Anti-Synchronization of the Hyperchaotic Liu and Hyperchaotic Qi Systems by Active Control

Transcription

1 Ati-Sychroizatio of the Hyperchaotic Liu ad Hyperchaotic Qi Systems by Active Cotrol Dr. V. Sudarapadia Professor, Research ad Developmet Cetre Vel Tech Dr. RR & Dr. SR Techical Uiversity, Cheai , INDIA R. Karthikeya Research Scholar, School of Electroics ad Electrical Egieerig, Sighaia Uiversity, Jhujhuu, Rajastha , INDIA ad Assistat Professor, Departmet of Electroics ad Istrumetatio Egieerig Vel Tech Dr. RR & Dr. SR Techical Uiversity, Avadi, Cheai , INDIA Abstract This paper ivestigates the problem of ati-sychroizatio of idetical hyperchaotic Liu systems (008), hyperchaotic Qi systems (008) ad o-idetical hyperchaotic Liu ad hyperchaotic Qi systems usig active oliear cotrol. Sufficiet coditios for achievig ati-sychroizatio of the idetical ad differet hyperchaotic Liu ad hyperchaotic Qi systems usig active oliear cotrol are derived based o Lyapuov stability theory. Sice the Lyapuov expoets are ot required for these calculatios, the active cotrol method is very effective ad coveiet to achieve ati-sychroizatio of idetical ad differet hyperchaotic Liu ad hyperchaotic Qi systems. Numerical simulatios are show to demostrate the effectiveess of the ati-sychroizatio schemes derived i this paper. Keywords-chaos; ati-sychroizatio; active cotrol; hyperchaotic Liu system; hyperchaoticqi system. I. INTRODUCTION Chaotic systems are dyamical systems that are highly sesitive to iitial coditios. The sesitive ature of chaotic systems is commoly called as the butterfly effect []. Sice the pioeerig work of Pecora ad Carroll [], chaos sychroizatio has bee studied extesively ad itesively i the last two decades [-7]. Chaos theory has bee explored i a variety of fields icludig physical systems [3], chemical systems [4] ad ecological systems [5], secure commuicatios [6-8] etc. I the recet years, various schemes such as PC method [], OGY method [9], active cotrol [0-], adaptive cotrol [3-4], time-delay feedback approach [5], backsteppig desig method [6], sampled-data feedback sychroizatio method [7], slidig mode cotrol [8], etc. have bee successfully applied for chaos sychroizatio. Recetly, active cotrol has bee applied to ati-sychroize idetical chaotic systems [9-0] ad differet hyperchaotic systems []. I most of the chaos ati-sychroizatio approaches, the master-slave or drive-respose formalism is used. If a particular chaotic system is called the master or drive system ad aother chaotic system is called the slave or respose system, the the idea of the ati-sychroizatio is to use the output of the master system to cotrol the slave system so that the states of the slave system have the same amplitude but opposite sigs as the states of the master system asymptotically. I other words, the sum of the states of the master ad slave systems are expected to coverge to zero asymptotically whe ati-sychroizatio appears. I this paper, we derive ew results for the ati-sychroizatio of idetical Liu systems (004), idetical Che systems (999) ad o-idetical Liu ad Che chaotic systems usig the active oliear cotrol method. The stability results derived i this paper are established usig Lyapuov stability theory. This paper has bee orgaized as follows. I Sectio II, we give the problem statemet ad our methodology. I Sectio III, we discuss the chaos ati-sychroizatio of two idetical hyperchaotic Liu systems ([], 008). I Sectio IV, we discuss the chaos ati-sychroizatio of two idetical hyperchaotic Qi ISSN : Vol. 3 No. 6 Jue 0 438

2 systems ([3], 008). I Sectio V, we discuss the ati-sychroizatio of hyperchaotic Liu ad hyperchaotic Qi systems. I Sectio VI, we summarize the mai results obtaied i this paper. II. PROBLEM STATEMENT AND OUR METHODOLOGY USING ACTIVE CONTROL Cosider the chaotic system described by () x = Ax + f ( x) x R is the state of the system, A is the where matrix of the system parameters ad f : R R is the oliear part of the system. We cosider the system () as the master or drive system. As the slave or respose system, we cosider the followig chaotic system described by the dyamics () y = By+ g( y) + u where y R is the state of the system, B is the matrix of the system parameters, g : R oliear part of the system adu R is the cotroller of the slave system. If A= Bad f = g, the x ad y are the states of two idetical chaotic systems. R is the If A B or f g, the x ad y are the states of two differet chaotic systems. I the oliear feedback cotrol approach, we desig a feedback cotroller u, which ati-sychroizes the states of the master system () ad the slave system () for all iitial coditios x(0), y(0) R. If we defie the ati-sychroizatio error as e= y+ x, (3) the the error dyamics is obtaied as = By+ Ax+ g( y) + f( x) + u (4) Thus, the global chaos ati-sychroizatio problem is essetially to fid a feedback cotroller u so as to stabilize the error dyamics (4) for all iitial coditios e(0) R. Hece, we fid a feedback cotroller u so that lim et ( ) = 0 for all e(0) R. (5) t We take as a cadidate Lyapuov fuctio Ve () = epe T, (6) where P is a positive defiite matrix. Note that V : R R is a positive defiite fuctio by costructio. We assume that the parameters of the master ad slave system are kow ad that the states of both systems () ad () are measurable. If we fid a feedback cotroller u so that Ve () = eqe T, (7) where Q is a positive defiite matrix, the V : R R is a egative defiite fuctio. Thus, by Lyapuov stability theory [4], the error dyamics (4) is globally expoetially stable ad hece the coditio (5) will be satisfied. ISSN : Vol. 3 No. 6 Jue 0 439

3 Hece, the states of the master system () ad the slave system () will be globally ad expoetially atisychroized for all iitial coditios x(0), y(0) R. III. ANTI-SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC LIU SYSTEMS A. Theoretical Results I this sectio, we apply the active cotrol method for the global chaos ati-sychroizatio of idetical hyperchaotic Liu systems. The hyperchaotic Liu system is oe of the paradigms of the four-dimesioal hyperchaotic systems discovered by L Liu, C. Liu ad Y. Zhag ([], 008). Thus, the master system is described by the hyperchaotic Liu dyamics = a( x x) = bx+ xx 3 x4 = xx cx + x = dx + x 3 4 where x, x, x3, x 4 are state variables of the system ad abcdare,,, positive, costat parameters of the system. The four-dimesioal system (8) is hyperchaotic whe the parameter values are take as a= 0, b= 35, c=.4 ad d = 5. The state orbits of the hyperchaotic Liu system (8) are illustrated i Fig.. (8) Figure. State Portrait of the Hyperchaotic Liu System The slave system is described by the cotrolled hyperchaotic Liu dyamics ISSN : Vol. 3 No. 6 Jue 0 440

4 = a( y y) + u = by+ yy3 y4 + u = yy cy + y + u = dy + y + u where y, y, y3, y4are state variables ad u, u, u3, u4are the cotrollers to be desiged. (9) The ati-sychroizatio error is defied by e = y + x, ( i =,,3,4) (0) i i i The error dyamics is easily obtaied as = a( e e) + u = be e4 + yy3+ xx3+ u = ce + e y y x x + u = de + e + u We choose the active oliear cotroller as u = ae u = be e + e4 yy3 xx3 u3 = e4 + yy + xx u = de e e 4 4 Substitutig () ito (), we obtai the liear error system = ae = e 3 = ce3 = e 4 4 Next, we cosider the quadratic Lyapuov fuctio defied by T V() e = e e= ( e + e + e3 + e4), (4) Differetiatig V alog the trajectories of (3), we get (5) V() e = ae e ce e, R 4. which is a egative defiite fuctio o 4 Thus, the error dyamics (3) is globally expoetially stable for all iitial coditios e(0) R. Hece, we obtai the followig result. Theorem. The idetical hyperchaotic Liu systems (8) ad (9) are globally ad expoetially atisychroized for all iitial coditios by the active oliear cotroller defied by (). B. Numerical Results 6 For the umerical simulatios, the fourth-order Ruge-Kutta method with time-step h = 0 is used to solve the hyperchaotic Liu systems (8) ad (9) with the active cotroller u give by () usig MATLAB. I the hyperchaotic case, the parameter values are () () (3) ISSN : Vol. 3 No. 6 Jue 0 44

5 The iitial values of the master system () are take as x (0) = 36, x (0) = 4, x (0) = 5, x (0) = 8 ad the iitial values of the slave system () are take as y (0) =, y (0) = 8, y (0) = 34, y (0) = 7 Fig. illustrates the complete sychroizatio of the idetical hyperchaotic Liu systems (8) ad (9). Figure. Ati-Sychroizatio of Idetical Hyperchaotic Liu Systems IV. ANTI-SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC QI SYSTEMS A. Theoretical Results I this sectio, we apply the active cotrol method for the global chaos ati-sychroizatio of idetical hyperchaotic Qi systems. The hyperchaotic Qi system is oe of the paradigms of the four-dimesioal hyperchaotic systems discovered by G. Qi, M.A. Wyk, B.J. Wyk ad G. Che ([3], 008). Thus, the master system is described by the hyperchaotic Qi dyamics = α( x x) + xx3 = β ( x+ x) xx3 = γx εx + x x = δ x + fx + x x (6) ISSN : Vol. 3 No. 6 Jue 0 44

6 x, x, x, x are state variables of the system ad α, βγδε,,,, f where the system. The four-dimesioal system (6) is hyperchaotic whe the parameter values are take as α = 50, β = 4, γ = 3, δ = 8, ε = 33 ad f = 30. The state orbits of the hyperchaotic Qi system (6) are illustrated i Fig. 3. are positive, costat parameters of Figure 3. State Orbits of the Hyperchaotic Qi System The slave system is described by the cotrolled hyperchaotic Qi dyamics = α( y y) + yy3+ u = β ( y+ y) yy3+ u = γ y εy + y y + u = δ y + fy + y y + u where y, y, y3, y4are state variables ad u, u, u3, u4are the cotrollers to be desiged. (7) The ati-sychroizatio error is defied by e = y + x, ( i =,,3,4) (8) i i i The error dyamics is easily obtaied as = α( e e ) + y y + x x + u 3 3 = β ( e + e ) y y x x + u 3 3 = γe εe + y y + x x + u = δ e + fe + y y + x x + u We choose the active oliear cotroller as (9) ISSN : Vol. 3 No. 6 Jue 0 443

7 u = αe ( a α)( x x) yy3 u = βe ( β + ) e ( b β) x + βx + x + y y x x u3 = εe4 ( γ c) x3 ( ε + ) x4 yy + xx u = fe dx x + fx dx y y Substitutig (0) ito (9), we obtai the liear error system = αe = e 3 = γe3 e = δe 4 4 Next, we cosider the quadratic Lyapuov fuctio defied by T V() e = e e= ( e + e + e3 + e4), () Differetiatig V alog the trajectories of (), we get (3) Ve () = αe e γ e δ e, R 4. which is a egative defiite fuctio o 4 Thus, the error dyamics () is globally expoetially stable for all iitial coditios e(0) R. Hece, we obtai the followig result. Theorem. The idetical hyperchaotic Qi systems (6) ad (7) are globally ad expoetially atisychroized for all iitial coditios by the active oliear cotroller defied by (0). B. Numerical Results 6 For the umerical simulatios, the fourth-order Ruge-Kutta method with time-step h = 0 is used to solve the hyperchaotic Qi chaotic systems (6) ad (7) with the active cotroller u give by (0) usig MATLAB. Also, the parameter values are take as α = 50, β = 4, γ = 3, δ = 8, ε = 33ad f = 30. (0) () ISSN : Vol. 3 No. 6 Jue 0 444

8 Figure 4. Ati-Sychroizatio of Idetical Hyperchaotic Qi Systems The iitial values of the master system () are take as x (0) =, x (0) = 44, x (0) = 3, x (0) = 6 ad the iitial values of the slave system () are take as y (0) = 4, y (0) = 38, y (0) = 7, y (0) = 5 Fig. 4 illustrates the complete sychroizatio of the idetical hyperchaotic Qi systems () ad (). V. ANTI-SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC QI SYSTEMS A. Theoretical Results I this sectio, we apply the active cotrol method for the global chaos ati-sychroizatio of o-idetical hyperchaotic Liu ad hyperchaotic Qi systems. We take the hyperchaotic Li system ([], 008) as the master system ad the hyperchaotic Qi system ([3], 008) s the slave system. Thus, the master system is described by the hyperchaotic Liu dyamics = a( x x) = bx+ xx 3 x4 = xx cx + x = dx + x 3 4 where x, x, x3, x 4 are state variables ad abcdare,,, positive, costat parameters of the system. The slave system is described by the cotrolled hyperchaotic Qi dyamics (4) ISSN : Vol. 3 No. 6 Jue 0 445

9 = α( y y) + yy3 + u = β ( y+ y) yy3+ u = γ y εy + y y + u = δ y + fy + y y + u where y, y, y3, y4are state variables, α, βγδε,,,, f are positive, costat parameters of the system ad u, u, u3, u4are the cotrollers to be desiged. The ati-sychroizatio error is defied by e = y+ x e = y + x e = y + x e = y + x The error dyamics is easily obtaied as α α ( ) ( ) e = ( e e) + ( a )( x x) + yy3 + u e = β e+ e + b β x βx x4 yy3+ xx3 + u e = γe εe + ( γ c) x + ( ε + ) x + y y x x + u e = δe + fe + dx + x fx + δx + y y + u We choose the active oliear cotroller as u = αe ( a α)( x x) yy3 u = βe ( β + ) e ( b β) x + βx + x + y y x x u3 = εe4 ( γ c) x3 ( ε + ) x4 yy + xx u = fe dx x + fx δ x y y Substitutig (8) ito (7), we obtai the liear error system = ae = e 3 = ce3 e = e 4 4 Next, we cosider the quadratic Lyapuov fuctio defied by T V() e = e e= ( e + e + e3 + e4), (30) R 4. which is a positive defiite fuctio o Differetiatig V alog the trajectories of (9), we get V () e = ae e ce e, (3) which is a egative defiite fuctio o R 4. Thus, the error dyamics (9) is globally expoetially stable for all iitial coditios e(0) R. Hece, we obtai the followig result. (5) (6) (7) (8) (9) ISSN : Vol. 3 No. 6 Jue 0 446

10 Theorem 3. The o-idetical hyperchaotic Liu system (4) ad hyperchaotic Qi system (5) are globally ad expoetially ati-sychroized for all iitial coditios by the active oliear cotroller defied by (8). B. Numerical Results For the umerical simulatios, the fourth-order Ruge-Kutta method with time-step h = 0 is used to solve the hyperchaotic systems (4) ad (5) with the active cotroller u give by (8) usig MATLAB. For the hyperchaotic Liu system (4), the parameter values are take as a= 0, b= 35, c=.4 ad d = 5. For the hyperchaotic Qi system (5), the parameter values are take as α = 50, β = 4, γ = 3, δ = 8, ε = 33ad f = 30. The iitial values of the master system () are take as x (0) = 5, x (0) = 0, x (0) = 38, x (0) = 48 ad the iitial values of the slave system () are take as y (0) = 4, y (0) = 0, y (0) = 34, y (0) = Fig. 5 illustrates the complete sychroizatio of the o-idetical hyperchaotic Liu system (4) ad hyperchaotic Qi system (5). 6 Figure 5. Ati-Sychroizatio of No-Idetical Hyperchaotic Liu ad Hyperchaotic Qi Systems VI. CONCLUSIONS I this paper, we have applied active cotrol method for the derivatio of state feedback cotrol laws so as to achieve global chaos ati-sychroizatio of idetical hyperchaotic Liu systems (008), idetical hyperchaotic Qi systems (008) ad o-idetical hyperchaotic Liu ad Qi systems. Our ati-sychroizatio results have ISSN : Vol. 3 No. 6 Jue 0 447

11 bee proved usig Lyapuov stability theory. Sice the Lyapuov expoets are ot required for these calculatios, the active cotrol method is very effective ad coveiet to achieve global chaos atisychroizatio for the idetical ad o-idetical hyperchaotic Liu ad Qi systems. Numerical simulatios have bee show to demostrate the effectiveess of the ati-sychroizatio schemes derived i this paper. REFERENCES [] K.T. Alligood, T. Sauer ad J.A. Yorke, Chaos: A Itroductio to Dyamical Systems, Spriger, New York, 997. [] L.M. Pecora ad T.L. Carroll, Sychroizatio i chaotic systems, Physical Review Letters, vol. 64, pp. 8-84, 990. [3] M. Lakshmaa ad K. Murali, Noliear Oscillators: Cotrollig ad Sychroizatio, World Scietific, Sigapore, 996. [4] S.K. Ha, C. Kerrer ad Y. Kuramoto, Dephasig ad burstig i coupled eural oscillators, Physical Review Letters, vol. 75, pp , 995. [5] B. Blasius, A. Huppert ad L. Stoe, Complex dyamics ad phase sychroizatio i spatially exteded ecological system, Nature, vol. 399, pp , 999. [6] K.M. Cuomo ad A.V. Oppeheim, Circuit implemetatio of sychroized chaos with applicatios to commuicatios, Physical Review Letters, vol. 7, pp , 993. [7] L. Kocarev ad U. Parlitz, Geeral approach for chaotic sychroizatio with applicatios to commuicatio, Physical Review Letters, vol. 74, pp , 995. [8] Y. Tao, Chaotic secure commuicatio systems history ad ew results, Telecommuicatio Review, vol. 9, pp , 999. [9] E. Ott, C. Grebogi ad J.A. Yorke, Cotrollig chaos, Physical Review Letters, vol. 64, pp , 990. [0] M.C. Ho ad Y.C. Hug, Sychroizatio of two differet chaotic systems usig geeralized active cotrol, Physics Letters A, vol. 30, pp , 00. [] L. Huag, R. Feg ad M. Wag, Sychroizatio of chaotic systems via oliaer cotrol, Physics Letters A, vol. 30, pp. 7-75, 005. [] H.K. Che, Global chaos sychroizatio of ew chaotic systems via oliear cotrol, Chaos, Solitos ad Fractals, vol. 3, pp. 45-5, 005. [3] J. Lu, X. Wu, X. Ha ad J. Lü, Adaptive feedback sychroizatio of a uified chaotic system, Physics Letters A, vol. 39, pp , 004. [4] S.H. Che ad J. Lü, Sychroizatio of a ucertai uified system via adaptive cotrol, Chaos, Solitos ad Fractals, vol. 4, pp , 00. [5] J.H. Park ad O.M. Kwo, A ovel criterio for delayed feedback cotrol of time-delay chaotic systems, Chaos, Solitos ad Fractals, vol. 7, pp , 003. [6] X. Wu ad J. Lü, Parameter idetificatio ad backsteppig cotrol of ucertai Lü system, Chaos, Solitos ad Fractals, vol. 8, pp. 7-79, 003. [7] J. Zhao ad J. Lü, Usig sampled-data feedback cotrol ad liear feedback sychroizatio i a ew hyperchaotic system, Chaos, Solitos ad Fractals, vol. 35, pp , 006. [8] H.T. Yau, Desig of adaptive slidig mode cotroller for chaos sychroizatio with ucertaities, Chaos, Solitos ad Fractals, vol., pp , 004. [9] G.H. Li, Sychroizatio ad ati-sychroizatio of Colpitts oscillators usig active cotrol, Chaos, Solitos ad Fractals, vol. 6, pp , 005. [0] J. Hu, S. Che ad L. Che, Adaptive cotrol for ati-sychroizatio of Chua s chaotic system, Phys. Lett. A, vol. 339, pp , 005. [] X. Zhag ad H. Zhu, Ati-sychroizatio of two differet hyperchaotic systems via active ad adaptive cotrol, Iterat. J. Noliear Sciece, vol. 6, pp. 6-3, 008. [] L. Liu, C. Liu ad Y. Zhag, Aalysis of a ovel four-dimesioal hyperchaotic system, Chiese Joural of Physics, vol. 46, o. 4, pp , 008. [3] G. Qi, M.A. Wyk, B.J. Wyk ad G. Che, O a ew hyperchaotic system, Physics Letters A, vol. 37, pp. 4-36, 008. [4] W. Hah, The Stability of Motio, Spriger, New York, 967. AUTHORS PROFILE Dr. V. Sudarapadia was bor o July 5, 967 at Uttamapalayam, Thei district, Tamil Nadu, Idia. He obtaied his D.Sc. degree i Electrical ad Systems Egieerig from Washigto Uiversity, USA i 996 He is workig as Professor (Systems ad Cotrol Egieerig), Research ad Developmet Cetre at Vel Tech Dr. RR & Dr. SR Techical Uiversity, Cheai, Tamil Nadu, Idia. He has published graduate-level books titled, Numerical Liear Algebra ad Probability, Statistics ad Queueig Theory with PHI Learig Private Limited, Idia. He has published over 30 refereed iteratioal joural publicatios. He has published 90 papers i Natioal Cofereces ad 45 papers i Iteratioal Cofereces. He is the Editor-i-Chief of Iteratioal Joural of Mathematics ad Scietific Computig ad Iteratioal Joural of Mathematical Scieces ad Applicatios. He is a Associate Editor of Iteratioal Joural o Cotrol Theory ad Applicatios, Iteratioal Joural of Advaces i Sciece ad Techology, Iteratioal Joural of Computer Iformatio Systems, Joural of Electroics ad Electrical Egieerig, etc. His research iterests are i the areas of Liear ad Noliear Cotrol Systems, Chaos Theory, Dyamical Systems ad Stability Theory, Optimal Cotrol, Operatios Research, Soft Computig, Modellig ad Scietific Computig, Numerical Methods, etc. He has delivered several Key Note Lectures o Noliear Cotrol Systems, Chaos ad Cotrol, Scietific Modellig ad Computig with SCILAB, etc. Mr. R. Karthikeya was bor o Dec., 978 at Cheai, Tamil Nadu, Idia. He is curretly pursuig Ph.D. i the School of Electroics ad Electrical Egieerig, Sighaia Uiversity, Rajastha, Idia. He obtaied M.E. degree i Embedded System Techologies from Viayaka Missios Uiversity, Tamil Nadu, Idia i 007. He obtaied B.E. degree i Electroics ad Commuicatios Egieerig from Uiversity of Madras, Idia i 000. 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12 He is also workig as a Assistat Professor of the Departmet of Electroics ad Istrumetatio Egieerig at Vel Tech Dr. RR & Dr. SR Techical Uiversity, Avadi, Cheai, Tamil Nadu, Idia. He has published eight papers i refereed Iteratioal Jourals. He has published several papers o Embedded Cotrol Systems, Chaos & Cotrol i Natioal ad Iteratioal Cofereces. He is a reviewer for Joural of Supercomputig, IEEE ISEA, jourals published by World Cogress of Sciece ad Techology, Joural of Digital Iformatio Maagemet, etc. His curret research iterests are Embedded Systems, Robotics, Commuicatios ad Cotrol Systems. ISSN : Vol. 3 No. 6 Jue 0 449