ADAPTIVESYNCHRONIZER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZHENGAND HYPERCHAOTIC YU SYSTEMS

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This paper derives new adaptive synchronizers for the hybrid synchronization of hyperchaotic Zheng …

This paper derives new adaptive synchronizers for the hybrid synchronization of hyperchaotic Zheng
systems (2010) and hyperchaotic Yu systems (2012). 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 not known and we handle this complicate problem using adaptive control. The
main results of this research work are established via adaptive control theory andLyapunov stability
theory. MATLAB plotsusing classical fourth-order Runge-Kutta method have been depictedfor the new
adaptive hybrid synchronization results for the hyperchaotic Zheng and hyperchaotic Yu systems.

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  • 1. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 2013DOI:10.5121/ijitcs.2013.3202 11ADAPTIVESYNCHRONIZER DESIGN FOR THEHYBRID SYNCHRONIZATION OF HYPERCHAOTICZHENGAND HYPERCHAOTIC YU SYSTEMSSundarapandian VaidyanathanResearch and Development Centre, Vel Tech Dr. RR & Dr. SR Technical UniversityAvadi, Chennai-600 062, Tamil Nadu, INDIAsundarvtu@gmail.comABSTRACTThis paper derives new adaptive synchronizers for the hybrid synchronization of hyperchaotic Zhengsystems (2010) and hyperchaotic Yu systems (2012). In the hybrid synchronization design of master andslave systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while theother part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in theprocess of synchronization. The research problem gets even more complicated, when the parameters of thehyperchaotic systems are not known and we handle this complicate problem using adaptive control. Themain results of this research work are established via adaptive control theory andLyapunov stabilitytheory. MATLAB plotsusing classical fourth-order Runge-Kutta method have been depictedfor the newadaptive hybrid synchronization results for the hyperchaotic Zheng and hyperchaotic Yu systems.KEYWORDSHybrid Synchronization, Adaptive Control, Chaos, Hyperchaos, Hyperchaotic Systems.1. INTRODUCTIONSince thediscovery by the German scientist,O.E.Rössler ([1], 1979), hyperchaotic systems havefound many applicationsin areas like neural networks [2],oscillators [3], communication [4-5],encryption [6], synchronization [7], etc. In chaos theory, hyperchaotic system is usually definedas a chaotic system having two or more positive Lyapunov exponents. Hyperchaotic systems havemany attractive features like high efficiency, high capacity, high security, etc.For the synchronization of chaotic systems, there are many methods available in the chaosliterature like OGY method [8], PC method [9],backstepping method [10-12], sliding controlmethod [13-15], active control method [16-17], adaptive control method [18-19], sampled-datafeedback control [20], time-delay feedback method [21], etc.In the hybrid synchronization of a pair of chaotic systems called the master and slave systems,one part of the systems, viz. the odd states, are completely synchronized (CS), while the otherpart of the systems, viz. the even states, are anti-synchronized so that CS and AS co-exist in theprocess of synchronization of the two systems.This paper focuses upon adaptive controller design for the hybrid synchronization of hyperchaoticZheng systems ([22], 2010) and hyperchaotic Yusystems ([23], 2012) with unknown parameters.
  • 2. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201312The main results derived in this paper have been proved using adaptive control theory [24]andLyapunov stability theory [25]..2. ADAPTIVE CONTROL METHODOLOGYFOR 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 erroris 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)whereQ 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. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 2013133. 4-D HYPERCHAOTIC SYSTEMSThe 4-D hyperchaotic Zhengsystem ([22], 2010) has the dynamics1 2 1 42 1 2 1 3 423 1 34 2( )x a x x xx bx cx x x xx x rxx dx= − += + + += − −= −(8)where , , , ,a b c r d are constant, positive parameters of the system.The 4-D Zheng system (8) exhibits a hyperchaotic attractor for the parametric values20, 14, 10.6, 4, 2.8a b c d r= = = = = (9)The Lyapunov exponents of the system (8) for the parametric values in (9) are1 2 3 41.8892, 0.2268, 0, 14.4130L L L L= = = = − (10)Since there are two positive Lyapunov exponents in (10), the Zheng system (8) is hyperchaoticfor the parametric values (9).The strange attractor of the hyperchaotic Zheng system is displayed in Figure 1.The 4-D hyperchaotic Yu system ([23], 2012) has the dynamics1 21 2 12 1 1 3 2 43 34 1( )x xx x xx x x x x xx e xx x = −= − + += −= −(11)where , , , ,     are constant, positive parameters of the system.The 4-D Yu system (11) exhibits a hyperchaotic attractor for the parametric values10, 40, 1, 3, 8    = = = = = (12)The Lyapunov exponents of the system (11) for the parametric values in (12) are1 2 3 41.6877, 0.1214, 0, 13.7271L L L L= = = = − (13)Since there are two positive Lyapunov exponents in (13), theYusystem (11) is hyperchaotic forthe parametric values (12).The strange attractor of the hyperchaotic Yu system is displayed in Figure 2.
  • 4. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201314Figure 1. The State Portrait of the HyperchaoticZhengSystemFigure 2. The State Portrait of the Hyperchaotic Yu System
  • 5. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 2013154. ADAPTIVECONTROL DESIGN FOR THE HYBRIDSYNCHRONIZATION OFHYPERCHAOTIC ZHENG SYSTEMSIn this section, we design an adaptivesynchronizer for the hybrid synchronization of two identicalhyperchaotic Zheng systems (2010) with unknown parameters.The hyperchaotic Zheng system is taken as the master system, whose dynamics isgiven by1 2 1 42 1 2 1 3 423 1 34 2( )x a x x xx bx cx x x xx x rxx dx= − += + + += − −= −(14)where , , , ,a b c d r are unknown parameters of the system and 4x∈ R is the state of the system.The hyperchaotic Zheng system is also taken as the slave system, whose dynamics is given by1 2 1 4 12 1 2 1 3 4 223 1 3 34 2 4( )y a y y y uy by cy y y y uy y ry uy dy u= − + += + + + += − − += − +(15)where 4y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptivecontrollers 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 2 4 1 3 1 3 22 23 3 1 1 34 2 4( )( )e a y x e y x ue b y x ce e y y x x ue re y x ue de u= − − + − += + + + + + += − − + += − +(17)Next, we choose a nonlinear controller for achieving hybrid synchronization as1 2 2 1 4 4 1 12 1 1 2 4 1 3 1 3 2 22 23 3 1 1 3 34 2 4 4ˆ( )( )ˆ ˆ( )( ) ( )ˆ( )ˆ( )u a t y x e y x k eu b t y x c t e e y y x x k eu r t e y x k eu d t e k e= − − − − + −= − + − − − − −= + − −= −(18)
  • 6. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201316In 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 determined as1 2 2 1 1 12 1 1 2 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 c c t e k ee r r 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)Differentiating (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 2 2 23 3 3 34 2 4 4( )( )ab crde e y x e k ee e y x e e 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 12 22 2 4 3ˆˆ( ) ( )ˆˆ ˆa 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 r  = − − − − + − − − + + −        + − + − − + − −      (24)In view of Eq. (24), we take the parameter update law as21 2 2 1 5 2 1 1 6 2 722 4 8 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 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 of
  • 7. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201317the identical hyperchaotic Zheng systems (14) and (15), where ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t r t areestimates of the unknown parameters , , , , ,a b c d r respectively. In addition, the parameterestimation errors , , , ,a b c d re e e e e 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.This completes the proof. Next, we use MATLAB to demonstrate our hybrid synchronization results.The classical fourthorder Runge-Kutta method with time-step 810h −= has been applied to solvethe hyperchaotic Zheng systems (14) and (15) with the adaptive nonlinear controller(18) and theparameter update law (25). The feedback gains arechosen as 5, ( 1,2, ,9).ik i= = The parameters of the hyperchaotic Zheng systems are taken as in the hyperchaotic case, i.e.20, 14, 10.6, 4, 2.8a b c d r= = = = =For simulations, the initial conditions of the hyperchaotic Zheng system (14) are chosen as1 2 3 4(0) 24, (0) 15, (0) 6, (0) 18x x x x= = − = − =Also, the initial conditions of the hyperchaotic Zheng system (15) are chosen as1 2 3 4(0) 12, (0) 9, (0) 26, (0) 6y y y y= = − = = −Also, the initial conditions of the parameter estimates are chosen asˆ ˆˆ ˆ ˆ(0) 9, (0) 7, (0) 8, (0) 2, (0) 5a b c d r= = − = = = −Figure 3 depicts the hybrid synchronization of the identical hyperchaoticZheng 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 e
  • 8. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201318Figure 3.Hybrid Synchronization of Identical Hyperchaotic Zheng SystemsFigure 4. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e e
  • 9. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201319Figure 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 YU SYSTEMSIn this section, we design an adaptive controller for the hybrid synchronization of two identicalhyperchaotic Yusystems (2012) with unknown parameters.The hyperchaotic Yusystem is taken as the master system, whose dynamics is given by1 21 2 12 1 1 3 2 43 34 1( )x xx x xx x x x x xx e xx x = −= − + += −= −(27)where , , , ,     are unknown parameters of the system and 4x∈ R is the state of the system.
  • 10. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201320The hyperchaotic Yu system is also taken as the slave system, whose dynamics is given by1 21 2 1 12 1 1 3 2 4 23 3 34 1 4( )y yy y y uy y y y y y uy e y uy y u = − += − + + += − += − +(28)Where 4y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptivecontrollers 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 12 2 23 3 34 4 4e y xe y xe y xe y x= −= += −= +(29)A simple calculation gives the error dynamics1 2 1 21 2 2 1 12 1 1 2 4 1 3 1 3 23 3 34 1 1 4( )( )( )y y x xe y x e ue y x e e y y x x ue e e e ue y x u = − − += + + + − − += − + − += − + +(30)Next, we choose a nonlinear controller for achieving hybrid synchronization as1 2 1 21 2 2 1 1 12 1 1 2 4 1 3 1 3 2 23 3 3 34 1 1 4 4ˆ( )( )ˆ ˆ( )( ) ( )ˆ( )ˆ( )( )y y x xu t y x e k eu t y x t e e y y x x k eu t e e e k eu t y 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 1 1 12 1 1 2 2 23 3 3 34 1 1 4 4ˆ( ( ))( )ˆ ˆ( ( ))( ) ( ( ))ˆ( ( ))ˆ( ( ))( )e t y y k ee t y x t e k ee t e k ee t y x k e      = − − −= − + + − −= − − −= − − + −(32)
  • 11. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201321Next, we define the parameter estimation errors asˆ( ) ( )ˆ( ) ( )ˆ( ) ( )ˆ( ) ( )ˆ( ) ( )e t te t te t te t te t t     = −= −= −= −= −(33)Differentiating (33) with respect to ,t we getˆ 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 1 1 4 4( )( )( )e e y x e k ee e y x e e k ee e e k ee e y x 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 21 1 2 2 3 3 4 4 1 2 2 1 2 1 12 22 3 4 1 1ˆˆ( ) ( )ˆˆ ˆ( )V k e k e k e k e e e y x e e e y xe e e e e e y x        = − − − − + − − − + + −        + − + − − + − + −      (37)In view of Eq. (37), we take the parameter update law as21 2 2 1 5 2 1 1 6 2 723 8 4 1 1 9ˆˆ ˆ( ) , ( ) ,ˆ ˆ, ( )e y x e k e e y x k e e k ee k e e y x 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 Yu 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.
  • 12. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201322Proof.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. 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 Yu systems (27) and (28) with the adaptive nonlinear controller(31) and theparameter update law (38). The feedback gains are taken as5, ( 1,2, ,9).ik i= = The parameters of the hyperchaotic Yu systems are taken as in the hyperchaotic case, i.e.10, 40, 1, 3, 8    = = = = =For simulations, the initial conditions of the hyperchaoticYu system (27) are chosen as1 2 3 4(0) 4, (0) 2, (0) 8, (0) 10x x x x= = − = = −Also, the initial conditions of the hyperchaotic Yusystem (28) are chosen as1 2 3 4(0) 16, (0) 8, (0) 12, (0) 6y y y y= = = = −Also, the initial conditions of the parameter estimates are chosen asˆ ˆˆ ˆ ˆ(0) 17, (0) 7, (0) 12, (0) 5, (0) 6    = = − = = − =Figure 6depicts the hybrid synchronization of the identical hyperchaoticYu systems.Figure 7depicts the time-history of the hybrid synchronization errors 1 2 3 4, , , .e e e eFigure 8depicts the time-history of the parameter estimation errors , , , , .e e e e e    
  • 13. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201323Figure 6.Hybrid Synchronization of Identical Hyperchaotic Yu SystemsFigure 7. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e e
  • 14. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201324Figure 8. Time-History of the Parameter Estimation Errors , , , ,e e e e e    6. ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATIONDESIGN OF HYPERCHAOTIC ZHENG AND HYPERCHAOTIC YU SYSTEMSIn this section, we design an adaptive controller for the hybrid synchronizationofhyperchaoticZheng system (2010) and hyperchaotic Yusystem (2012) with unknownparameters.The hyperchaotic Zhengsystem is taken as the master system, whose dynamics is given by1 2 1 42 1 2 1 3 423 1 34 2( )x a x x xx bx cx x x xx x rxx dx= − += + + += − −= −(40)where , , , ,a b c d r are unknown parameters of the system.The hyperchaotic Yu system is also taken as the slave system, whose dynamics is given by1 21 2 1 12 1 1 3 2 4 23 3 34 1 4( )y yy y y uy y y y y y uy e y uy y u = − += − + + += − += − +(41)where , , , ,     are unknown parametersand 1 2 3 4, , ,u u u u are the adaptivecontrollers.
  • 15. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201325For 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 21 2 1 2 1 4 12 1 2 4 1 2 1 3 1 3 223 3 3 1 34 1 2 4( ) ( )y ye y y a x x x ue y y e bx cx y y x x ue y rx e x ue y dx u = − − − − += + + + + − + += − + + + += − − +(43)Next, we choose a nonlinear controller for achieving hybrid synchronization as1 21 2 1 2 1 4 1 12 1 2 4 1 2 1 3 1 3 2 223 3 3 1 3 34 1 2 4 4ˆ ˆ( )( ) ( )( )ˆˆ ˆ ˆ( ) ( ) ( ) ( )ˆ ˆ( ) ( )ˆˆ( ) ( )y yu t y y a t x x x k eu t y t y e b t x c t x y y x x k eu t y r t x e x k eu t y d t 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 obtained as1 2 1 2 1 1 12 1 2 1 2 2 23 3 3 3 34 1 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 c c t x k ee t y r r 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 getˆ 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)
  • 16. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201326In view of (46), we can simplify the error dynamics (45) as1 2 1 2 1 1 12 1 2 1 2 2 23 3 3 3 34 1 2 4 4( ) ( )ab crde e y y e x x k ee e y e y e x e 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 412a b c d rV e e e e e e e e e e e e e e    = + + + + + + + + + + + + + (49)When 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 2 24 2 3 3 1 2 1 2 12 2 3 3ˆˆ 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 r e e y y e e ye e y e e y e         = − − − − + − − − + − + −          + − − + − + − − + −           + − + − − + −      4 1ˆe y  − (50)In view of Eq. (50), we take the parameter update law as1 2 1 5 2 1 6 2 2 74 2 8 3 3 9 1 2 1 102 1 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 k e e y y k ee y k e e y k e  = − − + = + = += − + = + = − += + = +   3 3 134 1 14ˆˆe 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 Zheng system (40) hyperchaotic Yu system (41), where ˆ( ),a t ˆ( ),b t ˆ( ),c t ˆ( ),d tˆ( ),r t ˆ( ),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 zeroexponentially for all initial conditions.Proof.We prove the above result using Lyapunov stability theory [25].Substituting the parameterupdate law (51) into (50), we get2 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.R
  • 17. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201327This 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. For simulations, theclassical fourth order Runge-Kutta method with time-step 810h −= has beenapplied to solve the hyperchaotic Li systems (27) and (28) with the adaptive nonlinearcontroller(31) and the parameter update law (38). The feedback gains are taken as5, ( 1,2, ,14).ik i= =  The parameters of the hyperchaotic Zheng and hyperchaotic Yu systemsare takenas 20,a = 14,b = 10.6,c = 4,d = 2.8,r = 10, = 40, = 1, = 3 = and 8. =For simulations, the initial conditions of the hyperchaotic Zheng system (40) are chosen as1 2 3 4(0) 4, (0) 9, (0) 1, (0) 4x x x x= = = =Also, the initial conditions of the hyperchaotic Yu system (41) are chosen as1 2 3 4(0) 8, (0) 3, (0) 1, (0) 2y y y y= = = =Also, the initial conditions of the parameter estimates are chosen asˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ(0) 2, (0) 6, (0) 3, (0) 3, (0) 1, (0) 7, (0) 4, (0) 9, (0) 5, (0) 4a b c d r     = = = = − = − = = = = =Figure 9depicts the hybrid synchronization of hyperchaoticZheng and hyperchaotic Yu systems.Figure 10depicts the time-history of the hybrid synchronization errors 1 2 3 4, , , .e e e e Figure11depicts the time-history of the parameter estimation errors , , , , .a b c d re e e e e Figure 12depicts thetime-history of the parameter estimation errors , , , , .e e e e e    Figure 9.Hybrid Synchronization of Hyperchaotic Xu and Lu Systems
  • 18. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201328Figure 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
  • 19. International Journal of Information Technology Convergence and Services (IJITCS) Vol.3, No.2, April 201329Figure 12. Time-History of the Parameter Estimation Errors , , , ,e e e e e    7. CONCLUSIONSThis paper derived new results for the active synchronizer design for achieving hybridsynchronization of hyperchaoticZhengsystems (2010) and hyperchaotic Yu systems (2012).Using Lyapunov control theory,adaptive control laws were derived for globally hybridsynchronizing the states of identical hyperchaotic Zheng systems, identical hyperchaotic Yusystems and non-identical hyperchaotic Zheng and Yu systems. Numerical simulations usingMATLABwere shown to validate and illustrate the hybrid synchronization results forhyperchaotic Zheng and Yu 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.
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