HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL

HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600 062, Tamil Nadu, INDIA sundarvtu@gmail.com ABSTRACT This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems. KEYWORDS Adaptive Control, Chaos, Hybrid Synchronization, Lorenz-Stenflo System, Qi System. 1. INTRODUCTION Chaotic systems are nonlinear dynamical systems that are highly sensitive to initial conditions. The sensitive nature of chaotic systems is commonly called as the butterfly effect [1]. Chaos is an interesting nonlinear phenomenon and has been extensively and intensively studied in the last two decades [1-30]. Chaos theory has been applied in many scientific disciplines such as Mathematics, Computer Science, Microbiology, Biology, Ecology, Economics, Population Dynamics and Robotics. In 1990, Pecora and Carroll [2] deployed control techniques to synchronize two identical chaotic systems and showed that it was possible for some chaotic systems to be completely synchronized. From then on, chaos synchronization has been widely explored in a variety of fields including physical systems [3], chemical systems [4-5], ecological systems [6], secure communications [7-9], etc. In most of the chaos synchronization approaches, the master-slave or drive-response formalism is used. If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of the synchronization is to use the DOI:10.5121/ijitca.2012.2102 15

output of the master system to control the slave system so that the output of the slave system tracks the output of the master system asymptotically. Since the seminal work by Pecora and Carroll [2], a variety of impressive approaches have been proposed for the synchronization of chaotic systems such as the OGY method [10], active control method [11-15], adaptive control method [16-22], sampled-data feedback synchronization method [23], time-delay feedback method [24], backstepping method [25], sliding mode control method [26-30], etc. So far, many types of synchronization phenomenon have been presented such as complete synchronization [2], phase synchronization [31], generalized synchronization [32], antisynchronization [33-34], projective synchronization [35], generalized projective synchronization [36-37], etc. Complete synchronization (CS) is characterized by the equality of state variables evolving in time, while anti-synchronization (AS) is characterized by the disappearance of the sum of relevant variables evolving in time. Projective synchronization (PS) is characterized by the fact that the master and slave systems could be synchronized up to a scaling factor, whereas in generalized projective synchronization (GPS), the responses of the synchronized dynamical states synchronize up to a constant scaling matrix α. It is easy to see that the complete synchronization (CS) and anti-synchronization (AS) are special cases of the generalized projective synchronization (GPS) where the scaling matrix α = I and α = I, respectively. In hybrid synchronization of chaotic systems [15], one part of the system is synchronized and the other part is anti-synchronized so that the complete synchronization (CS) and antisynchronization (AS) coexist in the system. The coexistence of CS and AS is highly useful in secure communication and chaotic encryption schemes. In this paper, we investigate the hybrid chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo systems ([38], 2001), identical Qi systems ([39], 2005) and nonidentical Lorenz-Stenflo and Qi systems. We deploy adaptive control method because the system parameters of the 4-D chaotic systems are assumed to be unknown. The adaptive nonlinear controllers are derived using Lyapunov stability theory for the hybrid chaos synchronization of the two 4-D chaotic systems when the system parameters are unknown. This paper is organized as follows. In Section 2, we provide a description of the 4-D chaotic systems addressed in this paper, viz. Lorenz-Stenflo systems (2001) and Qi systems (2005). In Section 3, we discuss the hybrid chaos synchronization of identical Lorenz-Stenflo systems. In Section 4, we discuss the hybrid chaos synchronization of identical Qi systems. In Section 5, we derive results for the hybrid synchronization of non-identical Lorenz-Stenflo and Qi systems. 2. SYSTEMS DESCRIPTION The Lorenz-Stenflo system ([38], 2001) is described by = α( x x ) + γ x 1 2 1 4 = x ( r x ) x 2 1 3 2 = x x β x 3 1 2 3 = x α x 4 1 4 (1) where x1, x2, x3, x 4 are the state variables and α, β, γ, r are positive constant parameters of the 16

system. The LS system (1) is chaotic when the parameter values are taken as α = 2.0, β = 0.7, γ = 1.5 and r = 26.0. The state orbits of the chaotic LS system (1) are shown in Figure 1. The Qi system ([29], 2005) is described by = a( x x ) + x x x 1 2 1 2 3 4 = b( x + x ) x x x 2 1 2 1 3 4 = cx + x x x 3 3 1 2 4 = dx + x x x 4 4 1 2 3 (2) where x1, x2, x3, x 4 are the state variables and a, b, c, d are positive constant parameters of the system. The system (2) is hyperchaotic when the parameter values are taken as a = 30, b = 10, c = 1 and d = 10. The state orbits of the chaotic Qi system (2) are shown in Figure 2. Figure 1. State Orbits of the Lorenz-Stenflo Chaotic System 17

Figure 2. State Orbits of the Qi Chaotic System 3. HYBRID SYNCHRONIZATION OF IDENTICAL LORENZ-STENFLO SYSTEMS 3.1 Theoretical Results In this section, we discuss the hybrid synchronization of identical Lorenz-Stenflo systems ([38], 2001), where the parameters of the master and slave systems are unknown. As the master system, we consider the LS dynamics described by = α( x x ) + γ x 1 2 1 4 = x ( r x ) x 2 1 3 2 = x x β x 3 1 2 3 = x α x 4 1 4 (3) where x1, x2, x3, x4 are the states and α, β, γ,r are unknown real constant parameters of the system. 18

As the slave system, we consider the controlled LS dynamics described by = α( y y ) + γ y + u 1 2 1 4 1 = y ( r y ) y + u 2 1 3 2 2 = y y β y + u 3 1 2 3 3 = y α y + u 4 1 4 4 (4) where y1, y2, y3, y4 are the states and u1, u2, u3, u4 are the nonlinear controllers to be designed. The hybrid chaos synchronization error is defined by e = y x 1 1 1 e = y + x 2 2 2 e = y x 3 3 3 e = y + x 4 4 4 (5) The error dynamics is easily obtained as = α( e e 2 x ) + γ ( e 2 x ) + u 1 2 1 2 4 4 1 = e + r( e + 2 x ) x x y y + u 2 2 1 1 1 3 1 3 2 = βe + y y x x + u 3 3 1 2 1 2 3 = ( e + 2 x ) αe + u 4 1 1 4 4 (6) Let us now define the adaptive control functions u ( t) = ˆ α ( e e 2 x ) ˆ γ ( e 2 x ) k e 1 2 1 2 4 4 1 1 u ( t) = e rˆ ( e + 2 x ) + y y + x x k e 2 2 1 1 1 3 1 3 2 2 u ( t) = ˆ βe y y + x x k e 3 3 1 2 1 2 3 3 u ( t) = e + 2x + ˆ αe k e 4 1 1 4 4 4 (7) where ˆ α, ˆ β, ˆ γ and ˆr are estimates of α, β, γ and r, respectively, and ki,( i = 1, 2,3, 4) are positive constants. Substituting (7) into (6), the error dynamics simplifies to = ( α ˆ α )( e e 2 x ) + ( γ ˆ γ )( e 2 x ) k e 1 2 1 2 4 4 1 1 = ( r rˆ )( e + 2 x ) k e 2 1 1 2 2 = ( β ˆ β ) e k e 3 3 3 3 = ( α ˆ α) e k e 4 4 4 4 (8) Let us now define the parameter estimation errors as e = α ˆ α, e = β ˆ β, e = γ ˆ γ and e = r rˆ. (9) α β γ r 19

Substituting (9) into (8), we obtain the error dynamics as = e ( e e 2 x ) + e ( e 2 x ) k e 1 α 2 1 2 γ 4 4 1 1 = e ( e + 2 x ) k e 2 r 1 1 2 2 = e e k e 3 β 3 3 3 = e e k e 4 α 4 4 4 (10) For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov approach is used. We consider the quadratic Lyapunov function defined by 1 2 2 2 2 2 2 2 2 V ( e1, e2, e3, e4, eα, eβ, eγ, er ) = ( e1 + e2 + e3 + e4 + eα + eβ + eγ + er ) (11) 2 which is a positive definite function on We also note that R 8. ˆ ˆ& = & α, = β, = ˆ& γ and = rˆ & (12) α β γ Differentiating (11) along the trajectories of (10) and using (12), we obtain r 2 2 2 2 2 2 V& = k1e1 k2e2 k3e3 k4e ˆ ˆ 4 + e α e1 ( e2 e1 2 x2 ) e4 & α e & β e3 β + + e [ e1 ( e4 2 x4) ] e ˆ γ & γ + r e2 ( e1 + 2 x1 ) r& (13) In view of Eq. (13), the estimated parameters are updated by the following law: ˆ & α = e ( e e 2 x ) e + k e & ˆ β = e + k e 2 1 2 1 2 4 5 α 2 3 6 β & ˆ γ = e ( e 2 x ) + k e 1 4 4 7 r& ˆ = e ( e + 2 x ) + k er 2 1 1 8 γ (14) where k5, k6, k7 and k8 are positive constants. Substituting (14) into (12), we obtain V& = k e k e k e k e k e k e k e k e (15) 2 2 2 2 2 2 2 2 1 1 2 2 3 3 4 4 5 α 6 β 7 γ 8 r which is a negative definite function on R 8. 20

Thus, by Lyapunov stability theory [40], it is immediate that the hybrid synchronization error ei,( i = 1,2,3,4) and the parameter estimation error eα, eβ, eγ, er decay to zero exponentially with time. Hence, we have proved the following result. Theorem 1. The identical hyperchaotic Lorenz-Stenflo systems (3) and (4) with unknown parameters are globally and exponentially hybrid synchronized via the adaptive control law (7), where the update law for the parameter estimates is given by (14) and ki,( i = 1, 2, K,8) are positive constants. 3.2 Numerical Results 6 For the numerical simulations, the fourth-order Runge-Kutta method with time-step h = 10 is used to solve the 4-D chaotic systems (3) and (4) with the adaptive control law (14) and the parameter update law (14) using MATLAB. We take k i = 4 for i = 1, 2, K,8. For the Lorenz-Stenflo systems (3) and (4), the parameter values are taken as α = 2.0, β = 0.7, γ = 1.5 and r = 26.0. Suppose that the initial values of the parameter estimates are aˆ (0) = 6, bˆ (0) = 5, rˆ (0) = 2, dˆ (0) = 7. The initial values of the master system (3) are taken as x (0) = 4, x (0) = 10, x (0) = 15, x (0) = 19. 1 2 3 4 The initial values of the slave system (4) are taken as y (0) = 21, y (0) = 6, y (0) = 3, y (0) = 27. 1 2 3 4 Figure 3 depicts the hybrid-synchronization of the identical Lorenz-Stenflo systems (3) and (4). It may also be noted that the odd states of the two systems are completely synchronized, while the even states of the two systems are anti-synchronized. Figure 4 shows that the estimated values of the parameters, viz. ˆ α, ˆ β, ˆ γ and ˆr converge to the system parameters α = 2.0, β = 0.7, γ = 1.5 and r = 26.0. 21

Figure 3. Hybrid-Synchronization of Lorenz-Stenflo Chaotic Systems Figure 4. Parameter Estimates ˆ α( t), ˆ β ( t), ˆ γ ( t), rˆ ( t) 22

4. HYBRID SYNCHRONIZATION OF IDENTICAL QI SYSTEMS 4.1 Theoretical Results In this section, we discuss the hybrid synchronization of identical Qi systems ([39], 2005), where the parameters of the master and slave systems are unknown. As the master system, we consider the Qi dynamics described by = a( x x ) + x x x 1 2 1 2 3 4 = b( x + x ) x x x 2 1 2 1 3 4 = cx + x x x 3 3 1 2 4 = dx + x x x 4 4 1 2 3 (16) where x1, x2, x3, x4are the states and a, b, c, d are unknown real constant parameters of the system. As the slave system, we consider the controlled Qi dynamics described by = a( y y ) + y y y + u 1 2 1 2 3 4 1 = b( y + y ) y y y + u 2 1 2 1 3 4 2 = cy + y y y + u 3 3 1 2 4 3 = dy + y y y + u 4 4 1 2 3 4 (17) where y1, y2, y3, y4 are the states and u1, u2, u3, u4 are the nonlinear controllers to be designed. The hybrid synchronization error is defined by e = y x 1 1 1 e = y + x 2 2 2 e = y x 3 3 3 e = y + x 4 4 4 (18) The error dynamics is easily obtained as = a( e e 2 x ) + y y y x x x + u 1 2 1 2 2 3 4 2 3 4 1 = b( e + e + 2 x ) y y y x x x + u 2 1 2 1 1 3 4 1 3 4 2 = ce + y y y x x x + u 3 3 1 2 4 1 2 4 3 = de + y y y + x x x + u 4 4 1 2 3 1 2 3 4 (19) 23

Let us now define the adaptive control functions u ( t) = aˆ ( e e 2 x ) y y y + x x x k e 1 2 1 2 2 3 4 2 3 4 1 1 u ( t) = bˆ ( e + e + 2 x ) + y y y + x x x k e 2 1 2 1 1 3 4 1 3 4 2 2 u ( t) = ce ˆ y y y + x x x k e 3 3 1 2 4 1 2 4 3 3 u ( t) = de ˆ y y y x x x k e 4 4 1 2 3 1 2 3 4 4 (20) where aˆ, bˆ, cˆ and ˆd are estimates of a, b, c and d, respectively, and ki,( i = 1,2,3,4) are positive constants. Substituting (20) into (19), the error dynamics simplifies to = ( a aˆ )( e e 2 x ) k e 1 2 1 2 1 1 = ( b bˆ )( e + e + 2 x ) k e 2 1 2 1 2 2 = ( c cˆ ) e k e 3 3 3 3 = ( d dˆ ) e k e 4 4 4 4 (21) Let us now define the parameter estimation errors as e = a aˆ, e = b bˆ, e = c cˆ and e = d dˆ. (22) a b c Substituting (22) into (21), we obtain the error dynamics as d = e ( e e 2 x ) k e 1 a 2 1 2 1 1 = e ( e + e + 2 x ) k e 2 b 1 2 1 2 2 = e e k e 3 c 3 3 3 = e e k e 4 d 4 4 4 (23) For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov approach is used. We consider the quadratic Lyapunov function defined by 1 2 2 2 2 2 2 2 2 V ( e1, e2, e3, e4, ea, eb, ec, ed ) = ( e1 + e2 + e3 + e4 + ea + eb + ec + ed ) (24) 2 which is a positive definite function on We also note that R 8. ˆ ˆ& = a&, = b, = cˆ & and = dˆ & (25) a b c d 24

Differentiating (24) along the trajectories of (23) and using (25), we obtain 2 2 2 2 V& = k1e1 k2e2 k3e3 k ˆ 4e4 + e a e1 ( e2 e1 2 x2) a& ˆ 2 2 e & + ˆ ˆ b e2 ( e1 + e2 + 2 x1 ) b + e c e3 c& e & + d e4 d (26) In view of Eq. (26), the estimated parameters are updated by the following law: aˆ & = e1 ( e2 e1 2 x2) + k5e ˆ & b = e2 ( e1 + e2 + 2 x1 ) + k6e 2 c& ˆ = e3 + k7ec & dˆ = e + k e 2 4 8 d a b (27) where k5, k6, k7 and k8 are positive constants. Substituting (27) into (26), we obtain V& = k e k e k e k e k e k e k e k e (28) 2 2 2 2 2 2 2 2 1 1 2 2 3 3 4 4 5 a 6 b 7 c 8 d which is a negative definite function on R 8. Thus, by Lyapunov stability theory [40], it is immediate that the synchronization error ei,( i = 1,2,3,4) and the parameter estimation error ea, eb, ec, ed decay to zero exponentially with time. Hence, we have proved the following result. Theorem 2. The identical Qi systems (16) and (17) with unknown parameters are globally and exponentially hybrid synchronized by the adaptive control law (20), where the update law for the parameter estimates is given by (27) and ki,( i = 1,2, K,8) are positive constants. 4.2 Numerical Results 6 For the numerical simulations, the fourth-order Runge-Kutta method with time-step h = 10 is used to solve the chaotic systems (16) and (17) with the adaptive control law (14) and the parameter update law (27) using MATLAB. We take k i = 4 for i = 1, 2, K,8. For the Qi systems (16) and (17), the parameter values are taken as a = 30, b = 10, c = 1, d = 10. Suppose that the initial values of the parameter estimates are aˆ (0) = 6, bˆ (0) = 2, cˆ (0) = 9, dˆ (0) = 5 25

The initial values of the master system (16) are taken as x (0) = 4, x (0) = 8, x (0) = 17, x (0) = 20. 1 2 3 4 The initial values of the slave system (17) are taken as y (0) = 10, y (0) = 21, y (0) = 7, y (0) = 15. 1 2 3 4 Figure 5 depicts the complete synchronization of the identical Qi systems (16) and (17). Figure 6 shows that the estimated values of the parameters, viz. system parameters aˆ, bˆ, cˆ and ˆd converge to the a = 30, b = 10, c = 1, d = 10. Figure 5. Anti-Synchronization of the Qi Systems 26

Figure 6. Parameter Estimates aˆ ( t), bˆ ( t), cˆ ( t), dˆ ( t ) 5. HYBRID SYNCHRONIZATION OF NON-IDENTICAL LORENZ-STENFLO AND QI SYSTEMS 5.1 Theoretical Results In this section, we discuss the adaptive synchronization of non-identical Lorenz-Stenflo system ([38], 2001) and Qi system ([39], 2005), where the parameters of the master and slave systems are unknown. As the master system, we consider the LS dynamics described by = α( x x ) + γ x 1 2 1 4 = x ( r x ) x 2 1 3 2 = x x β x 3 1 2 3 = x α x 4 1 4 (29) where x1, x2, x3, x4 are the states and α, β, γ,r are unknown real constant parameters of the system. 27

As the slave system, we consider the controlled Qi dynamics described by = a( y y ) + y y y + u 1 2 1 2 3 4 1 = b( y + y ) y y y + u 2 1 2 1 3 4 2 = cy + y y y + u 3 3 1 2 4 3 = dy + y y y + u 4 4 1 2 3 4 (30) where y1, y2, y3, y4 are the states, a, b, c, d are unknown real constant parameters of the system u, u, u, u are the nonlinear controllers to be designed. and 1 2 3 4 The hybrid chaos synchronization error is defined by e = y x 1 1 1 e = y + x 2 2 2 e = y x 3 3 3 e = y + x 4 4 4 (31) The error dynamics is easily obtained as = a( y y ) α( x x ) γ x + y y y + u 1 2 1 2 1 4 2 3 4 1 = b( y + y ) x + rx x x y y y + u 2 1 2 2 1 1 3 1 3 4 2 = cy + β x + y y y x x + u 3 3 3 1 2 4 1 2 3 = dy α x x + y y y + u 4 4 4 1 1 2 3 4 (32) Let us now define the adaptive control functions u ( t) = aˆ ( y y ) + ˆ α( x x ) + ˆ γ x y y y k e 1 2 1 2 1 4 2 3 4 1 1 u ( t) = bˆ ( y + y ) + x rx ˆ + x x + y y y k e 2 1 2 2 1 1 3 1 3 4 2 2 u ( t) = cy ˆ ˆ β x y y y + x x k e 3 3 3 1 2 4 1 2 3 3 u ( t) = dy ˆ + ˆ α x + x y y y k e 4 4 4 1 1 2 3 4 4 (33) where aˆ, bˆ, cˆ, dˆ, ˆ α, ˆ β, ˆ γ and ˆr are estimates of a, b, c, d, α, β, γ and r, respectively, and ki,( i = 1,2,3,4) are positive constants. Substituting (33) into (32), the error dynamics simplifies to = ( a aˆ )( y y ) ( α ˆ α )( x x ) ( γ ˆ γ ) x k e 1 2 1 2 1 4 1 1 = ( b bˆ )( y + y ) + ( r rˆ ) x k e 2 1 2 1 2 2 = ( c cˆ ) y + ( β ˆ β ) x k e 3 3 3 3 3 = ( d dˆ ) y ( α ˆ α ) x k e 4 4 4 4 4 (34) 28

Let us now define the parameter estimation errors as e = a aˆ, e = b bˆ, e = c cˆ, e = d dˆ a b c d e = α ˆ α, e = β ˆ β, e = γ ˆ γ, e = r rˆ α β γ r (35) Substituting (35) into (32), we obtain the error dynamics as = e ( y y ) e ( x x ) e x k e 1 a 2 1 α 2 1 γ 4 1 1 = e ( y + y ) + e x k e 2 b 1 2 r 1 2 2 = e y + e x k e 3 c 3 β 3 3 3 = e y e x k e 4 d 4 α 4 4 4 (36) For the derivation of the update law for adjusting the estimates of the parameters, the Lyapunov approach is used. We consider the quadratic Lyapunov function defined by 1 2 2 2 2 2 2 2 2 2 2 2 2 V = ( e1 + e2 + e3 + e4 + ea + eb + ec + ed + eα + eβ + eγ + er ) (37) 2 which is a positive definite function on We also note that R 12. = aˆ &, = bˆ &, = cˆ &, = dˆ &, = ˆ& α, = ˆ & β, = ˆ& γ, = rˆ & (38) a b c d α β γ r Differentiating (37) along the trajectories of (36) and using (38), we obtain 2 2 2 2 V& = k1e1 k2e2 k3e3 k4e4 + e ˆ ˆ a e1 ( y2 y1 ) a& e & b e2 ( y1 y2) b + + + e ˆ ˆ ˆ c e3 y3 c& e & + d e4 y4 d + e α e1 ( x2 x1 ) e4x4 & α e e ˆ& + ˆ ˆ β 3x3 β + e γ e1 x4 & γ + e r e2x1 r& (39) In view of Eq. (39), the estimated parameters are updated by the following law: a& ˆ = e ( y y ) + k e, & ˆ α = e ( x x ) e x + k e & bˆ & = e ( y + y ) + k e, ˆ β = e x + k e 1 2 1 5 a 1 2 1 4 4 9 α 2 1 2 6 b 3 3 10 β c& ˆ = e y + k e, & ˆ γ = e x + k e 3 3 7 c 1 4 11 γ & dˆ = e y + k e, r& ˆ = e x + k e 4 4 8 d 2 1 12 r (40) where k5, k6, k7, k8, k9, k10, k11 and k12 are positive constants. 29

Substituting (40) into (39), we obtain & 2 2 2 2 2 2 = 1 1 2 2 3 3 4 4 k5ea k6eb V k e k e k e k e k e k e 2 2 7 c 8 d k e k e 2 2 9 α 10 β k e k e (41) 2 2 11 γ 12 r which is a negative definite function on R 12. Thus, by Lyapunov stability theory [30], it is immediate that the synchronization error ei,( i = 1,2,3,4) and the parameter estimation error decay to zero exponentially with time. Hence, we have proved the following result. Theorem 3. The non-identical Lorenz-Stenflo system (29) and Qi system (30) with unknown parameters are globally and exponentially hybrid synchronized by the adaptive control law (33), where the update law for the parameter estimates is given by (40) and ki,( i = 1,2, K,12) are positive constants. 5.2 Numerical Results 6 For the numerical simulations, the fourth-order Runge-Kutta method with time-step h = 10 is used to solve the chaotic systems (29) and (30) with the adaptive control law (27) and the parameter update law (40) using MATLAB. We take k i = 4 for i = 1, 2, K,12. For the Lorenz-Stenflo system (29) and Qi system (30), the parameter values are taken as a = 30, b = 10, c = 1, d = 10, α = 2.0, β = 0.7, γ = 1.5, r = 26. (42) Suppose that the initial values of the parameter estimates are aˆ (0) = 2, bˆ (0) = 6, cˆ (0) = 1, dˆ (0) = 4, ˆ α (0) = 5, ˆ β (0) = 8, ˆ γ (0) = 7, rˆ (0) = 11 The initial values of the master system (29) are taken as x (0) = 6, x (0) = 15, x (0) = 20, x (0) = 9. 1 2 3 4 The initial values of the slave system (30) are taken as y (0) = 20, y (0) = 18, y (0) = 14, y (0) = 27. 1 2 3 4 Figure 7 depicts the complete synchronization of the non-identical Lorenz-Stenflo and Qi systems. Figure 8 shows that the estimated values of the parameters, viz. aˆ, bˆ, cˆ, dˆ, ˆ α, ˆ β, ˆ γ and ˆr converge to the original values of the parameters given in (42). 30

Figure 7. Complete Synchronization of Lorenz-Stenflo and Qi Systems Figure 8. Parameter Estimates aˆ ( t), bˆ ( t), cˆ ( t), dˆ ( t), ˆ α ( t), ˆ β ( t), ˆ γ ( t), rˆ ( t) 31

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[35] Qiang, J. (2007) Projective synchronization of a new hyperchaotic Lorenz system, Phys. Lett. A, Vol. 370, pp 40-45. [36] Jian-Ping, Y. & Chang-Pin, L. (2006) Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system, J. Shanghai Univ., Vol. 10, pp 299-304. [37] Li, R.H., Xu, W. & Li, S. (2007) Adaptive generalized projective synchronization in different chaotic systems based on parameter identification, Phys. Lett. A, Vol. 367, pp 199-206. [38] Stenflo, L. (2001) Generalized Lorenz equation for acoustic waves in the atmosphere, Physica Scripta, Vol. 53, No. 1, pp 177-180. [39] Qi, G., Chen, Z., Du, S. & Yuan, Z. (2005) On a four-dimensional chaotic system, Chaos, Solitons and Fractals, Vol. 23, pp 1671-1682. [40] Hahn, W. (1967) The Stability of Motion, Springer, New York. Author Dr. V. Sundarapandian is a Professor (Systems and Control Engineering), Research and Development Centre at Vel Tech Dr. RR & Dr. SR Technical University, Chennai, India. His current research areas are: Linear and Nonlinear Control Systems, Chaos Theory, Dynamical Systems and Stability Theory, Soft Computing, Operations Research, Numerical Analysis and Scientific Computing, Population Biology, etc. He has published over 175 research articles in international journals and two text-books with Prentice-Hall of India, New Delhi, India. He has published over 50 papers in International Conferences and 100 papers in National Conferences. He is the Editor-in-Chief of the AIRCC control journals International Journal of Instrumentation and Control Systems, International Journal of Control Theory and Computer Modeling, International Journal of Information Technology, Control and Automation. He is an Associate Editor of the journals International Journal of Information Sciences and Techniques, International Journal of Control Theory and Applications, International Journal of Computer Information Systems, International Journal of Advances in Science and Technology. He has delivered several Key Note Lectures on Control Systems, Chaos Theory, Scientific Computing, MATLAB, SCILAB, etc. 34