ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMS WITH UNKNOWN PARAMETERS VIA ADAPTIVE CONTROL

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In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems …

In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.

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  • 1. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013DOI : 10.5121/ijcseit.2013.3203 31ANTI-SYNCHRONIZATION OF HYPERCHAOTICWANG AND HYPERCHAOTIC LI SYSTEMS WITHUNKNOWN PARAMETERS VIA ADAPTIVE CONTROLSundarapandian Vaidyanathan11Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical UniversityAvadi, Chennai-600 062, Tamil Nadu, INDIAsundarvtu@gmail.comABSTRACTIn chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systemscalled drive and response systems. In this problem, the design goal is to drive the sum of their respectivestates to zero asymptotically. This problem gets even more complicated and requires special attention whenthe parameters of the drive and response systems are unknown. This paper uses adaptive control theoryand Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wangsystem (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems arenonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,secure transmission and secure communication. The main results derived in this paper are validated anddemonstrated with MATLAB simulations.KEYWORDSHyperchaos, Hyperchaotic Systems, Adaptive Control, Anti-Synchronization.1. INTRODUCTIONHyperchaotic systems are typically defined as nonlinear chaotic systems having two or morepositive Lyapunov exponents. They are applicable in several areas like lasers [1], chemicalreactions [2], neural networks [3], oscillators [4], data encryption [5], secure communication [6-8], etc.In chaos theory, the anti-synchronization problem deals with a pair of chaotic systems called thedrive and response systems, where the design goal is to render the respective states to be same inmagnitude, but opposite in sign, or in other words, to drive the sum of the respective states to zeroasymptotically [9].There are several methods available in the literature to tackle the problem of synchronization andanti-synchronization of chaotic systems like active control method [10-12], adaptive controlmethod [13-15], backstepping method [16-19], sliding control method [20-22] etc.This paper derives new results for the adaptive controller design for the anti-synchronization ofhyperchaotic Wang systems ([23], 2008) and hyperchaotic Li systems ([24], 2005) with unknownparameters. Lyapunov stability theory [25] has been applied to prove the main results of thispaper. Numerical simulations have been shown using MATLAB to illustrate the results.
  • 2. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013322. THE PROBLEM OF ANTI-SYNCHRONIZATION OF CHAOTIC SYSTEMSIn chaos synchronization problem, the drive 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.Also, the response system is described by the chaotic dynamics( )y By g y u= + + (2)where B is the n n× matrix of the system parameters, : n ng →R R is the nonlinear part andnu ∈R is the active controller to be designed.For the pair of chaotic systems (1) and (2), the design goal of the anti-synchronization problem isto construct a feedback controller ,u which anti-synchronizes their states for all (0), (0) .nx y ∈RThe anti-synchronization error is defined as,e y x= + (3)The error dynamics is obtained as( ) ( )e By Ax g y f x u= + + + + (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 u so 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 [26], the error dynamics (4) is globally exponentially stable.When the system parameters in (1) and (2) are unknown, we apply adaptive control theory toconstruct a parameter update law for determining the estimates of the unknown parameters.
  • 3. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013333. HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMSThe hyperchaotic Wang system ([23], 2008) is given by1 2 1 2 32 1 1 3 2 43 3 1 24 4 1 3( )0.50.5x a x x x xx cx x x x xx dx x xx bx x x= − += − − −= − += +(8)where , , ,a b c d are constant, positive parameters of the system.The Wang system (8) depicts a hyperchaotic attractor for the parametric values40, 1.7, 88, 3a b c d= = = = (9)The Lyapunov exponents of the system (8) are determined as1 2 3 43.2553, 1.4252, 0, 46.9794   = = = = − (10)Since there are two positive Lyapunov exponents in (10), the Wang system (8) is hyperchaotic forthe parametric values (9).Figure 1 shows the phase portrait of the hyperchaotic Wang system.The hyperchaotic Li system ([24], 2005) 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 rx = − += − += − += +(11)where , , , ,r    are constant, positive parameters of the system.The Li system (11) depicts a hyperchaotic attractor for the parametric values35, 3, 12, 7, 0.58r   = = = = = (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). Figure 2 shows the phase portrait of the hyperchaotic Li system.
  • 4. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201334Figure 1. Hyperchaotic Attractor of the Hyperchaotic Wang SystemFigure 2. Hyperchaotic Attractor of the Hyperchaotic Li System4. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG SYSTEMS VIAADAPTIVE CONTROLIn this section, we derive new results for designing a controller for the anti-synchronization ofidentical hyperchaotic Wang systems (2008) with unknown parameters via adaptive control.The drive system is the hyperchaotic Wang dynamics given by
  • 5. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013351 2 1 2 32 1 1 3 2 43 3 1 24 4 1 3( )0.50.5x a x x x xx cx x x x xx dx x xx bx x x= − += − − −= − += +(14)where , , ,a b c d are unknown parameters of the system and 4x∈ R is the state.The response system is the controlled hyperchaotic Wang dynamics given by1 2 1 2 3 12 1 1 3 2 4 23 3 1 2 34 4 1 3 4( )0.50.5y a y y y y uy cy y y y y uy dy y y uy by y y u= − + += − − − += − + += + +(15)where 4y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed.For the anti-synchronization, the error e is defined as1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (16)Then we derive the error dynamics as1 2 1 2 3 2 3 12 1 2 4 1 3 1 3 23 3 1 2 1 2 34 4 1 3 1 3 4( )0.50.5( )e a e e y y x x ue ce e e y y x x ue de y y x x ue be y y x x u= − + + += − − − − += − + + += + + +(17)The adaptive controller to solve the anti-synchronization problem is taken as1 2 1 2 3 2 3 1 12 1 2 4 1 3 1 3 2 23 3 1 2 1 2 3 34 4 1 3 1 3 4 4ˆ( )( )ˆ( ) 0.5ˆ( )ˆ( ) 0.5( )u a t e e y y x x k eu c t e e e y y x x k eu d t e y y x x k eu b 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( ), ( ), ( ), ( )a t b t c t d t are estimates for theunknown parameters , , ,a b c d respectively.By the substitution of (18) into (17), the error dynamics is obtained as1 2 1 1 12 1 2 23 3 3 34 4 4 4ˆ( ( ))( )ˆ( ( ))ˆ( ( ))ˆ( ( ))e a a t e e k ee c c t e k ee d d t e k ee b b t e k e= − − −= − −= − − −= − −(19)
  • 6. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201336Next, we define the parameter estimation errors asˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a a t e t b b t e t c c t e t d d t= − = − = − = − (20)Upon differentiation, we getˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a t e t b t e t c t e t d t= − = − = − = −      (21)Substituting (20) into the error dynamics (19), we obtain1 2 1 1 12 1 2 23 3 3 34 4 4 4( )acdbe e e e k ee e e k ee e e k ee e e k e= − −= −= − −= −(22)We consider the candidate Lyapunov function( )2 2 2 2 2 2 2 21 2 3 412a b c dV e e e e e e e e= + + + + + + + (23)Differentiating (23) along the dynamics (21) and (22), we obtain( )( ) ( )2 2 2 2 21 1 2 2 3 3 4 4 1 2 1 421 2 3ˆˆ( )ˆˆa bc dV k e k e k e k e e e e e a e e be e e c e e d = − − − − + − − + − + − + − −(24)In view of (24), we choose the following parameter update law:1 2 1 524 61 2 723 8ˆ ( )ˆˆˆabcda e e e k eb e k ec e e k ed e k e= − += += += − +(25)Next, we prove the following main result of this section.Theorem 4.1 The adaptive control law defined by Eq. (18) along with the parameter update lawdefined by Eq. (25), where ,( 1,2, ,8)ik i =  are positive constants, render global and exponentialanti-synchronization of the identical hyperchaotic Wang systems (14) and (15) with unknownparameters for all initial conditions 4(0), (0) .x y ∈ R In addition, the parameter estimation errors( ), ( ), ( ), ( )a b c de t e t e t e t globally and exponentially converge to zero for all initial conditions.
  • 7. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201337Proof. The proof is via Lyapunov stability theory [25] by taking V defined by Eq. (23) as thecandidate Lyapunov function. Substituting the parameter update law (25) into (24), we get2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 8( ) a b c dV e k e k ek e k e k e k e k e k e= − −− − − − − − (26)which is a negative definite function on8.R This completes the proof. Next, we demonstrate our adaptive anti-synchronization results with MATLAB simulations. Theclassical fourth order R-K method with time-step 810h −= has been used to solve the hyperchaoticWang systems (14) and (15) with the adaptive controller defined by (18) and parameter updatelaw defined by (25).The feedback gains in the adaptive controller (18) are taken as 5, ( 1, ,8).ik i= = The parameters of the hyperchaotic Wang systems are taken as in the hyperchaotic case, i.e.40, 1.7, 88, 3a b c d= = = =For simulations, the initial conditions of the drive system (14) are taken as1 2 3 4(0) 37, (0) 16, (0) 14, (0) 11x x x x= = − = =Also, the initial conditions of the response system (15) are taken as1 2 3 4(0) 21, (0) 32, (0) 28, (0) 8y y y y= = = − = −Also, the initial conditions of the parameter estimates are taken asˆ ˆˆ ˆ(0) 12, (0) 4, (0) 6, (0) 5a b c d= = = − =Figure 3 depicts the anti-synchronization of the identical hyperchaotic Wang systems.Figure 4 depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e eFigure 5 depicts the time-history of the parameter estimation errors , , , .a b c de e e e
  • 8. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201338Figure 3. Anti-Synchronization of Identical Hyperchaotic Wang SystemsFigure 4. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
  • 9. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201339Figure 5. Time-History of the Parameter Estimation Errors , , ,a b c de e e e5. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC LI SYSTEMS VIA ADAPTIVECONTROLIn this section, we derive new results for designing a controller for the anti-synchronization ofidentical hyperchaotic Li systems (2005) with unknown parameters via adaptive control.The drive system is the hyperchaotic Li dynamics 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 rx = − += − += − += +(27)where , , , , r    are unknown parameters of the system and 4x∈ R is the state.The response system is the controlled hyperchaotic Li dynamics given by1 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 ry u = − + += − + += − + += + +(28)where 4y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed.For the anti-synchronization, the error e is defined as1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (29)
  • 10. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201340Then we derive the error dynamics as1 2 1 4 12 1 2 1 3 1 3 23 3 1 2 1 2 34 4 2 3 2 3 4( )e e e e ue e e y y x x ue e y y x x ue re y y x x u = − + += + − − += − + + += + + +(30)The adaptive controller to solve the anti-synchronization problem is taken as1 2 1 4 1 12 1 2 1 3 1 3 2 23 3 1 2 1 2 3 34 4 2 3 2 3 4 4ˆ( )( )ˆ ˆ( ) ( )ˆ( )ˆ( )u t e e e k eu t e t e y y x x k eu t e y y x x k eu r t e y y x x k e = − − − −= − − + + −= − − −= − − − −(31)In Eq. (31), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )t t t t r t    are estimates forthe unknown parameters , , , , r    respectively.By the substitution of (31) into (30), the error dynamics is obtained as1 2 1 1 12 1 2 2 23 3 3 34 4 4 4ˆ( ( ))( )ˆ ˆ( ( )) ( ( ))ˆ( ( ))ˆ( ( ))e t e e k ee t e t e k ee t e k ee r r t e k e     = − − −= − + − −= − − −= − −(32)Next, we define the parameter estimation errors asˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )re t t e t t e t t e t t e t r r t          = − = − = − = − = − (33)Upon differentiation, we getˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )re t t e t t e t t e t t e t r t      = − = − = − = − = −        (34)Substituting (33) into the error dynamics (32), we obtain1 2 1 1 12 1 2 2 23 3 3 34 4 4 4( )re e e e k ee e e e e k ee e e k ee e e k e = − −= + −= − −= −(35)We consider the candidate Lyapunov function
  • 11. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201341( )2 2 2 2 2 2 2 2 21 2 3 412rV e e e e e e e e e   = + + + + + + + + (36)Differentiating (36) along the dynamics (34) and (35), we obtain( )( ) ( ) ( )2 2 2 2 21 1 2 2 3 3 4 4 1 2 1 32 22 1 2 4ˆˆ( )ˆˆ ˆrV k e k e k e k e e e e e e ee e e e e e e r     = − − − − + − − + − − + − + − + − (37)In view of (37), we choose the following parameter update law:1 2 1 523 622 71 2 824 9ˆ ( )ˆˆˆˆare e e k ee k ee k ee e k er e k e= − += − += += += +(38)Next, we prove the following main result of this section.Theorem 5.1 The adaptive control law defined by Eq. (31) along with the parameter update lawdefined by Eq. (38), where ,( 1,2, ,9)ik i =  are positive constants, render global and exponentialanti-synchronization of the identical hyperchaotic Li systems (27) and (28) with unknownparameters for all initial conditions 4(0), (0) .x y ∈ R In addition, the parameter estimation errors( ), ( ), ( ), ( ), ( )re t e t e t e t e t    globally and exponentially converge to zero for all initialconditions.Proof. The proof is via Lyapunov stability theory [25] by taking V defined by Eq. (36) as thecandidate Lyapunov function. 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 9( ) rV e k e k ek e k e k e k e k e k e k e   = − −− − − − − − − (39)which is a negative definite function on9.R This completes the proof. Next, we demonstrate our adaptive anti-synchronization results with MATLAB simulations. Theclassical fourth order R-K method with time-step 810h −= has been used to solve the hyperchaoticLi systems (27) and (28) with the adaptive controller defined by (31) and parameter update lawdefined by (38). The feedback gains in the adaptive controller (31) are taken as5, ( 1, ,9).ik i= = The parameters of the hyperchaotic Li systems are taken as in the hyperchaotic case, i.e.35, 3, 12, 7, 0.58r   = = = = =For simulations, the initial conditions of the drive system (27) are taken as
  • 12. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013421 2 3 4(0) 7, (0) 26, (0) 12, (0) 14x x x x= = − = =Also, the initial conditions of the response system (28) are taken as1 2 3 4(0) 21, (0) 28, (0) 18, (0) 29y y y y= − = = − =Also, the initial conditions of the parameter estimates are taken asˆ ˆˆ ˆ ˆ(0) 7, (0) 15, (0) 5, (0) 4, (0) 3r   = = = = = −Figure 6 depicts the anti-synchronization of the identical hyperchaotic Li systems.Figure 7 depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e eFigure 8 depicts the time-history of the parameter estimation errors , , , , .re e e e e   Figure 6. Anti-Synchronization of Identical Hyperchaotic Li Systems
  • 13. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201343Figure 7. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e eFigure 8. Time-History of the Parameter Estimation Errors , , , , re e e e e   6.ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG ANDHYPERCHAOTIC LI SYSTEMS VIA ADAPTIVE CONTROLIn this section, we derive new results for designing a controller for the anti-synchronization ofnon-identical hyperchaotic Wang system (2009) and hyperchaotic Li system (2005) withunknown parameters via adaptive control.The drive system is the hyperchaotic Wang dynamics given by
  • 14. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April2013441 2 1 2 32 1 1 3 2 43 3 1 24 4 1 3( )0.50.5x a x x x xx cx x x x xx dx x xx bx x x= − += − − −= − += +(40)where , , ,a b c d are unknown parameters of the system and 4x ∈ R is the state.The response system is the controlled hyperchaotic Li dynamics given by1 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 ry u = − + += − + += − + += + +(41)where , , , , r    are unknown parameters, 4y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptivecontrollers to be designed.For the anti-synchronization, the error e is defined as1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (42)Then we derive the error dynamics as1 2 1 2 1 4 2 3 12 1 1 2 2 4 1 3 1 3 23 3 3 1 2 1 2 34 4 4 1 3 2 3 4( ) ( )0.50.5e a x x y y y x x ue cx y x y x x x y y ue dx y y y x x ue bx ry x x y y u = − + − + + += + − + − − − += − − + + += + + + +(43)The adaptive controller to solve the anti-synchronization problem is taken as1 2 1 2 1 4 2 3 1 12 1 1 2 2 4 1 3 1 3 2 23 3 3 1 2 1 2 3 34 4 4 1 3 2 3 4 4ˆˆ( )( ) ( )( )ˆ ˆˆ( ) ( ) ( ) 0.5ˆ ˆ( ) ( )ˆ ˆ( ) ( ) 0.5u a t x x t y y y x x k eu c t x t y x t y x x x y y k eu d t x t y y y x x k eu b t x r t y x x y y k e = − − − − − − −= − − + − + + + −= + − − −= − − − − −(44)In Eq. (44), , ( 1,2,3,4)ik i = are positive gains and ˆ( ),a t ˆ( ),b t ˆ( ),c t ˆ( ),d t ˆ( ),t ˆ( ),t ˆ( ),tˆ( ),t ˆ( )r t are estimates for the unknown parameters , , , , , , , ,a b c d r    respectively.By the substitution of (44) into (43), the error dynamics is obtained as1 2 1 2 1 1 12 1 1 2 2 23 3 3 3 34 4 4 4 4ˆˆ( ( ))( ) ( ( ))( )ˆ ˆˆ( ( )) ( ( )) ( ( ))ˆ ˆ( ( )) ( ( ))ˆ ˆ( ( )) ( ( ))e a a t x x t y y k ee c c t x t y t y k ee d d t x t y k ee b b t x r r t y k e     = − − + − − −= − + − + − −= − − − − −= − + − −(45)
  • 15. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201345Next, we define the parameter estimation errors asˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )ˆ 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 t e t t e t t e t t e t r r t          = − = − = − = −= − = − = − = − = −(46)Upon differentiation, we getˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c dre t a t e t b t e t c t e t d te t t e t t e t t e t t e t r t      = − = − = − = −= − = − = − = − = −            (47)Substituting (46) into the error dynamics (45), we obtain1 2 1 2 1 1 12 1 1 2 2 23 3 3 3 34 4 4 4 4( ) ( )acdb re e x x e y y k ee e x e y e y k ee e x e y k ee e x e y k e = − + − −= + + −= − − −= + −(48)We consider the candidate Lyapunov function( )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   = + + + + + + + + + + + + (49)Differentiating (49) along the dynamics (47) and (48), we obtain( ) ( )( ) ( )( ) ( ) ( )2 2 2 21 1 2 2 3 3 4 4 1 2 1 4 4 2 13 3 1 2 1 3 32 2 2 1 4 4ˆˆ ˆ( )ˆ ˆˆ+ ( )ˆˆ ˆa b cdrV 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 y y e e ye e y e e y e e y r     = − − − − + − − + − + −  − − + − − + − − + − + − + −   (50)In view of (50), we choose the following parameter update law:1 2 1 5 1 2 1 94 4 6 3 3 102 1 7 2 2 113 3 8 2 1 12ˆˆ ( ) , ( )ˆ ˆ,ˆˆ ,ˆ ˆ,a abcda e x x k e e y y k eb e x k e e y k ec e x k e e y k ed e x k e e y k= − + = − += + = − += + = += − + = +  4 4 13ˆ rer e y k e= +(51)Next, we prove the following main result of this section.
  • 16. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201346Theorem 6.1 The adaptive control law defined by Eq. (44) along with the parameter update lawdefined by Eq. (51), where ,( 1,2, ,13)ik i =  are positive constants, render global andexponential anti-synchronization of the non-identical hyperchaotic Wang system (40) andhyperchaotic Li system (41) with unknown parameters for all initial conditions 4(0), (0) .x y ∈ RIn addition, all the parameter estimation errors globally and exponentially converge to zero for allinitial conditions.Proof. The proof is via Lyapunov stability theory [25] by taking V defined by Eq. (49) as thecandidate Lyapunov function. Substituting the parameter update law (51) into (50), we get2 2 2 2 2 2 2 21 1 2 2 3 3 4 4 5 6 7 82 2 2 2 29 10 11 12 13( ) a b c drV e k e k ek ek e k e k e k e k e k ek e k e k e k e   = − −−− − − − − −− − − −(52)which is a negative definite function on13.R This completes the proof. Next, we demonstrate our adaptive anti-synchronization results with MATLAB simulations. Theclassical fourth order R-K method with time-step 810h −= has been used to solve the hyperchaoticsystems (40) and (41) with the adaptive controller defined by (44) and parameter update lawdefined by (51).The feedback gains in the adaptive controller (44) are taken as 5, ( 1, ,13).ik i= = The parameters of the hyperchaotic Wang and Li systems are taken as in the hyperchaotic case,i.e.40, 1.7, 88, 3, 35, 3, 12, 7, 0. 58a b c d r   = = = = = = = = =For simulations, the initial conditions of the drive system (40) are taken as1 2 3 4(0) 12, (0) 34, (0) 31, (0) 14x x x x= = − = =Also, the initial conditions of the response system (41) are taken as1 2 3 4(0) 25, (0) 18, (0) 12, (0) 29y y y y= − = = − =Also, the initial conditions of the parameter estimates are taken asˆ ˆˆ ˆ(0) 21, (0) 14, (0) 26, (0) 16ˆ ˆˆ ˆ ˆ(0) 17, (0) 22 (0) 15, (0) 11, (0) 7a b c dr   = = = − == = − = = = −Figure 9 depicts the anti-synchronization of the hyperchaotic Wang and hyperchaotic Li systems.Figure 10 depicts the time-history of the anti-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 , , , , .re e e e e   
  • 17. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201347Figure 9. Anti-Synchronization of Hyperchaotic Wang and Hyperchaotic Li SystemsFigure 10. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
  • 18. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201348Figure 11. Time-History of the Parameter Estimation Errors , , ,a b c de e e eFigure 12. Time-History of the Parameter Estimation Errors , , , , re e e e e   7. CONCLUSIONSThis paper has used adaptive control theory and Lyapunov stability theory so as to solve the anti-synchronization problem for the anti-synchronization of hyperchaotic Wang system (2008) andhyperchaotic Li system (2005) with unknown parameters. Hyperchaotic systems are chaoticsystems with two or more positive Lyapunov exponents and they have viable applications likechemical reactions, neural networks, secure communication, data encryption, neural networks,etc. MATLAB simulations were depicted to illustrate the various adaptive anti-synchronizationresults derived in this paper for the hyperchaotic Wang and Li systems.
  • 19. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201349REFERENCES[1] Misra, A.P., Ghosh, D. & Chowdhury, A.R. (2008) “A novel hyperchaos in the quantum Zakharovsystem for plasmas,” Physics Letters A, Vol. 372, No. 9, pp 1469-1476.[2] Eiswirth, M., Kruel, T.M., Ertl, G. & Schneider, F.W. (1992) “Hyperchaos in a chemical reaction,”Chemical Physics Letters, Vol. 193, No. 4, pp 305-310.[3] Huang, Y. & Yang, X.S. (2006) “Hyperchaos and bifurcation in a new class of four-dimensionalHopfield neural networks,” Neurocomputing, Vol. 69, pp 13-15.[4] 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.[5] 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.[6] Tao, Y. (1999) “Chaotic secure communication systems – history and new results”, Telecommun.Review, Vol. 9, pp 597-634.[7] 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.[8] Nana, B., Woafo, P. & Domngang, S. (2009) ‘Chaotic synchronization with experimentalapplications to secure communications”, Comm. Nonlinear Sci. Numerical Simulation, Vol. 14, No.5, pp 2266-2276.[9] Sundarapandian, V. & Karthikeyan, R. (2011) “Anti-synchronization of Pan and Liu chaotic systemsby active nonlinear control,” International Journal of Engineering Science and Technology, Vol. 3,No. 5, pp 3596-3604.[10] Huang, L. Feng, R. & Wang, M. (2004) “Synchronization of chaotic systems via nonlinear control,”Physics Letters A, Vol. 320, No. 4, pp 271-275.[11] Lei, Y., Xu, W. & Zheng, H. (2005) “Synchronization of two chaotic nonlinear gyros using activecontrol,” Physics Letters A, Vol. 343, pp 153-158.[12] Sarasu, P. & Sundarapandian, V. (2011) “Active controller design for generalized projectivesynchronization of four-scroll chaotic systems”, International Journal of System Signal Control andEngineering Application, Vol. 4, No. 2, pp 26-33.[13] Sarasu, P. & Sundarapandian, V. (2012) “Generalized projective synchronization of two-scrollsystems via adaptive control,” International Journal of Soft Computing, Vol. 7, No. 4, pp 146-156.[14] Sundarapandian, V. (2012) “Adaptive control and synchronization of a generalized Lotka-Volterrasystem,” Vol. 1, No. 1, pp 1-12.[15] Sundarapandian, V. (2013) “Adaptive controller and synchronizer design for hyperchaotic Zhousystem with unknown parameters,” Vol. 1, No. 1, pp 18-32.[16] Bowong, S. & Kakmeni, F.M.M. (2004) “Synchronization of uncertain chaotic systems viabackstepping approach,” Chaos, Solitons & Fractals, Vol. 21, No. 4, pp 999-1011.[17] Suresh, R. & Sundarapandian, V. (2012) “Hybrid synchronization of n-scroll Chua and Lur’e chaoticsystems via backstepping control with novel feedback”, Arch. Control Sciences, Vol. 22, No. 3, pp255-278.[18] 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.[19] Sundarapandian, V. (2013) “Anti-synchronizing backstepping design for Arneodo chaotic system”,International Journal on Bioinformatics and Biosciences, Vol. 3, No. 1, pp 21-33.[20] Senejohnny, D.M. & Delavari, H. (2012) “Active sliding observer scheme based fractional chaossynchronization,” Comm. Nonlinear Sci. Numerical Simulation, Vol. 17, No. 11, pp 4373-4383.[21] Sundarapandian, V. (2012) “Anti-synchronization of hyperchaotic Xu systems via sliding modecontrol”, International Journal of Embedded Systems, Vol. 2, No. 2, pp 51-61.[22] Sundarapandian, V. (2013) “Anti-synchronizing sliding controller design for identical Pan systems,”International Journal of Computational Science and Information Technology, Vol. 1, No. 1, pp 1-9.[23] Wang, J. & Chen, Z. (2008) “A novel hyperchaotic system and its complex dynamics,” InternationalJournal of Bifurcation and Chaos, Vol. 18, No. 11, pp 3309-3324.[24] Li, Y.,Tang, W.K.S. & Chen, G. (2005) “Generating hyperchaos via state feedback control,”International Journal of Bifurcation and Chaos, Vol. 15, No. 10, pp 3367-3375.[25] Hahn, W. (1967) The Stability of Motion, Springer, Berlin.
  • 20. International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.3, No.2,April201350AuthorDr. V. Sundarapandian earned his D.Sc. in Electrical and Systems Engineering fromWashington University, St. Louis, USA in May 1996. He is Professor and Dean of theR & D Centre at Vel Tech Dr. RR & Dr. SR Technical University, Chennai, TamilNadu, India. So far, he has published over 300 research works in refereedinternational journals. He has also published over 200 research papers in National andInternational Conferences. He has delivered Key Note Addresses at manyInternational Conferences with IEEE and Springer Proceedings. He is an India Chairof AIRCC. He is the Editor-in-Chief of the AIRCC Control Journals – InternationalJournal of Instrumentation and Control Systems, International Journal of ControlTheory and Computer Modeling, International Journal of Information Technology, Control andAutomation, International Journal of Chaos, Control, Modelling and Simulation, and International Journalof Information Technology, Modeling and Computing. His research interests are Control Systems, ChaosTheory, Soft Computing, Operations Research, Mathematical Modelling and Scientific Computing. Hehas published four text-books and conducted many workshops on Scientific Computing, MATLAB andSCILAB.