Modified Projective Synchronization of Different Hyperchaotic Systems

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1 ISSN , England, UK Journal of Informaion and Compuing Science Vol., No., 009, pp Modified Projecive Synchronizaion of Differen Hyperchaoic Sysems HongLan Zhu, +, XueBing Zhang Huaiyin Insiue of Technology,Huaian, 3003, PR China, Huaian College of Informaion Technology, Huaian, 3003, PR China. (Received May 30, 008, acceped Sepember 9, 008) Absrac. This paper presens modified projecive synchronizaion of wo differen hyperchaoic sysems using acive and adapive conrol mehod. The proposed echnique is applied o achieve chaos modified projecive synchronizaion for hyperchaoic LÜ sysem and hyperchaoic Rössler sysem. Numerical simulaions resuls are presened o demonsrae he effeciveness of he mehod. Keywords: Modified projecive synchronizaion; Acive conrol; Adapive conrol; Hyperchaoic. Inroducion Since he idea of synchronizing chaoic sysems was inroduced by Pecora and Carroll [] in 990, chaos synchronizaion has received increasing aenion due o is poenial applicaions in secure communicaion, ecological sysems, sysem idenificaion, ec. The concep of synchronizaion has been exended o he scope, such as complee synchronizaion[], generalized synchronizaion (GS)[3-6], phase synchronizaion[7-8], lag synchronizaion[9-0], and even ani-synchronizaion[-3].the generalized synchronizaion implies he esablishmen of funcional relaions beween drive and response sysems. More recenly, Li[] consider a new GS mehod, called modified projecive synchronizaion(mps), where he responses of he synchronized dynamical saes synchronize up o a consan scaling marix. Our aenion in he presen paper is o sudy modified projecive synchronizaion of differen hyperchaoic sysems.this paper is organized as follows: Secion describes he hyperchaoic LÜ sysem and hyperchaoic Rössler sysem. In Secion 3 and Secion he modified projecive synchronizaion of he hyperchaoic LÜ sysem and hyperchaoic Rössler sysem is achieved via acive and adapive conrol. In Secion 5 numerical resuls demonsrae he effeciveness of he proposed conrol echnique is presened. Finally, Secion 6 conains a summarized conclusion of he resuls.. Sysems descripion LÜ sysem as a ypical ransacion sysem, found by LÜ and Chen, which connecs he Lorenz and Chen aracors and represens he ransiion from one o he oher.recenly, Chen Ai-min ohers proposed hyperchaoic LÜ sysem[5].the hyperchaoic LÜ sysem is described by: x = ax ( x) + x x = cx xx3 x 3 = xx bx3 x = xx3+ dx The sysem is based on LÜ sysem,he firs equaion wih a conroller, he second wih hree equaion non-conroller.when a = 36, b= 3, c = 0 and d ake oher differen values,sysem performance of he () + Corresponding auhor. address: zhuhonglan@gmail.com. Published by World Academic Press, World Academic Union

2 3 HongLan Zhu, e al: Modified Projecive Synchronizaion of Differen Hyperchaoic Sysems differen dynamics :when.03 d 0.6, sysem is a periodic orbi ;when 0.6 < d 0.35,sysem is chaoic aracor ;and when 0.35 < d.30,here are wo index greaer han zero sysem is hyperchaoic aracor.the hyperchaoic aracor is shown in Fig.. The hyperchaoic Rössler sysem is sudied by Rössler OE[6].The sysem is described by: y = y y3 y = y+ ay + y y 3 = b+ yy3 y = cy 3+ dy () Where y, y, y3, y are sae variables, and a, b, c, d are real consans. When a = 0.5, b = 3, c = 0.5, d = 0.05 he sysem is hyperchaoic.the hyperchaoic aracor is shown in Fig.. Fig.:Hyperchaoic aracors for he LÜ sysem. (a) Hyperchaoic aracor in ( x, x, x 3) space; (b) hyperchaoicaracor in ( x, x3, x ) space. Fig.: Hyperchaoic aracors for he Rössler sysem. (a) Hyperchaoi aracor in ( y, y, y 3) space; (b) hyperchaoic aracor in ( y, y3, y) space. 3. Modified projecive synchronizaion of wo hyperchaoic sysems via acive conrol To observe he modified projecive synchronizaion behavior in hyperchaoic LÜ and hyperchaoic Rössler sysems, we assume ha LÜ sysem drives he Rössler sysem. Therefore, we define he maser as he hyperchaoc LÜ sysem () and slave sysems as follows. JIC for conribuion: edior@jic.org.uk

3 Journal of Informaion and Compuing Science, (009), pp y = y y3+ u() y = y+ ay + y + u() y 3 = b+ yy3+ u3() y = cy3+ dy + u() where u, u, u3, uare four conrol funcions o be designed; in order o deermine he conrol funcions o realize modified projecive synchronizaion beween sysems () and (3).Le us define he saes of he MPS errors for he slave sysem (3) ha is o be conrolled and he conrolling sysem (3) as: e = y αx e = y αx e3 = y3 α3x3 e = y α x Where α, α, α 3, α are consan.then wo chaoic sysems can be synchronized in he sense of MPS, i.e., lim e = 0, lim e = 0, lim e = 0, lim e = 0. 3 From Eq. (), we have he following error dynamics: e = e e3 αx α3x3 αa( x x) αx + u e = e + ae + e + α x + aα x + α x α cx + α x x + u e 3 = b+ ye3+ yα3x3 α3xx + α3bx3+ u3 e = ce 3 cα3x3+ de + dαx αxx3 αdx + u Then, by defining he acive conrol inpus u, u, u, u as follows: his leads o: 3 3 u = αx + α3x3+ αa( x x) + αx + v u = αx aαx αx + αcx αxx3+ v u3 = yα3x3+ α3xx α3bx3 b+ v3 u =+ cα3x3 ( d d) αx + αxx 3+ v e = e e3+ v e = e+ ae + e + v (7) e 3 = ye3+ v3 e = ce 3+ de + v The error sysem (7) o be conrolled is a linear sysem wih conrol inpu v, v, v3 and v as he funcion of he error saes e, e, e3and e As saed,as long as lim e ( ) = 0, modified projecive synchronizaion beween he driver and response sysems is realized, ha is, he hyperchaoic LÜ sysem and hyperchaoic Rössler sysem are Modified projecive synchronizaion under a acive conrol. According o he original mehod of acive conrol, v, v, v3 and v can be rewrien as v e v e = A (8) v 3 e 3 v e (3) () (5) (6) JIC for subscripion: publishing@wau.org.uk

4 36 HongLan Zhu, e al: Modified Projecive Synchronizaion of Differen Hyperchaoic Sysems Where A is a consan marix. In order o make he closed loop sysem (7) sable, he proper choice of he elemens of he marix A is such ha he sysem (7) mus have all eigenvalues wih negaive real pars. Le A λ λ a 0 0 = 0 0 λ3 y c λ d λ > 0, λ > 0, λ > 0, λ > 0. So he eigenvalues of he closed loop sysem (7) are: wih 3 λ < 0, λ < 0, λ < 0, λ < 0.Thus modified projecive synchronizaion beween hyperchaoic LÜ 3 sysem and hyperchaoic Rössler sysem is achieved.. Modified projecive synchronizaion of wo hyperchaoic sysems via adapive conrol The purpose of his secion is o inroduce a adapive conrol law for resolving he modified projecive synchronizaion beween he hyperchaoic LÜ sysem and hyperchaoic Rössler sysem wih complee uncerain sysem parameers. In order o observe he modified projecive synchronizaion behavior in he hyperchaoic LÜ sysem and hyperchaoic Rössler sysem,we assume ha hyperchaoic LÜ sysem drives he hyperchaoic Rössler sysem. Therefore, we define he maser sysem and slave sysems as follows And x = ax ( x) + x x = cx xx3 x 3 = xx bx3 x = xx3+ dx y = y y3+ u y = y+ ay + y + u y 3 = b+ yy3+ u3 y = cy3+ dy + u where u, u, u3, uare four conrol funcions o be designed; in order o deermine he conrol funcions o realize modified projecive synchronizaion beween sysems (0) and ().Le us define he saes of he MPS errors for he slave sysem () ha is o be conrolled and he conrolling sysem (3) as: e = y αx e = y αx e3 = y3 α3x3 e = y α x Where α, α, α 3, α are consan.then wo chaoic sysems can be synchronized in he sense of MPS, i.e., lim e = 0, lim e = 0, lim e = 0, lim e = 0. 3 From Eq. (), we have he following error dynamics: (9) (0) () () JIC for conribuion: edior@jic.org.uk

5 Journal of Informaion and Compuing Science, (009), pp e = e e3 αx α3x3 αa( x x) αx + u e = e+ ae + e + αx+ aαx + αx αcx + αxx3+ u (3) e 3 = b+ yy3 α3xx + α3bx3+ u3 e = ce 3 cα3x3+ de + dαx αxx3 αdx + u For sysems (0) and () wihou conrols u, u, u3, u, hen he rajecories of wo sysems will quickly separae from each oher and become irrelevan. However, when conrols are applied, he wo sysems will approach modified projecive synchronizaion for any iniial condiions hrough conrol funcions. Wih his mind, we propose he following Theorem. Theorem Le us now define he adapive conrol funcions u, u, u3, u as u = αx + α3x3+ αx + aˆ α( x x) ke u= αx αx αxx3 ae ˆ aˆαx ˆ + αcx e ke () ˆ ˆ u3 = b yy3+ α3xx bα3x3 k3e3 u ˆ ˆ ˆ ˆ ˆ = ce 3+ cα3x3 de dαx + αxx3 + dαx ke Where ki ( i=,,3,) is he conrol gains of posiive scalars, abcda ˆ, ˆ, ˆ, ˆ, ˆ, b ˆ, cˆ, d ˆ are esimaes of abcda,,,,, b, c, d respecively and he parameers adapive laws of abcda,,,,, b, c, d as below aˆ = αe( x x) ˆ b= α3x3e3 cˆ = αxe ˆ d = αxe (5) aˆ = e + αxe bˆ = e3 cˆ = e3e α3x3e ˆ d = e + αxe Then he uncerain maer sysem and response sysem is modified projecive synchronized. Proof : According o maser sysem (0) and he conrolled slave sysem (), we ge he error dynamical sysem (3) can be described by e = a α ( x x) e ke e = a ( e + αx) + c αx + e ke (6) e 3 = b b α 3x3 k3e3 e = c ( e3+ α3x3) d e d αx + d αx ke Where a = aˆ a, b = bˆ b, b = cˆ c, d = dˆ d, a ˆ ˆ = a a, b = b b, c = c c, d = d d. Consider a Lyapunov funcion as V = ( e + e + e3 + e + a + b + c + d + a + b + c + d ) (7) The differenial of he Lyapunov funcion along he rajecory of error sysem (7) is JIC for subscripion: publishing@wau.org.uk

6 38 HongLan Zhu, e al: Modified Projecive Synchronizaion of Differen Hyperchaoic Sysems 3 3 d V = e e + e e + e e + e e + aa + bb+ cc + dd + a a + b b + c c + d = e( a α( x x) e ke ) + e( a ( e + αx) c αx + e ke) (8) + e3( b b α3x3 k3e3) + e( c ( e3 + α3x3) d e d αx + d αx ke) + aa + bb + cc + dd + a a + b b + c c + d d Subsiue (), (5) ino (8), we ge T V = k e k e k e k e = e Pe (9) Where P diag{ k, k, k, k } = Since V is a posiive decreasing funcion and V is negaive semidefinie, we canno immediaely obain ha he origin of error sysem (3) is asympoically sable. In fac, asv 0, hen e, e, e, e L 3 and abcd,,,, a, b, c, d T L. From he error sysem (3), we have e, e, e 3, e L.Since V = e Pe, hen. we have λ ( P) e d e Ped Vd = V (0) V ( ) V (0) 0 min 0. Where λmin( P) is he minimum 0 eigenvalue of posiive-definie marix P. Thus e, e, e 3, e L. According o he Barbala s Lemma, we have e 0, e 0, e 0, e 0. Thus in he closed-loop sysem 3 y α x, y α x, y α x, y α x, as, i.e. lim e( ) = 0. Therefore, he slave sysem () synchronizes he maser sysem (0) in he sense of MPS. This complees he proof. 5. Simulaion In his secion, we will show a series of numerical simulaions o demonsrae he effeciveness of he proposed conrol scheme. All simulaion procedures are coded and execued using he Malab sofware. Fourh order Runge-Kua inegraion mehod is used o solve wo sysems of differenial hyperchoaic sysems.in addiion, a ime sep size 0.00is employed. 5.. simulaions for acive conrol e, e, e, e Fig.3: Dynamics of modified projecive synchronizaion errors 3 beween hyperchaoic LÜ sysem and hyperchaoic Rössler sysem wih ime via acive conrol. We assume ha he conrol gain ( λ, λ, λ 3, λ ) = (,,, ). We will selec he parameers of hyperchaoic LÜ sysem a = 0.5, b = 3, c = 0, d = and he parameers of hyperchaoic Rössler sysem as a = 0.5, b = 3, c = 0.5, d = 0.05.Therefore, boh LÜ and Rössler sysems exhibi hyperchaoic behavior. The iniial values of he maser and slave sysems are ( x(0), x(0), x3(0), x (0)) = (0.3,.5,3.,0.) and ( y(0), y(0), y3(0), y (0)) = ( 0,0,0,5). So he iniial values ofhe error sysem JIC for conribuion: edior@jic.org.uk

7 Journal of Informaion and Compuing Science, (009), pp is ( e(0), e(0), e3(0), e (0)) = ( 0.3, 0.75,.6,5.). The modified projecive synchronizaion sysems () and (3) via acive conrol law are shown in Fig simulaions for adapive conrol We assume ha conrol gains ki = ( i=,,3,). We will selec he parameers of hyperchaoic LÜ sysem a = 0.5, b = 3, c = 0, d = and he parameers of hyperchaoic Rössler sysem a = 0.5, b = 3, c = 0.5, d = 0.05.Therefore, boh he hyperchaoic LÜ sysem and hyperchaoic Rössler sysem exhibi hyperchaoic behavior. As a es for verificaion of MPS of he sysem, le us ake α =, α = 0.3, α = 0.5, α = and he iniial values of he maser and slave sysems 3 are ( x(0), x(0), x3(0), x (0)) = (0.3,.5,3.,0.) and ( y(0), y(0), y3(0), y (0)) = ( 0,0,0,5).So he iniial values of he error ( e(0), e(0), e3(0), e (0)) = ( 0.3, 0.75,.6,5.).The modified projecive synchronizaion of sysems 0) and () are shown in Fig.. Fig.: Dynamics of synchronizaion errors e, e, e, e 3 beween he maser sysem and slave sysem wih ime via adapive conrol. abcda ˆ, ˆ, ˆ, ˆ, ˆ, b ˆ, cˆ, d ˆ Fig.5: Adapive parameers esimaion errors: In Fig. 3 and Fig. we can show ha he drive sysem and response sysem realize modified projecive synchronizaion quickly. A he same ime we can see he esimae values of unknown parameers abcda,,,,, b, c, d in Fig Conclusion In his Leer, he modified projecive synchronizaion of wo differen hyperchaoic sysems has been JIC for subscripion: publishing@wau.org.uk

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