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ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AND HYPERCHAOTIC LI SYSTEMS

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This paper derives new adaptive results for the hybrid synchronization of hyperchaotic Xi systems (2009)
and hyperchaotic Li systems (2005). In the hybrid synchronization design of master and slave systems, one
part of the systems, viz. their odd states, are completely synchronized (CS), while the other part, viz. their
even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process of
synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are unknown and we tackle this problem using adaptive control. The main results of
this research work are proved using adaptive control theory and Lyapunov stability theory. MATLAB
simulations using classical fourth-order Runge-Kutta method are shown for the new adaptive hybrid
synchronization results for the hyperchaotic Xu and hyperchaotic Li systems.

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Transcript of "ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AND HYPERCHAOTIC LI SYSTEMS"

2. 2. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201318main results derived in this paper have been proved using adaptive control theory [24] andLyapunov stability theory [25].2. ADAPTIVE CONTROL METHODOLOGY FOR HYBRID SYNCHRONIZATIONThe master system is described by the chaotic dynamics( )x Ax f x= + (1)where A is the n n× matrix of the system parameters and : n nf →R R is the nonlinear part.The slave system is described by the chaotic dynamics( )y By g y u= + + (2)where B is the n n× matrix of the system parameters and : n ng →R R is the nonlinear partFor the pair of chaotic systems (1) and (2), the hybrid synchronization error is defined as, if is odd, if is eveni iii iy x iey x i−= +(3)The error dynamics is obtained as11( ) ( ) ( ) if is odd( ) ( ) ( ) if is evennij j ij j i i iji nij j ij j i i ijb y a x g y f x u ieb y a x g y f x u i==− + − +=  + + + +∑∑ (4)The design goal is to find a feedback controller uso thatlim ( ) 0te t→∞= for all (0)e ∈Rn(5)Using the matrix method, we consider a candidate Lyapunov function( ) ,TV e e Pe= (6)where P is a positive definite matrix. It is noted that : nV →R R is a positive definite function.If we find a feedback controller uso that( ) ,TV e e Qe= − (7)where Q is a positive definite matrix, then : nV → R R is a negative definite function.Thus, by Lyapunov stability theory [25], the error dynamics (4) is globally exponentially stable.Hence, the states of the chaotic systems (1) and (2) will be globally and exponentiallyhybrid synchronized for all initial conditions (0), (0) .nx y ∈R When the system parameters areunknown, we use estimates for them and find a parameter update law using Lyapunov approach.
3. 3. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013193. HYPERCHAOTIC SYSTEMSThe hyperchaotic Xu system ([22], 2009) has the 4-D dynamics1 2 1 42 1 1 33 3 1 24 1 3 2( )x a x x xx bx rx xx cx x xx x x dx= − += += − −= −(8)where , , , ,a b c r d are constant, positive parameters of the system.The Xu system (8) exhibits a hyperchaotic attractor for the parametric values10, 40, 2.5, 16, 2a b c r d= = = = = (9)The Lyapunov exponents of the system (8) for the parametric values in (9) are1 2 3 41.0088, 0.1063, 0, 13.6191   = = = = − (10)Since there are two positive Lyapunov exponents in (10), the Xu system (8) is hyperchaotic forthe parametric values (9).The strange attractor of the hyperchaotic Xu system is displayed in Figure 1.The hyperchaotic Li system ([23], 2005) has the 4-D dynamics1 2 1 42 1 1 3 23 3 1 24 2 3 4( )x x x xx x x x xx x x xx x x x = − += − += − += +(11)where , , , ,     are constant, positive parameters of the system.The Li system (11) exhibits a hyperchaotic attractor for the parametric values35, 3, 12, 7, 0.58    = = = = = (12)The Lyapunov exponents of the system (11) for the parametric values in (12) are1 2 3 40.5011, 0.1858, 0, 26.1010   = = = = − (13)Since there are two positive Lyapunov exponents in (13), the Li system (11) is hyperchaotic forthe parametric values (12).The strange attractor of the hyperchaotic Li system is displayed in Figure 2.
4. 4. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201320Figure 1. The State Portrait of the Hyperchaotic Xu SystemFigure 2. The State Portrait of the Hyperchaotic Li System4. ADAPTIVE CONTROL DESIGN FOR THE HYBRID SYNCHRONIZATION OFHYPERCHAOTIC XU SYSTEMSIn this section, we design an adaptive controller for the hybrid synchronization of two identicalhyperchaotic Xu systems (2009) with unknown parameters.The hyperchaotic Xu system is taken as the master system, whose dynamics is given by1 2 1 42 1 1 33 3 1 24 1 3 2( )x a x x xx bx rx xx cx x xx x x dx= − += += − −= −(14)where , , , ,a b c d r are unknown parameters of the system and 4x∈ R is the state of the system.
5. 5. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201321The hyperchaotic Xu system is also taken as the slave system, whose dynamics is given by1 2 1 4 12 1 1 3 23 3 1 2 34 1 3 2 4( )y a y y y uy by ry y uy cy y y uy y y dy u= − + += + += − − += − +(15)where 4y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed usingestimates ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t r t of the unknown parameters , , , ,a b c d r , respectively.For the hybrid synchronization, the error e is defined as1 1 1 2 2 2 3 3 3 4 4 4, , ,e y x e y x e y x e y x= − = + = − = + (16)A simple calculation gives the error dynamics1 2 2 1 4 4 12 1 1 1 3 1 3 23 3 1 2 1 2 34 2 1 3 1 3 4( ) 2( ) ( )e a y x e e x ue b y x r y y x x ue ce y y x x ue de y y x x u= − − + − += + + + += − − + += − + + +(17)Next, we choose a nonlinear controller for achieving hybrid synchronization as1 2 2 1 4 4 1 12 1 1 1 3 1 3 2 23 3 1 2 1 2 3 34 2 1 3 1 3 4 4ˆ( )( ) 2ˆ ˆ( )( ) ( )( )ˆ( )ˆ( )u a t y x e e x k eu b t y x r t y y x x k eu c t e y y x x k eu d t e y y x x k e= − − − − + −= − + − + −= + − −= − − −(18)In Eq. (18), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t r t are estimates of theunknown parameters , , , ,a b c d r , respectively.By the substitution of (18) into (17), the error dynamics is simplified as1 2 2 1 1 12 1 1 1 3 1 3 2 23 3 3 34 2 4 4ˆ( ( ))( )ˆ ˆ( ( ))( ) ( ( ))( )ˆ( ( ))ˆ( ( ))e a a t y x e k ee b b t y x r r t y y x x k ee c c t e k ee d d t e k e= − − − −= − + + − + −= − − −= − − −(19)Next, we define the parameter estimation errors asˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c d re t a a t e t b b t e t c c t e t d d t e t r r t= − = − = − = − = − (20)
6. 6. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201322Differentiating (20) with respect to ,t we getˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c d re t a t e t b t e t c t e t d t e t r t= − = − = − = − = −        (21)In view of (20), we can simplify the error dynamics (19) as1 2 2 1 1 12 1 1 1 3 1 3 2 23 3 3 34 2 4 4( )( ) ( )ab rcde e y x e k ee e y x e y y x x k ee e e k ee e e k e= − − −= + + + −= − −= − −(22)We take the quadratic Lyapunov function( )2 2 2 2 2 2 2 2 21 2 3 41,2a b c d rV e e e e e e e e e= + + + + + + + + (23)which is a positive definite function on9.RWhen we differentiate (22) along the trajectories of (19) and (21), we get2 2 2 21 1 2 2 3 3 4 4 1 2 2 1 2 1 123 2 4 2 1 3 1 3ˆˆ( ) ( )ˆˆ ˆ( )a bc d rV k e k e k e k e e e y x e a e e y x be e c e e e d e e y y x x r  = − − − − + − − − + + −        + − − + − − + + −      (24)In view of Eq. (24), we take the parameter update law as21 2 2 1 5 2 1 1 6 3 72 4 8 2 1 3 1 3 9ˆˆ ˆ( ) , ( ) ,ˆ ˆ, ( )a b cd ra e y x e k e b e y x k e c e k ed e e k e r e y y x x k e= − − + = + + = − += − + = + +  (25)Theorem 4.1 The adaptive control law (18) along with the parameter update law (25), where,( 1,2, ,9)ik i =  are positive gains, achieves global and exponential hybrid synchronization ofthe identical hyperchaotic Xu systems (14) and (15), where ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t r t are estimatesof the unknown parameters , , , , ,a b c d r respectively. Moreover, all the parameter estimationerrors converge to zero exponentially for all initial conditions.Proof. We prove the above result using Lyapunov stability theory [25].Substituting the parameter update law (25) into (24), we get2 2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 8 9a b c d rV k e k e k e k e k e k e k e k e k e= − − − − − − − − − (26)which is a negative definite function on9.RThis shows that the hybrid synchronization errors 1 2 3 4( ), ( ), ( ), ( )e t e t e t e t and the parameterestimation errors ( ), ( ), ( ), ( ), ( )a b c d re t e t e t e t e t are globally exponentially stable for all initialconditions.
7. 7. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201323This completes the proof.Next, we demonstrate our hybrid synchronization results with MATLAB simulations.The classical fourth order Runge-Kutta method with time-step 810h −= has been applied to solvethe hyperchaotic Xu systems (14) and (15) with the adaptive nonlinear controller (18) and theparameter update law (25). The feedback gains are taken as 5, ( 1,2, ,9).ik i= = The parameters of the hyperchaotic Xu systems are taken as in the hyperchaotic case, i.e.10, 40, 2.5, 16, 2a b c r d= = = = =For simulations, the initial conditions of the hyperchaotic Xu system (14) are chosen as1 2 3 4(0) 11, (0) 7, (0) 9, (0) 5x x x x= = = = −Also, the initial conditions of the hyperchaotic Xu system (15) are chosen as1 2 3 4(0) 3, (0) 4, (0) 6, (0) 12y y y y= = = − =Also, the initial conditions of the parameter estimates are chosen asˆ ˆˆ ˆ ˆ(0) 4, (0) 11, (0) 2, (0) 5, (0) 16a b c d r= = = − = − =Figure 3 depicts the hybrid synchronization of the identical hyperchaotic Xu systems.Figure 4 depicts the time-history of the hybrid synchronization errors 1 2 3 4, , , .e e e eFigure 5 depicts the time-history of the parameter estimation errors , , , , .a b c d re e e e eFigure 3. Hybrid Synchronization of Identical Hyperchaotic Xu Systems
8. 8. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201324Figure 4. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e eFigure 5. Time-History of the Parameter Estimation Errors , , , ,a b c d re e e e e5. ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATIONDESIGN OF HYPERCHAOTIC LI SYSTEMSIn this section, we design an adaptive controller for the hybrid synchronization of two identicalhyperchaotic Li systems (2005) with unknown parameters.The hyperchaotic Li system is taken as the master system, whose dynamics is given by1 2 1 42 1 1 3 23 3 1 24 2 3 4( )x x x xx x x x xx x x xx x x x = − += − += − += +(27)where , , , ,     are unknown parameters of the system and 4x∈ R is the state of the system.The hyperchaotic Li system is also taken as the slave system, whose dynamics is given by
9. 9. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013251 2 1 4 12 1 1 3 2 23 3 1 2 34 2 3 4 4( )y y y y uy y y y y uy y y y uy y y y u = − + += − + += − + += + +(28)where 4y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed usingestimates ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )t t t t t     of the unknown parameters , , , , ,     respectively.For the hybrid synchronization, the error e is defined as1 1 1 2 2 2 3 3 3 4 4 4, , ,e y x e y x e y x e y x= − = + = − = + (29)A simple calculation gives the error dynamics1 2 2 1 4 4 12 1 1 2 1 3 1 3 23 3 1 2 1 2 34 4 2 3 2 3 4( ) 2( )e y x e e x ue y x e y y x x ue e y y x x ue e y y x x u = − − + − += + + − − += − + − += + + +(30)Next, we choose a nonlinear controller for achieving hybrid synchronization as1 2 2 1 4 4 1 12 1 1 2 1 3 1 3 2 23 3 1 2 1 2 3 34 4 2 3 2 3 4 4ˆ( )( ) 2ˆ ˆ( )( ) ( )ˆ( )ˆ( )u t y x e e x k eu t y x t e y y x x k eu t e y y x x k eu t e y y x x k e = − − − − + −= − + − + + −= − + −= − − − −(31)In Eq. (31), , ( 1,2,3,4)ik i = are positive gains.By the substitution of (31) into (30), the error dynamics is simplified as1 2 2 1 1 12 1 1 2 2 23 3 3 34 4 4 4ˆ( ( ))( )ˆ ˆ( ( ))( ) ( ( ))ˆ( ( ))ˆ( ( ))e t y x e k ee t y x t e k ee t e k ee t e k e      = − − − −= − + + − −= − − −= − −(32)Next, we define the parameter estimation errors asˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( )ˆ ˆ( ) ( ), ( ) ( )e t t e t t e t te t t e t t           = − = − = −= − = −(33)Differentiating (33) with respect to ,t we get
10. 10. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201326ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t e t t        = − = − = − = − = −        (34)In view of (33), we can simplify the error dynamics (32) as1 2 2 1 1 12 1 1 2 2 23 3 3 34 4 4 4( )( )e e y x e k ee e y x e e k ee e e k ee e e k e = − − −= + + −= − −= −(35)We take the quadratic Lyapunov function( )2 2 2 2 2 2 2 2 21 2 3 41,2V e e e e e e e e e    = + + + + + + + + (36)which is a positive definite function on9.RWhen we differentiate (35) along the trajectories of (32) and (33), we get2 2 2 2 21 1 2 2 3 3 4 4 1 2 2 1 32 22 2 1 1 4ˆˆ( )ˆˆ ˆ( )V k e k e k e k e e e y x e e ee e e e y x e e        = − − − − + − − − + − −        + − + + − + −      (37)In view of Eq. (37), we take the parameter update law as2 21 2 2 1 5 3 6 2 722 1 1 8 4 9ˆˆ ˆ( ) , ,ˆ ˆ( ) ,e y x e k e e k e e k ee y x k e e k e      = − − + = − + = += + + = +  (38)Theorem 5.1 The adaptive control law (31) along with the parameter update law (38), where,( 1,2, ,9)ik i =  are positive gains, achieves global and exponential hybrid synchronization ofthe identical hyperchaotic Li systems (27) and (28), where ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )t t t t t     areestimates of the unknown parameters , , , , ,     respectively. Moreover, all the parameterestimation errors converge to zero exponentially for all initial conditions.Proof. We prove the above result using Lyapunov stability theory [25].Substituting the parameter update law (38) into (37), we get2 2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 8 9V k e k e k e k e k e k e k e k e k e    = − − − − − − − − − (39)which is a negative definite function on9.RThis shows that the hybrid synchronization errors 1 2 3 4( ), ( ), ( ), ( )e t e t e t e t and the parameterestimation errors ( ), ( ), ( ), ( ), ( )e t e t e t e t e t     are globally exponentially stable for all initialconditions. This completes the proof. 
11. 11. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201327Next, we demonstrate our hybrid synchronization results with MATLAB simulations.The classical fourth order Runge-Kutta method with time-step 810h −= has been applied to solvethe hyperchaotic Li systems (27) and (28) with the adaptive nonlinear controller (31) and theparameter update law (38). The feedback gains are taken as 5, ( 1,2, ,9).ik i= = The parameters of the hyperchaotic Li systems are taken as in the hyperchaotic case, i.e.35, 3, 12, 7, 0.58    = = = = =For simulations, the initial conditions of the hyperchaotic Li system (27) are chosen as1 2 3 4(0) 6, (0) 7, (0) 15, (0) 22x x x x= = − = = −Also, the initial conditions of the hyperchaotic Li system (28) are chosen as1 2 3 4(0) 12, (0) 4, (0) 9, (0) 6y y y y= = = = −Also, the initial conditions of the parameter estimates are chosen asˆ ˆˆ ˆ ˆ(0) 7, (0) 8, (0) 2, (0) 9, (0) 12    = = = − = − =Figure 6 depicts the hybrid synchronization of the identical hyperchaotic Li systems.Figure 7 depicts the time-history of the hybrid synchronization errors 1 2 3 4, , , .e e e eFigure 8 depicts the time-history of the parameter estimation errors , , , , .e e e e e    Figure 6. Hybrid Synchronization of Identical Hyperchaotic Li Systems
12. 12. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201328Figure 7. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e eFigure 8. Time-History of the Parameter Estimation Errors , , , ,e e e e e    6. ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATIONDESIGN OF HYPERCHAOTIC XU AND HYPERCHAOTIC LI SYSTEMSIn this section, we design an adaptive controller for the hybrid synchronization of hyperchaoticXu system (2009) and hyperchaotic Li system (2005) with unknown parameters.The hyperchaotic Xu system is taken as the master system, whose dynamics is given by1 2 1 42 1 1 33 3 1 24 1 3 2( )x a x x xx bx rx xx cx x xx x x dx= − += += − −= −(40)where , , , ,a b c d r are unknown parameters of the system.The hyperchaotic Li system is also taken as the slave system, whose dynamics is given by
13. 13. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013291 2 1 4 12 1 1 3 2 23 3 1 2 34 2 3 4 4( )y y y y uy y y y y uy y y y uy y y y u = − + += − + += − + += + +(41)where , , , ,     are unknown parameters and 1 2 3 4, , ,u u u u are the adaptive controllers.For the hybrid synchronization, the error e is defined as1 1 1 2 2 2 3 3 3 4 4 4, , ,e y x e y x e y x e y x= − = + = − = + (42)A simple calculation gives the error dynamics1 2 1 2 1 4 4 12 1 2 1 1 3 1 3 23 3 3 1 2 1 2 34 4 2 2 3 2 3 4( ) ( )e y y a x x y x ue y y bx rx x y y ue y cx y y x x ue y dx y y x x u = − − − + − += + + + − += − + + + += − + + +(43)Next, we choose a nonlinear controller for achieving hybrid synchronization as1 2 1 2 1 4 4 1 12 1 2 1 1 3 1 3 2 23 3 3 1 2 1 2 3 34 4 2 2 3 2 3 4 4ˆ ˆ( )( ) ( )( )ˆˆ ˆ ˆ( ) ( ) ( ) ( )ˆ ˆ( ) ( )ˆˆ( ) ( )u t y y a t x x y x k eu t y t y b t x r t x x y y k eu t y c t x y y x x k eu t y d t x y y x x k e = − − + − − + −= − − − − + −= − − − −= − + − − −(44)where , ( 1,2,3,4)ik i = are positive gains.By the substitution of (44) into (43), the error dynamics is simplified as1 2 1 2 1 1 12 1 2 1 1 3 2 23 3 3 3 34 4 2 4 4ˆ ˆ( ( ))( ) ( ( ))( )ˆˆ ˆ ˆ( ( )) ( ( )) ( ( )) ( ( ))ˆ ˆ( ( )) ( ( ))ˆˆ( ( )) ( ( ))e t y y a a t x x k ee t y t y b b t x r r t x x k ee t y c c t x k ee t y d d t x k e      = − − − − − −= − + − + − + − −= − − + − −= − − − −(45)Next, we define the parameter estimation errors asˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )ˆˆ ˆˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )ˆ ˆ( ) ( ), ( ) ( )a b c dre t a a t e t b b t e t c c t e t d d te t r r t e t t e t t e t te t t e t t           = − = − = − = −= − = − = − = −= − = −(46)Differentiating (46) with respect to ,t we get
14. 14. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201330ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c d re t a t e t b t e t c t e t d t e t r te t t e t t e t t e t t e t t        = − = − = − = − = −= − = − = − = − = −              (47)In view of (46), we can simplify the error dynamics (45) as1 2 1 2 1 1 12 1 2 1 1 3 2 23 3 3 3 34 4 2 4 4( ) ( )ab rcde e y y e x x k ee e y e y e x e x x k ee e y e x k ee e y e x k e = − − − −= + + + −= − + −= − −(48)We take the quadratic Lyapunov function( )2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 3 41,2a b c d rV e e e e e e e e e e e e e e    = + + + + + + + + + + + + + (49)which is a positive definite function on 14.RWhen we differentiate (48) along the trajectories of (45) and (46), we get2 2 2 21 1 2 2 3 3 4 4 1 2 1 2 1 3 34 2 2 1 3 1 2 1 3 32 2 2 1ˆˆ ˆ( )ˆ ˆˆˆ ( )ˆˆa b cd rV k e k e k e k e e e x x a e e x b e e x ce e x d e e x x r e e y y e e ye e y e e y e        = − − − − + − − − + − + −          + − − + − + − − + − −           + − + − +      4 4ˆe y  − (50)In view of Eq. (50), we take the parameter update law as1 2 1 5 2 1 6 3 3 74 2 8 2 1 3 9 1 2 1 103 3 11 2 2 12ˆˆ ˆ( ) , , ,ˆ ˆˆ, , ( ) ,ˆ ˆ, ,a b cd ra e x x k e b e x k e c e x k ed e x k e r e x x k e e y y k ee y k e e y k e  = − − + = + = += − + = + = − += − + = +   2 1 134 4 14ˆ ,ˆe y k ee y k e= += +(51)Theorem 6.1 The adaptive control law (44) along with the parameter update law (51), where,( 1,2, ,14)ik i =  are positive gains, achieves global and exponential hybrid synchronization ofthe hyperchaotic Xu system (40) hyperchaotic Li system (41), where ˆ( ),a t ˆ( ),b t ˆ( ),c t ˆ( ),d t ˆ( ),r tˆ( ),t ˆ( ),t ˆ( ),t ˆ( ),t ˆ( )t are estimates of the unknown parameters , , , , ,a b c d r , , , , ,    respectively. Moreover, all the parameter estimation errors converge to zero exponentially for allinitial conditions.Proof. We prove the above result using Lyapunov stability theory [25].Substituting the parameter update law (51) into (50), we get
15. 15. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013312 2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 8 92 2 2 2 210 11 12 13 14a b c d rV k e k e k e k e k e k e k e k e k ek e k e k e k e k e    = − − − − − − − − −− − − − −(52)which is a negative definite function on 14.RThis shows that the hybrid synchronization errors 1 2 3 4( ), ( ), ( ), ( )e t e t e t e t and the parameterestimation errors ( ), ( ), ( ), ( ), ( ),a b c d re t e t e t e t e t ( ), ( ), ( ), ( ), ( )e t e t e t e t e t     are globallyexponentially stable for all initial conditions. This completes the proof. Next, we demonstrate our hybrid synchronization results with MATLAB simulations.The classical fourth order Runge-Kutta method with time-step 810h −= has been applied to solvethe hyperchaotic Li systems (27) and (28) with the adaptive nonlinear controller (31) and theparameter update law (38). The feedback gains are taken as 5, ( 1,2, ,9).ik i= = The parameters of the hyperchaotic Xu and Li systems are taken as in the hyperchaotic case, i.e.10, 40, 2.5, 16, 2, 35, 3, 12, 7, 0.58a b c r d     = = = = = = = = = =For simulations, the initial conditions of the hyperchaotic Xu system (27) are chosen as1 2 3 4(0) 12, (0) 4, (0) 8, (0) 20x x x x= = − = = −Also, the initial conditions of the hyperchaotic Li system (28) are chosen as1 2 3 4(0) 21, (0) 7, (0) 19, (0) 12y y y y= = = = −Also, the initial conditions of the parameter estimates are chosen asˆ ˆˆ ˆ ˆ(0) 7, (0) 14, (0) 12, (0) 8, (0) 15ˆ ˆˆ ˆ ˆ(0) 12, (0) 5, (0) 4, (0) 11, (0) 6a b c d r    = − = = = − == = = = = −Figure 9 depicts the hybrid synchronization of hyperchaotic Xu and hyperchaotic Li systems.Figure 10 depicts the time-history of the hybrid synchronization errors 1 2 3 4, , , .e e e eFigure 11 depicts the time-history of the parameter estimation errors , , , .a b c de e e eFigure 12 depicts the time-history of the parameter estimation errors , , , , .e e e e e    
16. 16. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201332Figure 9. Hybrid Synchronization of Hyperchaotic Xu and Lu SystemsFigure 10. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e eFigure 11. Time-History of the Parameter Estimation Errors , , ,a b c de e e e
17. 17. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April201333Figure 12. Time-History of the Parameter Estimation Errors , , , ,e e e e e    7. CONCLUSIONSIn this paper, using adaptive control method, we derived new results for the adaptive controllerdesign for the hybrid synchronization of hyperchaotic Xu systems (2009) and hyperchaotic Lisystems (2005) with unknown parameters. Using Lyapunov control theory, adaptive control lawswere derived for globally hybrid synchronizing the states of identical hyperchaotic Xu systems,identical hyperchaotic Li systems and non-identical hyperchaotic Xu and Li systems. MATLABsimulations were displayed in detail to demonstrate the adaptive hybrid synchronization resultsderived in this paper for hyperchaotic Xu and Li systems.REFERENCES[1] Rössler, O.E. (1979) “An equation for hyperchaos,” Physics Letters A, Vol. 71, pp 155-157.[2] Huang, Y. & Yang, X.S. (2006) “Hyperchaos and bifurcation in a new class of four-dimensionalHopfield neural networks,” Neurocomputing, Vol. 69, pp 13-15.[3] Machado, L.G., Savi, M.A. & Pacheco, P.M.C.L. (2003) “Nonlinear dynamics and chaos in coupledshape memory oscillators,” International Journal of Solids and Structures, Vol. 40, No. 19, pp. 5139-5156.[4] Tao, Y. (1999) “Chaotic secure communication systems – history and new results”, Telecommun.Review, Vol. 9, pp 597-634.[5] Li, C., Liao, X. & Wong, K.W. (2005) “Lag synchronization of hyperchaos with applications tosecure communications,” Chaos, Solitons & Fractals, Vol. 23, No. 1, pp 183-193.[6] Prokhorov, M.D. & Ponomarenko, V.I. (2008) “Encryption and decryption of information in chaoticcommunication systems governed by delay-differential equations,” Chaos, Solitons & Fractals, Vol.35, No. 5, pp 871-877.[7] Yassen, M.T. (2008) “Synchronization hyperchaos of hyperchaotic systems”, Chaos, Solitons andFractals, Vol. 37, pp 465-475.[8] Ott, E., Grebogi, C. & Yorke, J.A. (1990) “Controlling chaos”, Phys. Rev. Lett., Vol. 64, pp 1196-1199.[9] Pecora, L.M. & Carroll, T.L. (1990) “Synchronization in chaotic systems”, Phys. Rev. Lett., Vol. 64,pp 821-824.[10] Bowong, S. & Kakmeni, F.M.M. (2004) “Synchronization of uncertain chaotic systems viabackstepping approach,” Chaos, Solitons & Fractals, Vol. 21, No. 4, pp 999-1011.[11] Suresh, R, & Sundarapandian, V. (2012) “Global chaos synchronization of WINDMI and Coulletchaotic systems by backstepping control”, Far East J. Math. Sciences, Vol. 67, No. 2, pp 265-287.
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