems addressed in Thus, we have proved the following result. this paper. Theorem 3. The non-identical Liu-Chen-Liu system (34) and Liu-Su-Liu system (35) are exponentially and globally anti- © 2010, IJARCS All Rights Reserved 412

Dr. V. Sundarapandian et al , International Journal of Advanced Research in Computer Science, 1 (4), Nov. –Dec, 2010,408-413

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[8] K. Murali and M. Lakshmanan, “Secure communication using a compound signal from generalized synchronizable chaotic systems”, Phys. Lett. A, vol. 241, pp. 303-310, 1998. Dr. V. Sundarapandian is a Research [9] E. Ott, C. Grebogi and J.A. Yorke, “Controlling chaos”, Phys. Rev. Lett., vol. 64, pp. 1196-1199, 1990. Professor (Systems and Control Engineering) in the Research and Development Centre, Vel Tech Dr. RR & Dr. SR [10] M.C. Ho and Y.C. Hung, “Synchronization of two Technical University, Avadi, Chennai since September 2009. different chaotic systems by using generalized active Previously, he was a Professor and Academic Convenor at the control,” Phys. Lett. A, vol. 301, pp. 424-428, 2002. Indian Institute of Information Technology and Management- [11] L. Huang, R. Feng and M. Wang, “Synchronization of Kerala, Trivandrum. He earned his Doctor of Science Degree chaotic systems via nonlinear control,” Phys. Lett. A, vol. from the Department of Electrical and Systems Engineering, 320, pp. 271-275, 2004. Washington University, St. Louis, Missouri, USA in May [12] H.K. Chen, “Global chaos synchronization of new 1996. He has published over 90 research papers in refereed chaotic systems via nonlinear control,” Chaos, Solitons International Journals and presented over 100 research papers and Fractals, vol. 23, pp. 1245-1251, 2005. in National and International Conferences in India and abroad. He is an Associate Editor of the International Journal on

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