Global Chaos Synchronization of Hyperchaotic Lorenz and Hyperchaotic Chen Systems by Adaptive Control

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1 Global Chaos Synchronization of Hyperchaotic Lorenz and Hyperchaotic Chen Systems by Adaptive Control Dr. V. Sundarapandian Professor, Research and Development Centre Vel Tech Dr. RR & Dr. SR Technical University, Chennai , INDIA R. Karthikeyan Research Scholar, School of Electronics and Electrical Engineering, Singhania University, Jhunjhunu, Rajasthan- 55, INDIA and Assistant Professor, Department of Electronics and Instrumentation Engineering Vel Tech Dr. RR & Dr. SR Technical University, Avadi, Chennai , INDIA Abstract In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of identical hyperchaotic Lorenz systems (007), identical hyperchaotic Chen systems (00) and non-identical hyperchaotic Lorenz and hyperchaotic Chen systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive synchronizing schemes using the estimates of parameters for both master and slave systems. Our adaptive synchronization results 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 synchronize identical and non-identical hyperchaotic Lorenz and hyperchaotic Chen systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the hyperchaotic systems addressed in this paper. Keywords-chaos; synchronization; adaptive control; hyperchaotic Lorenz system; hyperchaotic Chen system. I. INTRODUCTION Chaotic systems are dynamical systems that are highly sensitive to initial conditions. The sensitive nature of chaotic systems is commonly called as the butterfly effect []. Since the seminal work of Pecora and Carroll [], chaos synchronization has been studied extensively in the last two decades [-7]. Chaos theory has been applied to a variety of fields like physical systems [], chemical systems [4], ecological systems [5], secure communications [6-8] etc. In the recent years, various schemes such as PC method [], OGY method [9], active control [0-], adaptive control [-4], time-delay feedback approach [5], backstepping design method [6], sampled-data feedback synchronization method [7], sliding mode control [8], etc. have been successfully applied for chaos synchronization. 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 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. In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of identical hyperchaotic Lorenz systems ([9], 007), identical hyperchaotic Chen systems ([0], 00) and nonidentical hyperchaotic Lorenz and hyperchaotic Chen systems. We assume that the parameters of the master and slave systems are unknown ISSN : Vol. No. 6 June 0 458

2 This paper has been organized as follows. In Section II, we discuss the adaptive synchronization of identical hyperchaotic Lorenz systems. In Section III, we discuss the adaptive synchronization of identical hyperchaotic Chen systems. In Section IV, we discuss the adaptive synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems. In Section V, we summarize the main results obtained in this paper. II. ADAPTIVE SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC LORENZ SYSTEMS A. Theoretical Results In this section, we discuss the adaptive synchronization of identical hyperchaotic Lorenz systems ([9], 006), where the parameters of the master and slave systems are unknown. As the master system, we consider the hyperchaotic Lorenz dynamics described by = σ ( x x) = ρ x x x4 xx = xx β x = rx x 4 where x, x, x, x4are the states and σ, β, ρ,r are unknown parameters of the system. () As the slave system, we consider the controlled hyperchaotic Lorenz dynamics described by = σ ( y y) + u = ρ y y y4 yy + u = yy β y+ u = ry y + u 4 4 where y, y, y, y4are the states and u, u, u, u4are the nonlinear controllers to be designed. () The four-dimensional system () is hyperchaotic when the parameter values are taken as σ = 0, β = 8 /, ρ = 8 and r = 0. The state orbits of the hyperchaotic Lorenz system () are shown in Fig.. ISSN : Vol. No. 6 June 0 459

3 where Fig. State Orbits of the Hyperchaotic Lorenz System The chaos synchronization error is defined by e = y x, ( i =,,,4) () i i i The error dynamics is easy obtained as = σ ( e e) + u = ρe e e y y + x x + u = βe + yy xx + u = r( y y x x ) + u Let us now define the adaptive control functions u(), t u(), t u() t and u () t 4 as u() t = σ ( e e) ke u () t = ρe + e + e + y y x x k e u () t = βe yy + xx ke u () t = r ( y y x x ) k e , σ β, ρ and r are estimates of σ, β, ρ and r respectively, and k,( i =,,,4) are positive constants. Substituting (5) into (4), the error dynamics simplifies to = ( σ σ)( e e) ke = ( ρ ρ) e ke = ( β β) e ke = ( r r )( y y x x ) k e i (6) (4) (5) ISSN : Vol. No. 6 June 0 460

4 Let us now define the parameter estimation errors as e = σ, σ e = β β, e = ρ ρ and e = r r (7) σ β ρ r Substituting (7) into (6), we obtain the error dynamics as = e ( e e ) ke σ = e e k e ρ = e e k e β = e ( y y x x ) k e 4 r 4 4 (8) 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 V ( e, e, e, e4, eσ, eβ, eρ, er) = ( e + e + e + e4 + eσ + eβ + eρ + er ) (9) R 8. which is a positive definite function on We also note that =, σ = β, = ρ and = r (0) σ β ρ r Differentiating (9) along the trajectories of (8) and using (0), we obtain V = ke ke ke k 4e4+ e σ e( e e) σ e β e β + + e ee ρ ρ + e r e4( yy xx) r In view of Eq. (), the estimated parameters are updated by the following law: σ = e ( e e ) + k e β = e + k e 5 σ 6 β ρ = ee + k e 7 r = e4( yy xx) + k8er ρ where k5, k6, k7and k8 are positive constants. () () Substituting () into (), we obtain V = ke k e k e k e k e k e k e k e () σ 6 β 7 ρ 8 r which is a negative definite function on R 8. Thus, by Lyapunov stability theory [], it is immediate that the synchronization error ei,( i =,,,4) and the parameter estimation error eσ, eβ, eρ, er decay to zero exponentially with time. Thus, it follows that the master system () and the slave system () are completely synchronized and that the parameter estimates converges to the original values of the system parameters. Hence, we have proved the following result. ISSN : Vol. No. 6 June 0 46

5 Theorem. The identical hyperchaotic Lorenz systems () and () with unknown parameters are globally and exponentially synchronized by the adaptive control law (5), where the update law for the parameter estimates is given by () and ki,( i =,,8) are positive constants. B. Numerical Results 6 For the numerical simulations, the fourth-order Runge-Kutta method with time-step h = 0 is used to solve the hyperchaotic systems () and () with the adaptive control law (5) and the parameter update law () using MATLAB. For the hyperchaotic Lorenz systems () and (), the parameter values are taken as σ = 0, β = 8 /, ρ = 8 and r = 0. Suppose that the initial values of the parameter estimates are (0) σ =, β(0) = 0, ρ(0) = 0 and r (0) = 5. The initial values of the master system () are taken as x(0) = 8, x(0) = 4, x(0) = 6 and x 4 (0) = 40. The initial values of the slave system () are taken as y (0) = 9, y (0) = 6, y (0) = 45 and y 4 (0) = 5. Fig. depicts the complete synchronization of the identical hyperchaotic systems () and (). Fig. shows that the estimated values of the parameters, viz. respectively. parameters σ = 0, β = 8 /, ρ = 8 and r = 0.,, σ β, ρ and r converge to the system Fig. Synchronization of the Identical Hyperchaotic Lorenz Systems ISSN : Vol. No. 6 June 0 46

6 Fig. Parameter Esimates σ(), t β(), t ρ (), t r() t III. ADAPTIVE SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC CHEN SYSTEMS A. Theoretical Results In this section, we discuss the adaptive synchronization of identical hyperchaotic Chen systems ([0], 008), where the parameters of the master and slave systems are unknown. As the master system, we consider the hyperchaotic Chen dynamics described by = a( x x) = 4x 0xx + cx + 4x4 = x bx = dx 4 where x, x, x, x4are the states and abcdare,,, unknown parameters of the system. (4) As the slave system, we consider the controlled hyperchaotic Chen dynamics described by = a( y y) + u = 4y 0yy+ cy + 4y4 + u = y by+ u = dy + u 4 4 where y, y, y, y4are the states and u, u, u, u4are the nonlinear controllers to be designed. (5) The four-dimensional system (4) is hyperchaotic when the parameter values are taken as ISSN : Vol. No. 6 June 0 46

7 a= 5, b=, c= and d =. The state orbits of the hyperchaotic Chen system (4) are shown in Fig. 4. where Fig. 4 State Orbits of the Hyperchaotic Chen System The chaos synchronization error is defined by e = y x i = (6) i i i, (,,,4) The error dynamics is easy obtained as = a( e e) + u = 4e 0( y y x x ) + ce + 4e + u = be + y x + u = de + u Let us now define the adaptive control functions u(), t u(), t u() t and u () t 4 as u() t = a ( e e) ke u ( t) = 4e + 0( y y x x ) ce 4e k e 4 () = + u t be y x k e u () t = de k e abcand,, d are estimates of abcand,, d respectively, and k,( i =,,,4) are positive constants. Substituting (8) into (7), the error dynamics simplifies to i (7) (8) ISSN : Vol. No. 6 June 0 464

8 = ( a a )( e e ) ke = ( c c) e ke = ( b b) e ke = ( d d ) e k e Let us now define the parameter estimation errors as e = a a, e = b b, e = c c and e = d d (0) a b c Substituting (0) into (9), we obtain the error dynamics as = e ( a e e ) ke = ece ke = ebe ke = e e k e 4 d 4 4 d 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 V ( e, e, e, e4, ea, eb, ec, ed) = ( e + e + e + e4 + ea + eb + ec + ed) () R 8. which is a positive definite function on We also note that = a, = b, = c and = d () a b c Differentiating () along the trajectories of () and using (), we obtain V = ke ke ke k4e4+ e a e( e e) a e b e b + + e c e c e + d ee 4 d In view of Eq. (4), the estimated parameters are updated by the following law: a = e( e e) + k5e b= e + k6eb c = e + k7ec d = ee + k e 4 8 where k5, k6, k7and k8 are positive constants. d a d Substituting (5) into (4), we obtain (6) V = ke k e k e k e k e k e k e k e a 6 b 7 c 8 d which is a negative definite function on R 8. Thus, by Lyapunov stability theory [], it is immediate that the synchronization error ei,( i =,,,4) and the parameter estimation error ea, eb, ec, e d decay to zero exponentially with time. (9) () (4) (5) ISSN : Vol. No. 6 June 0 465

9 Hence, we have proved the following result. Theorem. The identical hyperchaotic Chen systems (4) and (5) with unknown parameters are globally and exponentially synchronized by the adaptive control law (8), where the update law for the parameter estimates is given by (5) and,(,,8) i k i = are positive constants. B. Numerical Results 6 For the numerical simulations, the fourth-order Runge-Kutta method with time-step h = 0 is used to solve the hyperchaotic systems (4) and (5) with the adaptive control law (8) and the parameter update law (5) using MATLAB. For the hyperchaotic Chen systems (4) and (5), the parameter values are taken as a= 5, b=, c= and d =. Suppose that the initial values of the parameter estimates are a(0) =, b(0) = 8, c(0) = 0 and d (0) = 4. The initial values of the master system (4) are taken as x(0) = 6, x(0) = 5, x(0) = 4 and x 4 (0) = 0. The initial values of the slave system (5) are taken as y(0) = 9, y(0) = 6, y(0) = 5 and y 4 (0) =. Fig. 5 depicts the complete synchronization of the identical hyperchaotic systems (4) and (5).,, Fig. 6 shows that the estimated values of the parameters, viz. abcand d converge to the system respectively. parameters a= 5, b=, c= and d =, Fig. 5 Synchronization of the Identical Hyperchaotic Chen Systems ISSN : Vol. No. 6 June 0 466

10 Fig. 6 Parameter Estimates at (), bt (), ct (), dt () IV. ADAPTIVE SYNCHRONIZATION OF HYPERCHAOTIC LORENZ AND HYPERCHAOTIC CHEN SYSTEMS A. Theoretical Results In this section, we discuss the adaptive synchronization of non-identical hyperchaotic Lorenz system ([9], 006) and hyperchaotic Chen system ([0], 00). We consider the hyperchaotic Lorenz system as the master system and the hyperchaotic Chen system as the slave system and assume that the parameters of the master and slave systems are unknown. As the master system, we consider the hyperchaotic Lorenz dynamics described by = σ ( x x) = ρx x x4 xx = xx β x x = rx x 4 where x, x, x, x4are the states and σ, β, ρ,r are unknown parameters of the system. As the slave system, we consider the controlled hyperchaotic Chen dynamics described by = a( y y) + u = 4y 0yy+ cy + 4y4 + u = y by + u = dy + u 4 4 y, y, y, y are the states, abcdare,,, unknown parameters of the system and u, u, u, u 4 are the where 4 nonlinear controllers to be designed. (7) (8) ISSN : Vol. No. 6 June 0 467

11 The chaos synchronization error is defined by e = y x e = y x e = y x e = y x The error dynamics is easy obtained as = a( y y) σ ( x x) + u = 4y 0y y + cy + 4y ρx + x + x + x x + u = y by xx + β x+ u = dy rx x + u Let us now define the adaptive control functions u(), t u(), t u() t and u () t 4 as (9) (0) u() t = a( y y) + σ ( x x) ke u () t = 4y + 0y y cy 4y + ρx x x x x k e 4 4 () = + + β u t y by x x x k e u () t = dy + rx x k e where σ, β, ρ, rabc,,, and d are estimates of σ, β, ρ, rabc,,, and d respectively, and k i, ( i =,,,4) are positive constants. Substituting () into (0), the error dynamics simplifies to = ( a a)( y y) ( σ σ)( x x) ke e = ( c c ) y ( ρ ρ) x ke = ( b b) y+ ( β β) x ke = ( d d ) y ( r r ) x x k e Let us now define the parameter estimation errors as e = σ σ, e = β β, e = ρ ρ, e = r r σ β ρ e = a a, e = b b, e = c c, e = d d a b c d Substituting () into (), we obtain the error dynamics as = e ( a y y ) e ( ) σ x x ke = ec y eρ x ke = eb y + eβ x ke e = e y e x x k e 4 d r 4 4 r 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 V = ( e + e + e + e4 + eσ + eβ + eρ + er + ea + eb + ec + ed), (5) () () (4) () ISSN : Vol. No. 6 June 0 468

12 which is a positive definite function on We also note that R. = σ, = β, = ρ, = r, = a, = b, = c, = d (6) σ β ρ r a b c d Differentiating (5) along the trajectories of (4) and using (6), we obtain V = ke ke ke k4e4+ e σ e( x x) σ e β ex β + + e e ρ x ρ + e r e4xx r+ e a e( y y) a e + b ey b+ e c ey c + e d e4y d In view of Eq. (4), the estimated parameters are updated by the following law: σ = e( x x) + k5e, a σ = e( y y) + k9e β = ex + ke, b = ey + k e 6 β 0 b ρ = ex+ ke, c = ey + k e 7 ρ c r = e x x + k e, d = e y + k e 4 8 r 4 d a (7) (8) where ki,( i= 5,,) are positive constants. Substituting (8) into (7), we obtain V = ke k e k e k e k e k e k e k e k e k e k e k e (9) σ 6 β 7 ρ 8 r 9 a 0 b c d which is a negative definite function on R. Thus, by Lyapunov stability theory [], it is immediate that the synchronization error ei,( i =,,,4) and the parameter estimation errors e, e, e, e, e, e, e, e Hence, we have proved the following result. σ β ρ r a b c ddecay to zero exponentially with time. Theorem. The non-identical hyperchaotic Lorenz system (7) and hyperchaotic Chen system (8) with unknown parameters are globally and exponentially synchronized by the adaptive control law (), where the update law for the parameter estimates is given by (8) and,(,,) i k i= are positive constants. B. Numerical Results 6 For the numerical simulations, the fourth-order Runge-Kutta method with time-step h = 0 is used to solve the hyperchaotic systems (7) and (8) with the adaptive control law () and the parameter update law (8) using MATLAB. For the hyperchaotic Lorenz system (7) and hyperchaotic Chen system (8), the parameter values are taken as σ = 0, β = 8 /, ρ = 8, r = 0., a= 5, b=, c= and d =. Suppose that the initial values of the parameter estimates are σ(0) = 5, β(0) =, ρ(0) = 4, r(0) = 6, a(0) = 0, b (0) = 5, c(0) = 4 and d (0) = 8. The initial values of the master system (7) are taken as x(0) = 9, x(0) = 4, x(0) = 8 and x 4 (0) = 0. The initial values of the slave system (8) are taken as ISSN : Vol. No. 6 June 0 469

13 y(0) = 8, y(0) =, y(0) = 46 and y 4 (0) = 7. Fig. 7 depicts the complete synchronization of the non-identical hyperchaotic systems (7) and (8). Fig. 7 Synchronization of the Hyperchaotic Lorenz and Hyperchaotic Chen Systems Fig. 8 shows that the estimated values of the parameters, viz., σ, β ρ, rabc,,, system parameters σ = 0, β = 8 /, ρ = 8, r = 0., a= 5, b=, c= and d =, and d converge to the respectively. ISSN : Vol. No. 6 June 0 470

14 Fig. 8 Parameter Estimates σ(), t β(), t ρ (), t r(), t a(), t b(), t c(), t d() t V. CONCLUSIONS In this paper, we have applied adaptive control method for the global chaos synchronization of identical hyperchaotic Lorenz systems (006), identical hyperchaotic Chen systems (00) and non-identical hyperchaotic Lorenz and Chen systems with unknown parameters. The adaptive synchronization results derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive nonlinear control method is very effective and convenient to achieve global chaos synchronization for the uncertain hyperchaotic systems discussed in this paper. Numerical simulations are also shown for the synchronization of identical and non-identical uncertain hyperchaotic Lorenz and hyperchaotic Chen systems to demonstrate the effectiveness of the adaptive synchronization schemes derived in this paper. REFERENCES [] K.T. Alligood, T. Sauer and J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, New York, 997. [] L.M. Pecora and T.L. Carroll, Synchronization in chaotic systems, Physical Review Letters, vol. 64, pp. 8-84, 990. [] M. Lakshmanan and K. Murali, Nonlinear Oscillators: Controlling and Synchronization, World Scientific, Singapore, 996. [4] S.K. Han, C. Kerrer and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators, Physical Review Letters, vol. 75, pp. 90-9, 995. [5] B. Blasius, A. Huppert and L. Stone, Complex dynamics and phase synchronization in spatially extended ecological system, Nature, vol. 99, pp , 999. [6] K.M. Cuomo and A.V. Oppenheim, Circuit implementation of synchronized chaos with applications to communications, Physical Review Letters, vol. 7, pp , 99. [7] L. Kocarev and U. Parlitz, General approach for chaotic synchronization with applications to communication, Physical Review Letters, vol. 74, pp , 995. [8] Y. Tao, Chaotic secure communication systems history and new results, Telecommunication Review, vol. 9, pp , 999. [9] E. Ott, C. Grebogi and J.A. Yorke, Controlling chaos, Physical Review Letters, vol. 64, pp , 990. [0] M.C. Ho and Y.C. Hung, Synchronization of two different chaotic systems using generalized active control, Physics Letters A, vol. 0, pp , 00. ISSN : Vol. No. 6 June 0 47

15 [] L. Huang, R. Feng and M. Wang, Synchronization of chaotic systems via nonlinaer control, Physics Letters A, vol. 0, pp. 7-75, 005. [] H.K. Chen, Global chaos synchronization of new chaotic systems via nonlinear control, Chaos, Solitons and Fractals, vol., pp. 45-5, 005. [] J. Lu, X. Wu, X. Han and J. Lü, Adaptive feedback synchronization of a unified chaotic system, Physics Letters A, vol. 9, pp. 7-, 004. [4] S.H. Chen and J. Lü, Synchronization of an uncertain unified system via adaptive control, Chaos, Solitons and Fractals, vol. 4, pp , 00. [5] J.H. Park and O.M. Kwon, A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos, Solitons and Fractals, vol. 7, pp , 00. [6] X. Wu and J. Lü, Parameter identification and backstepping control of uncertain Lü system, Chaos, Solitons and Fractals, vol. 8, pp. 7-79, 00. [7] J. Zhao and J. Lü, Using sampled-data feedback control and linear feedback synchronization in a new hyperchaotic system, Chaos, Solitons and Fractals, vol. 5, pp. 76-8, 006. [8] H.T. Yau, Design of adaptive sliding mode controller for chaos synchronization with uncertainties, Chaos, Solitons and Fractals, vol., pp. 4-47, 004. [9] T. Gao, G. Chen, Z. Chen and S. Cang, The generation and circuit implementation of a new hyperchaos based upon Lorenz system, Physics Letters A, vol. 6, pp , 007. [0] J. Li-Xin, D. Hao and H. Meng, A new four-dimensional hyperchaotic Chen system and its generalized synchronization, Chinese Physics B, vol. 9, pp , 00. [] W. Hahn, The Stability of Motion, Springer, New York, 967. AUTHORS PROFILE Dr. V. Sundarapandian was born on July 5, 967 at Uttamapalayam, Theni district, Tamil Nadu, India. He obtained his D.Sc. degree in Electrical and Systems Engineering from Washington University, USA in 996 He is working as Professor (Systems and Control Engineering), Research and Development Centre at Vel Tech Dr. RR & Dr. SR Technical University, Chennai, Tamil Nadu, India. He has published graduate-level books titled, Numerical Linear Algebra and Probability, Statistics and Queueing Theory with PHI Learning Private Limited, India. He has published over 0 refereed international journal publications. He has published 90 papers in National Conferences and 45 papers in International Conferences. He is the Editor-in-Chief of International Journal of Mathematics and Scientific Computing and International Journal of Mathematical Sciences and Applications. He is an Associate Editor of International Journal on Control Theory and Applications, International Journal of Advances in Science and Technology, International Journal of Computer Information Systems, Journal of Electronics and Electrical Engineering, etc. His research interests are in the areas of Linear and Nonlinear Control Systems, Chaos Theory, Dynamical Systems and Stability Theory, Optimal Control, Operations Research, Soft Computing, Modelling and Scientific Computing, Numerical Methods, etc. He has delivered several Key Note Lectures on Nonlinear Control Systems, Chaos and Control, Scientific Modelling and Computing with SCILAB, etc. Mr. R. Karthikeyan was born on Dec., 978 at Chennai, Tamil Nadu, India. He is currently pursuing Ph.D. in the School of Electronics and Electrical Engineering, Singhania University, Rajasthan, India. He obtained M.E. degree in Embedded System Technologies from Vinayaka Missions University, Tamil Nadu, India in 007. He obtained B.E. degree in Electronics and Communications Engineering from University of Madras, India in 000. He is also working as an Assistant Professor of the Department of Electronics and Instrumentation Engineering at Vel Tech Dr. RR & Dr. SR Technical University, Avadi, Chennai, Tamil Nadu, India. He has published eight papers in refereed International Journals. He has published several papers on Embedded Control Systems, Chaos & Control in National and International Conferences. He is a reviewer for Journal of Supercomputing, IEEE ISEA, journals published by World Congress of Science and Technology, Journal of Digital Information Management, etc. His current research interests are Embedded Systems, Robotics, Communications and Control Systems. ISSN : Vol. No. 6 June 0 47