Anti-Synchronization of Tigan and Li Systems with Unknown Parameters via Adaptive Control

Abstract. In this paper, the adaptive nonlinear control method has been deployed to derive new results for the anti-synchronization of identical Tigan systems (2008), identical Li systems (2009) and nonidentical Tigan and Li systems. In adaptive anti-synchronization of identical chaotic systems, the parameters of the master and slave systems are unknown and the feedback control law has been derived using the estimates of the system parameters. In adaptive anti-synchronization of non-identical chaotic systems, the parameters of the master system are known, but the parameters of the slave system are unknown and accordingly, the feedback control law has been derived using the estimates of the parameters of the slave system. Our adaptive synchronization results derived in this paper for the uncertain Tigan and Li systems are established using Lyapunov stability theory. Numerical simulations are shown to demonstrate the effectiveness of the adaptive anti-synchronization schemes for the uncertain chaotic systems addressed in this paper.


Introduction
Chaotic systems are nonlinear systems that are highly sensitive to initial conditions.This sensitivity is popularly known as the butterfly effect [1].The first chaotic system was discovered by Lorenz [2] when he was studying weather patterns.
Since the pioneering work by Pecora and Carroll [3] chaos synchronization and antisynchronization problems have been studied extensively and intensively in the chaos literature .
In most of the chaos synchronization approaches, the master-slave or drive-response formalism is used.If a particular chaotic system is called a master or drive system and another chaotic system is called a slave or response system, then the goal of anti-synchronization is to use the output of the master system to control the slave system so that the states of the slave system have the same amplitude but opposite signs as the states of the master system asymptotically.
In this paper, we discuss the antisynchronization of identical hyperchaotic Tigan systems [32], identical Li systems [33], and nonidentical Tigan and Li systems.Our synchronization results are established using the Lyapunov stability theory [34].
In adaptive synchronization of identical chaotic systems, the parameters of the master and slave systems are unknown and we devise feedback control laws using the estimates of the system parameters.
In adaptive synchronization of non-identical chaotic systems, the parameters of the master system are known, but the parameters of the slave system are unknown and we devise feedback control laws using the estimates of the parameters of the slave system.
This paper has been organized as follows.In Section 2, we discuss the adaptive antisynchronization of identical Tigan systems [32].In Section 3, we discuss the adaptive antisynchronization of identical Li systems [33].In Section 4, we discuss the adaptive antisynchronization of non-identical Tigan and Li systems.In Section 5, we summarize the main results obtained in this paper.

Adaptive Anti-Synchronization of Identical
Tigan Systems This section details the adaptive antisynchronization of identical Tigan systems [32], when the parameters of the master and slave systems are unknown.

Theoretical Results
As the master system, we consider the Tigan dynamics described by where ,, x x x are the state variables and ,, abcare unknown parameters of the system.As the slave system, we consider the controlled Tigan dynamics described by where ,, y y y are the state variables and ,, u u u are the nonlinear control inputs to be designed.
The Tigan systems (1) and ( 2) are chaotic when the parameter values are chosen as The strange chaotic attractor of the system (1) is depicted in Figure 1.The anti-synchronization error is defined as , ( 1,2,3).
The error dynamics is obtained as e a e e u e c a e ay y ax x u e be y y x x u We define the adaptive control functions as a e e k e u t c a e ay y ay y k e u t be y y x x k e where ,, a b c are estimates of , , , abcrespectively ,, k k k are positive constants.Substituting ( 5) into (4), the closed-loop error dynamics is obtained as We define the parameter estimation errors as , , .
Using (7), the error dynamics is simplified as .
For the derivation of the update law for adjusting the estimates of parameters, the Lyapunov method is used.
We consider the quadratic Lyapunov function defined by V e e e e e e       which is a positive definite function on 6   .R We note that , , Differentiating ( 9) along the trajectories of (8) and noting (10), we find that In view of (11), the estimated parameters are updated by the following law: Now, we state and prove the following result.Theorem 1.The identical uncertain Tigan systems ( 1) and ( 2) are globally and exponentially antisynchronized by the adaptive control law (5), where the update law for the parameter estimates We know that V as defined in ( 9) is a positive definite function on 6   .R Substituting ( 12) into (11), we obtain , which is a negative definite function on 6   .R Hence, by the Lyapunov stability theory [34], it follows that ( ) 0

Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 6 10 h   is used to solve the two systems of differential equations ( 1) and ( 2) with the adaptive nonlinear controller (5) and update law of estimates (12).
We take The parameters of the Tigan systems are chosen so that the system (1) and ( 2) are chaotic, i.e.    ,, e e e   ,, u u u

Adaptive Anti-Synchronization of Identical Li Systems
This section details the adaptive antisynchronization of identical Li systems [33], when the parameters of the master and slave systems are unknown.

Theoretical Results
As the master system, we consider the Li dynamics described by ,, x x x are the state variables and ,,    are unknown parameters of the system.As the slave system, we consider the controlled Li dynamics described by ,, y y y are the state variables and ,, u u u are the nonlinear control inputs to be designed.
The Li systems ( 14) and ( 15) are chaotic when the parameter values are chosen as; 5,

     
The strange chaotic attractor of the system ( 14) is depicted in Figure 7.The anti-synchronization error is defined as , ( 1,2,3).
The error dynamics is obtained as e e e u e e y y x x u e e y y x x u We define the adaptive control functions as u t e e k e u t e y y x x k e u t e y y x x k e where ,,    are estimates of , , ,

  
respectively and ,, k k k are positive constants.Substituting ( 18) into ( 17), the closed-loop error dynamics is obtained as ) .
We define the parameter estimation errors as; , , .e e e Using (20), the error dynamics is simplified as For the derivation of the update law for adjusting the estimates of parameters, the Lyapunov method is used.
We consider the quadratic Lyapunov function defined by   In view of (24), the estimated parameters are updated by the following law: Now, we state and prove the following result.Theorem 2. The identical uncertain Li systems ( 14) and ( 15) are globally and exponentially antisynchronized by the adaptive control law (18), where the update law for the parameter estimates ,,    is given by (25) and , ( 1, 2, ,6)

t 
Proof.This resut is a simple consequence of the Lyapunov stability theory.We know that V as defined in ( 22) is a positive definite function on 6 .R Substituting (25) into (24), we obtain

Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 6 10 h   is used to solve the two systems of differential equations ( 14) and ( 15) with the adaptive nonlinear controller (18) and update law of estimates (25).
We take The parameters of the Li systems are chosen so that the system ( 14) and ( 15) are chaotic, i.e.

y 
Figure 8 shows the anti-synchronization of the Li systems ( 14) and ( 15). Figure 9 shows the time-history of the anti-synchronization errors ,, e e e   ,, u u u

Adaptive Anti-Synchronization of Tigan and Li Chaotic Systems
This section details the adaptive antisynchronization of Tigan and Li systems.Here, we consider the Tigan system [32] as the master system, whose parameters are known.Also, we consider the Li system [33] as the slave system, whose parameters are unknown.

Theoretical Results
As the master system, we consider the Tigan dynamics described by ,, x x x are the state variables and ,, abc are known parameters of the system.
As the slave system, we consider the controlled Li dynamics described by ,, u u u are the nonlinear control inputs to be designed.
The error dynamics is obtained as .
We define the adaptive control functions as ,, k k k are positive constants.Substituting (31) into (30), the closed-loop error dynamics is obtained as For the derivation of the update law for adjusting the estimates of parameters, the Lyapunov method is used.
We consider the quadratic Lyapunov function defined by   In view of (37), the estimated parameters are updated by the following law:  The parameters of the Li system (28) are chosen so that the system is chaotic, namely,
Figure 14 shows the time-history of the antisynchronization errors ,, e e e  ,, u u u

Conclusion
In this paper, the adaptive control method has been applied in the study of global chaos antisynchornization of identical Tigan systems [32] identical Li systems [33] and non-identical Tigan system with known parameters and the Li system with unknown parameters.For the adaptive antisynchronization of identical chaotic systems, it was assumed that the system parameters are unknown.For the adaptive anti-synchronization of different chaotic systems, it was assumed that the parameters of the master system are known, but the parameters of the slave system are unknown.Our therotical results have been fully established using the Lyapunov stability theory.Numerical simulations are also shown for the antisynchronization of identical and non-identical Tigan and Li chaotic systems to demonstrate the effectiveness of the adaptive anti-synchronization schemes derived in this paper.

Figure 1 .
Figure 1.Strange Attractor of the Tigan System are positive constants.
decay to zero exponentially as .tProof.This result is a simple consequence of the Lyapunov stability theory.
completes the proof.

Figure 3 Figure 4
Figure 3 shows the time-history of the antisynchronization errors 1 2 3. ,, e e e Figure 4 shows the time-history of the parameter estimates , , .a b c Figure 5 shows the time-history of the parameter estimation errors , , .

Figure 4 .
Figure 4. Time History of the Estimates ,, a b c

Figure 5 .Figure 6 .
Figure 5.Time History of the Estimation Errors

Figure 7 .
Figure 7. Strange Attractor of the Li System are positive constants.


decay to zero exponentially as .

Figure 10 .Figure 11 .
Figure 10.Time History of the Estimates

Figure 12 .
Figure 12.Time History of the Applied Control Inputs y y ax x u e y bx y y x x u k k k are positive constants.Now, we state and prove the following result.

Figure 13 .Figure 14 .
Figure 15 shows the time-history of the parameter estimates , , .   Figure 16 shows the time-history of the parameter estimation errors , , .e e e    Figure 17 shows the time-history of the applied control inputs 1 2 3, , .u u u