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1 Jornal of Uncertain Systems Vol8, No, pp90-00, 0 Online at: wwwjsorgk Projective Synchronization of Different Hyper-chaotic Systems by Active Nonlinear Control Ayb Khan, Ram Pravesh Prasadd,* Department of Mathematics, Zakir Hsain Delhi College, University of Delhi, New Delhi -0, Indiaa Department of Mathematics, University of Delhi, New Delhi-07, Indiaa Received October 0; Revised 7 Jly 0 Abstract In this paper, we discss projective synchronization of two identical and different d hyper-chaotic projective synchronizationn of two systems sing active nonlinear control method The proposed method is applied to achieve identical hyper-chaotic Chen and L systems Nmerical simlations illstrate the effectiveness and feasibility of the proposed method 0 World Academic Press, UK All rights reserved Keywords: projective synchronization, hyper-chaotic Chen and L systems, activee control Introdction The idea of chaos synchronization method was first reported by Ott [], Pecora and Carroll [,, 5] Chaos synchronization being an interesting phenomenon has been observed in mtally copled, ni-directionally copled, ven noise indced chaotic oscillators with wide applications in several fields sch as physical systems, chemical systems, ecological systems, secre commnications, biological systems, electronic circits,, etc In most of the chaos synchronization approaches, the master-slave orr drive-response formalismm is sed A particlar chaotic system is called thee master or drive system and another chaotic systemm is known as the slave or response system The idea of chaos synchronizatios on is to se the otpt of the master system to control the slave system so that the otptt of the slave system trackss the otpt of master system asymptomatically Sincee the seminal research by Pecora and Carroll on thee synchronization of chaotic systems, a nmber of impressive approaches have been proposed for chaos synchronization sch as OGY method, sample-data feedback method, time-delay feedback method, active control method, adaptive control method, back stepping method, sliding mode control method, etc So far, many types of chaos synchronization have been presentedd in the literatre, sch as complete synchronization [5], phase synchronization [5], generalized synchronization [8], anti-synchronization [, 5], projective synchronization [8], generalized projective synchronization [, ], etc Complete synchronization characterized by the convergence of two chaotic trajectories has been observed in mtally copled, nidirectionally copled, c andd even noise indced chaotic oscillators [, 7, ] Phase synchronization is characterized by the fact that thee phase difference between two chaotic systems is locked within pie and the amplitdes remains chaotic and ncorrelated [6,,, ], generalized synchronization establishes the fnctional relation between master and slave systems [] In copled partiallyy linear systems, Mainieri and Rehacek investigated the synchronization of two identical systems (drive and response) which synchronize p to scaling factor This type of chaotic synchronizationn is referred to as projective synchronization [0] The phenomenon of antihave stdied synchronization is noticeable in periodic chaotic systems [, ] Recently, Kimm et al [9] andd H et al [7] the anti-synchronization phenomena in i mtally copled and nidirectionallyy copled identical chaoticc systems In fact the anti-synchronization of two different chaotic systems has greater potential applications than that of the two identical chaotic systems Projective synchronization in thee form of chaos synchronization has been observed in partially linear chaotic system of for dimensions [] In projective synchronization, the phases aree locked and the amplitdes of the two copled systems synchronize p to a scaling factor The scaling factor is a constant transformation between the * Corresponding athor rammbh@gmailcom (RP Prasad)

Jornal of Uncertain Systems, Vol8, No,, pp90-00, 0 9 synchronized variables of the masterr and slave systems Projective synchronization is different from generalized synchronization [0, 6], for the

the master system drives thee slave systemm throgh a copling c variable, in a manner that the response of the slave system can atomatically track the response of the master systemm and move in the

synchronization the time variation of the state vector ratio of the two systems becomes zero, z there is the t ratio tendss to a fixed scaling factor The scaling factor that characterizess the

revealed the maniplation of the scaling factor by b introdcingg control scheme, throgh which the performance of the synchronized systemss can be managed in a preferred way The manscript is organised

2 Jornal of Uncertain Systems, Vol8, No,, pp90-00, 0 9 synchronized variables of the masterr and slave systems Projective synchronization is different from generalized synchronization [0, 6], for the slave system, projective synchronization is not asymptotically stable Projective synchronization can be identified as copling of two identical copies of partially linear chaotic systems together Let the master system drives thee slave systemm throgh a copling c variable, in a manner that the response of the slave system can atomatically track the response of the master systemm and move in the same dynamical pattern in a proportional scale Mainieri and Rehacek [] explained the mechanism of projective synchronization sing cylindrical co-ordinates for three dimensional systems In projectivee synchronization the time variation of the state vector ratio of the two systems becomes zero, z there is the t ratio tendss to a fixed scaling factor The scaling factor that characterizess the synchronized dynamics is, however, sensitive to initial conditions and defined on the chaotic variables Ths, the otcome of the synchronized dynamics becomes npredictable The stdies of [0, 9, ] have revealed the maniplation of the scaling factor by b introdcingg control scheme, throgh which the performance of the synchronized systemss can be managed in a preferred way The manscript is organised in seven sections Section deals with the system s description Section covers the methodology while Section and Section 5 deal with the projective synchronization of two identical hyper-chaotic Chen and L systems by active nonlinear control In Section 6, the t projective synchronization of two different hyper chaotic Chen and L systems by activee nonlinear control has beenn explained while w Section 7 is marked ass conclsion System Description The Chen system is given by x ay ( x) w y dx xz cy () z xy bz yz rw where x, y, z and w are state vectors and a a>0, b>0, c>0, d>0 andd r>0 are parameters of the system When a=5, b=, c=, d=7 and r=05, the system is chaotic which iss exhibited in Figre The L system is given by ay ( x) w xz cy () xy bz xz rw where x, y, z and w are state vectors and a a, b, c andd r are positive parameters of the system The system displays the chaotic behavior at a=6, b=, c=0 and r= which is clear in Figre Figre : The chaoticc behavior off Chen system at a=5, b=, c=, d=7, and r=05

9 A Khan and RP Prasad: Projective Synchronization of Different Hyper-chaotic Systems Methodology In projective synchronization the master and slavee vectors synchronize p too a scaling factor, that

continos non linear vector fnction, is the vector controller If f g, then x and y are the state vectors of two identical chaotic systems If f g, then x and y are the state vectors two different

as n () for all initial conditions x(0),, y(0) R inn the form of projective x () The error dynamics is Figre : The chaotic behavior of L system at a=6, b= =, c=0, and r= n n n where R is the state

3 9 A Khan and RP Prasad: Projective Synchronization of Different Hyper-chaotic Systems Methodology In projective synchronization the master and slavee vectors synchronize p too a scaling factor, that is, the vectors become proportional We define projective synchronization as follows Consider the chaoticc systems: f( x) () n n n where y R is the state vector of thee system (), and g : R R is the continos non linear vector fnction, is the vector controller If f g, then x and y are the state vectors of two identical chaotic systems If f g, then x and y are the state vectors two different chaotic systems In the nonlinear feedback controll approach, we design a feedback controller, which synchronizes the states of system ( ) and the slave system synchronization Defining the state error as n () for all initial conditions x(0),, y(0) R inn the form of projective x () The error dynamics is Figre : The chaotic behavior of L system at a=6, b= =, c=0, and r= n n n where R is the state vector of thee system and f : R R is the continos nonlinear vector fnction System () representing the master system and the slave system is g( y) () Ths, the projective synchronization problem is essentially to find a feedbackk controller so as to stabilize the error n dynamics for all initial conditions e(0) ) R However, to achieve the projective synchronization of () and (), we w design a feedback controller sch that n lim et ( ) lim t y x 0 for all e (0) R, is a scaling factor (5) t g( y) f( x ) Then the projective synchronization off the systems () and () is achieved Projective Synchronization of Two Identical Hyper-chaoticc Chen Systems We consider the two identical hyper chaotic Chenn systems described as the drive systemm with sbscript and the response system with sbscript : ()

4 Jornal of Uncertain Systems, Vol8, No, pp90-00, 0 9 ay ( x) w dx x z cy x y bz y z rw, ay ( x) dx x z cy x y bz y z rw () () For control fnctions i, i,,, can synchronize two identical systems in the sense of projective synchronization Here, we define the state errors as e x x y y z z, e w w In order to observe the projective synchronization between systems () and () The errors dynamics is developed as a( e e) e de ce xz xz () xy be xy y z re y z Frther, we chose the active control fnctions as V xz xz V x y y x V y z y z V Sbstitting () and (), we get a( e e) e V de ce V be V re V The system (5) to be controlled mst be a linear system with the control inpt fnction V [ V, V, V, V] T as the fnctions of the error states e When the error system (5) is stabilized by the feedback V, the error will converge to zero as t which implies that the systems () and () are globally synchronized Choosing V as ( V, V, V, V ) A( e, e, e, e ) T T where A is a constant matrix For the error system (5) to be asymptotically stable, the elements of the matrix A are chosen so as the error system (5) will have all eigen vales with negative real parts Varios choices of A are possible For thogh there are varios choices in the selection of matrix, the best possible choice is 0 a 0 d c 0 0 A, r ae e be e () (5) (6)

becomes linear and has eigen vales a, -,, -b, - This ensres that the error

the projective synchronization between two identicall hyper chaotic Chen

effectiveness of the proposed p method The system becomes hyper chaotic when

conditions of the master and slave systems are x (0) 0, y (0) 0, z(0) 06, w

factor is taken as Figre displays the dynamics of the synchronization errors

hyper-chaotic Chen systems are synchronized in the sense of projective

5 9 A Khan and RP Prasad: Projective Synchronization of Different Hyper-chaotic Systems With the choice of matrix A, the error systemm (5) becomes linear and has eigen vales a, -,, -b, - This ensres that the error states e, e, e converge to zero as t tends to infinity i and this implies the projective synchronization between two identicall hyper chaotic Chen systems Nmerical Reslts Nmerical reslts are presented to t demonstrate the effectiveness of the proposed p method The system becomes hyper chaotic when the parameters vales are takenn as a=5, b= =, c=, d=7 and r=05 The initial conditions of the master and slave systems are x (0) 0, y (0) 0, z(0) 06, w (0) 0, x (0), y (0) 0, z (0) 0, w (0), respectively Moreover, the scaling factor is taken as Figre displays the dynamics of the synchronization errors for the master and a slave systems This indicates that two identical hyper-chaotic Chen systems are synchronized in the sense of projective synchronization Figre : Synchronization errors of (a) e t (b) e t (c) e t (d) ) e t Figre : Synchronization error of e

6 Jornal of Uncertain Systems, Vol8, No, pp90-00, Projective Synchronization of Two Identical Hyper-Chaotic L Systems In this section we stdy projective synchronization between two identical hyper- chaotic L system by active nonlinear control We denote the master system with sbscript and slave system with sbscript in their following representations: ay ( x) w xzcy (5) xy bz x z rw, ay ( x) x z cy x y bz x z rw (5) For control fnctions i, i,,, are introdced to synchronize the two identical systems in the form of projective synchronization We define the state errors as e x x, e y y, e z z, e w w In order to observe the projective synchronization between systems (5) and (5), we obtain the errors dynamics a( e e) e xz ce xz (5) xy be xy x z re x z Choosing the active control fnctions Sbstitting (5) and (5), we get V xz xz V x y x y V x z x z V a( e e) e V ce V be V re V Ths, the system (55) to be controlled is a linear system with the control inpt fnction V [ V, V, V, V] T being the fnctions of the error states e When the error system (55) is stabilized by the feedback V, the error will converge to zero as t which implies that the systems (5) and (5) are globally synchronized We choose V as follows: ( V, V, V, V ) A( e, e, e, e ) T T where A is a constant matrix For the error system (55) to be asymptotically stable, the elements of the matrix A are chosen so as the error system (55) will have all eigen vales with negative real parts From the varios possible choices of A, the most adeqate choice is (5) (55)

96 A Khan and RP Prasad: Projective Synchronization of Different

system (56) it is obvios that t the eigen vales are -a, -,-b, - This ensres

andd this indicates that the projective synchronization between two

reslts are presented to t demonstrate the effectiveness of the proposed p

takenn as a=6, b=, c=0 andd r= also the initial conditions of the master

w (0) 0, respectively Moreover, the scaling factor is taken as Figre

slave systems This showss that two identical hyper-chaotic L systems can be

7 96 A Khan and RP Prasad: Projective Synchronization of Different Hyper-chaotic Systems 0 0 A 0 0 a 0 c ae e be e 0 0 r (56) From system (56) it is obvios that t the eigen vales are -a, -,-b, - This ensres that the error states e, e, e converge to zero as t tends to t infinity andd this indicates that the projective synchronization between two identical hyper chaotic L systems is achieved a Nmerical Reslts Nmerical reslts are presented to t demonstrate the effectiveness of the proposed p method The system becomes hyper chaotic when the parameters vales are takenn as a=6, b=, c=0 andd r= also the initial conditions of the master and slave systems are x (0) 5, y (0) 8, z(0), w(0), x (0 ), y (0), z (0) 8, w (0) 0, respectively Moreover, the scaling factor is taken as Figre displays the dynamics of the synchronization errors for the master and slave systems This showss that two identical hyper-chaotic L systems can be synchronized s in the sensee of projectivee synchronization Figre 5: Synchronization errors of (a) e t (b) e t (c) e t (d) e t

Jornal of Uncertain Systems, Vol8, No,, pp90-00, 0 97 6 Projective Synchronization of Two Different Hyper-chaoticc Systemss In this section we

the Chen system (master system) and L system (slave system) as The control fnctions i ( i,,, ) can synchronize two different hyper-chaotic Chen

dynamics as Choosing the active control fnctions Sbstitting (6) in (6), we get Figre 6: Synchronization errorr of e ay ( x) w xz cy xy bz xz rw

8 Jornal of Uncertain Systems, Vol8, No,, pp90-00, Projective Synchronization of Two Different Hyper-chaoticc Systemss In this section we stdy projective synchronizationn between hyper-chaotic Chen C and hyper chaotic L systems by active nonlinear control method Here,, we consider the Chen system (master system) and L system (slave system) as The control fnctions i ( i,,, ) can synchronize two different hyper-chaotic Chen and L systems in the sense of projective synchronization We define the state errors e x x, e y y, e z z, e w w and then we obtain the errors dynamics as Choosing the active control fnctions Sbstitting (6) in (6), we get Figre 6: Synchronization errorr of e ay ( x) w xz cy xy bz xz rw a( e e ) e x z x y be x y x z re y z V x z x y x z ce be re ay ( x) w dx x z cy xy bz yz rw, ce dx dx xz x y V y z V V V V xz ae ( e) e V V (6) (6) (6) (6) (65)

98 A Khan and RP Prasad: Projective Synchronization of Different Hyper-chaotic Systems Ths, the system (65) to be controlled is a linear system with the control inpt fnction V [ V, V, V, V] T and the

synchronized We choose V sch that T ( V, V, V, V ) A( e, e, e, e ) T where A is a constant matrix For the error system (65) to be asymptotically stable, the elements off the matrix A are chosen so

e be e For this the error system (65) becomes linear and has eigen vales a, -, -b, - This ensres that t the error states e, e,e converge to zero as t tends to infinity and this implies that the

two systems becomes hyper chaotic when the parameters valess are taken as a = 5, b =, c =, d = 7, r=05, and a=6, b=, c=0, r=, respectively, and the initial conditions of the master and slave systems

9 98 A Khan and RP Prasad: Projective Synchronization of Different Hyper-chaotic Systems Ths, the system (65) to be controlled is a linear system with the control inpt fnction V [ V, V, V, V] T and the fnctions of the errorr states e When the error system (65) iss stabilized byy the feedback V, the error will converge to zero as t whichh implies that the systems (6) and (6)) are globally synchronized We choose V sch that T ( V, V, V, V ) A( e, e, e, e ) T where A is a constant matrix For the error system (65) to be asymptotically stable, the elements off the matrix A are chosen so that the error system (65) will have all eigen vales with negative real parts Ot of varios choices of A the best choice is 0 a 0 A 0 c r Hence the error system (65) becomes into ae e be e For this the error system (65) becomes linear and has eigen vales a, -, -b, - This ensres that t the error states e, e,e converge to zero as t tends to infinity and this implies that the projective synchronization between two different hyper chaotic Chen and L systems is achieved Nmerical Reslts Nmerical reslts are presented to demonstrate the effectiveness of the proposed method The two systems becomes hyper chaotic when the parameters valess are taken as a = 5, b =, c =, d = 7, r=05, and a=6, b=, c=0, r=, respectively, and the initial conditions of the master and slave systems are x (0),, y (0) 0, z (0) 0, w (0), x (0) 6, y (0) 05, z (0), w (0) 9, respectively Moreover, the scaling factor is taken as Figre 5 displays the dynamics of the synchronizationn errors for thee master and slave systems This shows that two different hyper-chaotic Chen and L systems are synchronized in the sense s of projective synchronization (66) Figre 7: Synchronization errors of (a) e t (b) e t (c) e t (d) e t

Jornal of Uncertain Systems, Vol8, No,, pp90-00, 0 99 Figre 8: Synchronization errorr of e, e, e 7 Conclsion The projective synchronization between two identical hyper-chaotic systems and a two

10 Jornal of Uncertain Systems, Vol8, No,, pp90-00, 0 99 Figre 8: Synchronization errorr of e, e, e 7 Conclsion The projective synchronization between two identical hyper-chaotic systems and a two different hyper-chaotic systems via active nonlinear control method has been achieved The designed controller is applied to synchronize two indentical hyper-chaotic systems Moreover, the active controller is applied to t obtain the synchronization between hyper-chaotic Chen and L systems in the sense of projectivee synchronization Nmerical simlations show the effectively and feasibility of the proposed method Acknowledgements The athors are thankfl to the referees for valablee sggestions, leading to an overall improvement of thee paper References [] Belykh, VN, and LO Cha, New type of strange attractor from a geometric model of Cha s circits, International Jornal of Bifrcation and Chaos, vol, no, pp697 70, 99 [] Carroll, TL, and LM Pecora, Synchronizating chaotic circits, IEEE Transactions on Circits Systems, vol8, no, pp5 56, 99 [] Chen, S, Wang, F, and C Wang, Synchronizing strict-feedback and general strict-feedback chaotic systems via a single controller, Chaos, Solitons & Fractals, vol0, pp5, 00 [] Chiang, T, Lin, J, Liao, T, and J Yan, Anti-synchronization of ncertain nified chaotic c systems s with dead-zone nonlinearity, Nonlinear Analysis, vol68, pp69 67, 008 [5] Ge, ZM, and CC Chen, Phase synchronization of copled chaotic mltiple time scales s systems, Chaos Solitonss and Fractals, vol0,, pp69 67, 00 [6] Ho, MC, Hng, YC, and CH Cho, Phase and anti-phase synchronization of two chaotic systems by sing active control, Physicss Letters A, vol96, pp 8, 00 [7] H, J, Chen, S, and L Chen, Adaptive control for anti-synchronization of Chaa ss chaotic systems, Physics Letters A, vol9, pp55 60, 005 [8] Jia, Q, Projective synchronization of a new hyper-chaotic Lorenz system, Physics Letters L A, vol70, pp0 5, 007 [9] Kim, CM, Rim, S, Kye, WH, Ry, JW, and YJ Park, Anti-synchronization predictability andd eqivalence of nidirectionally copled of chaotic oscillators, Physics Letters A, vol0, pp9 6, 00 [0] Kocarev, L, and U Parlitz, Generalized synchronization, dynamical systems, Physical Review Letters, vol76, pp86 89, 996 [] Li, R-H, X, W, and S Li, Adaptive generalized projective synchronization in different d chaoticc systems basedd on parameter identification, Physicss Letters A, vol 67, pp99 06, 007 [] Mainieri, R, and J Rehacek, Projective synchronization in three dimensional chaotic systems, Physical Review Letters, vol8, pp0 05,999

11 00 A Khan and RP Prasad: Projective Synchronization of Different Hyper-chaotic Systems [] Michael, GR, Arkady, SP, and K Jrgen, From phase to Lag synchronization in copled chaotic oscillators, Physical Review Letters, vol78, pp9 96, 997 [] Ott, E, Grebogi, C, and JA Yorke, Controlling chaos, Physical Review Letters, vol6, pp96 99, 990 [5] Pecora, LM, and TL Carrol, Synchronization in chaotic systems, Physical Review Letters, vol6, pp8 8, 990 [6] Rlkov, NF, Sshchik, MM, Tsimring, LS, and HDI Abarbanel, Generalized synchronization of chaos in directionally copled chaotic systems, Physical Review E, vol5, pp980 99, 995 [7] Shinbrot, T, Grebogi, C, Ott, E, and JA Yorke, Using small pertrbations to control chaos, Natre, vol6, pp 7, 99 [8] Wang, YW, and ZH Gan, Generalized synchronization of continos chaotic systems, Chaos Solitons and Fractals, vol7, pp97 0, 006 [9] X, D, Control of projective synchronization in chaotic systems, Physical Review E, vol6, pp70 70, 00 [0] X, D, and Z Li, Controlled projective synchronization in nonpartially linear chaotic systems, International Jornal of Bifrcation and Chaos, vol, no6, pp95 0, 00 [] X, D, Li, Z, and SR Bishop, Maniplating the scaling factor of projective synchronization in three dimensional chaotic systems, Chaos, vol, no, pp9, 00 [] Yan, J-P, and C-P Li, Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system, Jornal of Shanghai University, vol0, pp99 0, 006 [] Yang, XS, On the existence of generalized synchronization in nidirectionally copled systems, Applied Mathematics and Comptation, vol, pp7 79, 00 [] Y, X, and Y Song, Chaos synchronization via controlling partial state of chaotic systems, International Jornal of Bifrcation and Chaos, vol no6, pp77 7, 00 [5] Zhang, X, and H Zh, Anti-synchronization of two different hyperchaotic systems via active and adaptive control, International Jornal of Nonlinear Science, vol6, pp6, 008

HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL

HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical