Research Article Dual Synchronization of Fractional-Order Chaotic Systems via a Linear Controller

Transcription

1 The Scienific World Journal Volume 213, Aricle ID , 6 pages hp://dx.doi.org/1155/213/ Research Aricle Dual Synchronizaion of Fracional-Order Chaoic Sysems via a Linear Conroller Jian Xiao, Zhen-zhen Ma, and Ye-hong Yang College of Mahemaics and Saisics, Chongqing Universiy, Chongqing 41331, China Correspondence should be addressed o Jian Xiao; xj4448@126.com Received 11 July 213; Acceped 13 Augus 213 Academic Ediors: N. Herisanu, T. Li, Y. Xia, and Q. Xie Copyrigh 213 Jian Xiao e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. The problem of he dual synchronizaion of wo differen fracional-order chaoic sysems is sudied. By a linear conroller, we realize he dual synchronizaion of fracional-order chaoic Finally, he proposed mehod is applied for dual synchronizaion of Van der Pol-Willis sysems and Van der Pol-Duffing The numerical simulaion shows he accuracy of he heory. 1. Inroducion In recen years, he opic of chaos synchronizaion has araced increasing aenion in many fields. The resul of synchronizaion of chaoic oscillaors is used in nonlinear oscillaors 1, circui experimen 2, secre communicaion 3, and someoherfields.in199,hefirsconcepofsynchronizaion was presened by Carroll and Perora 4.And here are many mehods abou chaos synchronizaion such as Lyapunov equaion 5, Perora-Carroll PC 4andbackseppingconrol 6. All of hese mehods are amid of he synchronizaion beween one maser and one slave sysem do no consis of he synchronizaion of mulimaser sysems and mulislave Dual synchronizaion is a special circumsance in synchronizaion of chaoic oscillaors. The firs idea of muliplexing chaos using synchronizaion was invesigaed in a small map and an elecronic circui model by Tsimring andsushchikin1996in7; hen he concep of dual synchronizaion was raised by Liu and Davids in 2 in 8, which concenraes on using a scalar signal o simulaneously synchronize wo differen pairs chaoic oscillaors, ha is, he synchronizaion beween wo maser sysems and wo slave Nowadays, here are many dual synchronizaion mehods, such as in 2 Liu and Davids inroduce he dual synchronizaion of 1-D discree chaoic sysems via specific classes of piecewise-linear maps wih condiional linear coupling in 8. The dual synchronizaion beween he Lorenz and Rossler sysems by he Lyapunov sabilizaion heory is invesigaed in 9. The oupu feedback sraegy is used o sudy he dual synchronizaion of wo differen 3-D coninuous chaoic sysems in 1. Then he dual synchronizaion in modulaed ime-delayed sysems is invesigaed by designing a delay feedback conroller in 11. All of hese works are amid of he dual synchronizaion of ineger-order chaoic sysems and do no consis of he dual synchronizaion of fracional-order chaoic In his paper, a new mehod of dual synchronizaion of fracional-order chaoic sysems is proposed, by a linear conroller; he dual synchronizaion of chaos is obained. The res of his paper is organized as follows: in Secion 2, we consruc a heory frame abou he dual synchronizaion of wo differen fracional-order chaoic By a linear conroller, we obain dual synchronizaion beween wo differen fracional-order chaoic sysems in Secion 3. In Secion 4, heproposedmehodisappliedodualsynchronizaion of Van der Pol-Willis sysems and Van der Pol-Duffing sysems for evaluaing he performance of he mehod and by numerical simulaion; he resul shows ha he conroller designed by he applicaion of his mehod is effecive. Finally, conclusions are drawn in Secion 5.

2 2 The Scienific World Journal 2. Problem Analysis We define he following wo sysems as wo maser Maser 1: d α x =f, x, 1 dα Maser 2: d α y =g, y, 2 where x=x 1,x 2,...,x n T and y=y 1,y 2,...,y m T are he sae vecors of he wo maser f CR + R n, R n and g CR + R m,r m are wo known funcions. α, 1 is he order of he wo maser By a linear combinaion of he wo maser sysems saes, a signal V m is give as V m = n i=n a i x i + m j=1 b j y j =a 1,a 2,...,a n x+b 1,b 2,...,b m y =Ax+By=A B x y =CT ξ, where A=a 1,a 2,...,a n T and B=b 1,b 2,...,b m T are wo known marices, and a i, b j, i = 1,2,...,n, j = 1,2,...,m cannobezeroahesameime.sop=a T B T T is a known marix. ξ = x T y T T is a combinaion of he wo maser sysems saes. The corresponding wo slave sysems are as follows: Slav: =f, X +U1, 4 Slave 2: =g, Y +U2, 5 where X=X 1,X 2,...,X n T and Y=Y 1,Y 2,...,Y m T are he sae vecors of he wo slave sysems, U 1 =u 1 1,u1 2,..., u 1 n T and U 2 =u 2 1,u2 2,...,u2 m T are vecors of manipulaed variables, and α, 1 is he order of he wo slave Similarly, by a linear combinaion of wo slave sysems saes, asignalv s is generaed as follows: V s = n i=n a i X i + m j=1 b j Y j =a 1,a 2,...,a n X+b 1,b 2,...,b m Y =AX+BY=A B X Y =CT η, where η = X T Y T T is a combinaion of he wo slave sysems saes. 3 6 The error signal for dual synchronizaion is e=v s V m =A B X x Y y =CT η ξ. 7 The main goal is o synchronize he maser sysems and he slave sysems is equivalen o lim X x =, lim where is he Euclidian norm. 3. Dual Synchronizaion Sraegy Y y =, 8 Lemma 1. Considering he fracional-order sysem D α z = Qz, z =z, 9 where <α 1, z R n,andq R n n ;hensysem9 is sable if and only if argλ i Q απ/2, i = 1, 2,...,where argλ i Q denoes he argumen of he eigenvalue λ i of Q. Theorem 2. The dual synchronizaion of fracional-order chaoic sysems beween he maser sysems and he slave sysemsisachievedifandonlyifhefollowingcondiionsaisfies arg eig G +KCT απ 2, 1 where G is he coefficien marix of maser sysems and K is a conrol gain vecor. Proof. We can rewri and2 in he following form by defining Ψ=f g T : d α x d α y f, x = g,y, Similarly, 4 and5 canberewrienas f, X +U1 d =, 2 g, Y +U d α ξ =Ψ, ξ. 11 dα α η =Ψ,η+U, dα 12 where U = U 1 T U 2 T T, one defines U = U1 = U 2 K 1e K 2 e and E=X x Y y. Equaion 12 is ransformed ino = f, X +K 1e, 13 g, Y +K 2 e

3 The Scienific World Journal 3 so he error sysem is ransformed ino d α E = dα x dα y The error sysem is obained as d α η dα ξ = dα η dα ξ. 14 dα =Ψ,η+Ke Ψ, ξ =Ψ,η Ψ, ξ +Ke =Ψ, ξ + E Ψ, ξ +Kη ξ =Ψ, ξ + E Ψ, ξ +KC T E. 15 Using he firs-order Taylor expansion, he funcion Ψ is rewrien as Ψ, ξ + E Ψ, ξ = Ψ, ξ E+h.o. =G E+h.o., ξ 16 where h.o. denoes he higher order erms of he series. We subsiu6ino15and yield d α E =G E+h.o. +KCT E=G +KC T E+h.o.. 17 We can ransfer h7ino d α E =G +KCT E 18 according o Lemma 1, we can know ha he error sysem is asympoically sable a zero if and only if he following condiion is saisfied arg eig G +KCT απ The Example Analysis and Numerical Simulaions Example 3 dual synchronizaion of Van der Pol-Willis sysems. In he firs example, we can use he proposed mehod o achieve he dual synchronizaion of he Van der Pol sysem and he Willis sysem. Maser 1: Van der Pol sysem d α x 1 =x 1 γx 3 1 βx 2 +f 1 cos, d α x 2 =lx 1 mx 2 +n. 2 Maser 2: Willis sysem d α y 2 d α y 1 =y 2, =ay 1 +by 2 1 +cy3 1 +dy 2 +f 2 cos. Sohecorrespondingslavesysemsareasfollows: Slav: Slave 2: 2 1 =X 1 γx 3 1 βx 2 +f 1 cos +k 1 e, 2 =lx 1 mx 2 +n+k 2 e, 1 =Y 2 +k 3 e, =ay 1 +by 2 1 +cy3 1 +dy 2 +f 2 cos +k 4 e, where e=a 1 +a 2 e 2 +b 1 e 3 +b 2, =X 1 x 1, e 2 =X 2 x 2, e 3 =Y 1 y 1,and =Y 2 y 2. The G marix of he maser sysems is achieved as G = 1 3γx 2 1 β l lm 1, 24 a+2by 1 +3cy 2 1 d so he corresponding error marix are as follows: = d α d α e 2 d α e 3 d α 1 3γx 2 1 +a 1k 1 β + a 2 k 1 b 1 k 1 b 2 k 1 l+a 1 k 2 lm+a 2 k 2 b 1 k 2 b 2 k 2 a 1 k 3 a 2 k 3 b 1 k 3 1+b 2 k 3 e 2. e 3 a 1 k 4 a 2 k 4 a+2by 1 +3cy 2 1 +b 1k 4 d+b 2 k 4 25 We should choose he appropriae parameers so ha all he eigenvalues of he Jacobian marix of 25 saisfy Maignon condiion; ha is, he eigenvalues evaluaed a he equilibrium poin are saisfied: arg eig G +KCT > απ 2. 26

4 4 The Scienific World Journal e 2.2 e e 2 Figur: Error signals beween he pair of Van der Pol sysem. e 3 Figure 2: Error signals beween he pair of Willis sysem. The eigenvalue equaion of he equilibrium poin is locally asympoically sable. From wha we have discussed above, we can know ha A and B are wo known marices; he parameer K can be appropriaely seleced for saisfying he Maignon condiion. Dual synchronizaion of he Van der Pol sysem and he Willis sysem is simulaed. The sysem parameers are se o be γ=1/3, β=1, f 1 =.74, l =, m =.8, n =.7, a =.9, b=3, c= 2, d =, f 2 =, A=1,1,1, B=1,1,1,andα=1,so G +KC T = 1 x 2 1 +k k 1 k 1 k 1 + k k 2 k 2 k 2 k 3 k 3 k 3 1+k 3 k 4 k y 1 6y 2 1 +k 4 + k If 295 < k 1 < 13, k 2 =, k 3 = 1,andk 4 = 4, which saisfy 18, he eigenvalue equaion of he equilibrium poin is locally asympoically sable. We choose k 1 = 21, k 2 =, k 3 = 1,andk 4 = 4. Theiniialcondiions of he maser sysem 1 and he maser sysem 2 are aken as x 1 =, x 2 =.2 and y 1 =.2, y 2 =.3; he iniialcondiionsofheslavesysem1andheslavesysem2 are aken as X 1 =.3, X 2 =.4 and Y 1 =.5, Y 2 =.6, so he iniial condiions of he error sysem are se o be =.2, e 2 =.2, e 3 =.3, and =.3. InFigures1 and 2, wecanseehaallerror variables have converged o zero; ha is, we achieve he dual synchronizaion beween he Van der Pol and he Willis Exampl dual synchronizaion of Van der Pol and Duffing sysems. For Exampl, he dual synchronizaion of Van der Pol and Duffing sysems is invesigaed. Maser 1: Van der Pol sysem d α x 1 =x 1 γx 3 1 βx 2 +f 1 cos, d α x 2 Maser 2: Duffing sysem =lx 1 mx 2 +n. 28 d α y 1 =y 2, 29 d α y 2 =ay 1 +by 3 1 +cy 2 +f 2 cos. So he corresponding slave sysems are Slav: 1 =X 1 γx 3 1 βx 2 +f 1 cos +k 1 e, 3 2 =lx 1 mx 2 +n+k 2 e, Slave 2: 1 =Y 2 +k 3 e, 31 2 =ay 1 +by 3 1 +cy 2 +f 2 cos +k 4 e, where e=a 1 +a 2 e 2 +b 1 e 3 +b 2, =X 1 x 1, e 2 =X 2 x 2, e 3 =Y 1 y 1,and =Y 2 y 2.

5 The Scienific World Journal 5 e e 2 Figure 3: Error signals beween he pair of Van der Pol sysem. e e 3 Figur: Error signals beween he pair of Duffing sysem. The G marix of he maser sysems is achieved as 1 3γx 2 1 β l lm G = a+3by 2 1 c So he corresponding error marix are as follows: = d α d α e 2 d α e 3 d α 1 3γx 2 1 +a 1k 1 β + a 2 k 1 b 1 k 1 b 2 k 1 e 2 e. 3 l+a 1 k 2 lm+a 2 k 2 b 1 k 2 b 2 k 2 a 1 k 3 a 2 k 3 b 1 k 3 1+b 2 k 3 a 1 k 4 a 2 k 4 a+3by 2 1 +b 1k 4 c+b 2 k 4 33 We should choose he appropriae parameers so ha all he eigenvalues of he Jacobian marix of 33 saisfy Maignon condiion; ha is, he eigenvalues evaluaed a he equilibrium poin are saisfied: arg eig G +KCT > απ The eigenvalue equaion of he equilibrium poin is locally asympoically sable. Because A and B are wo known marices, he parameer K canbeappropriaelyselecedfor saisfying he Maignon condiion. Accordingowhawehavesudiedabove,parameersare se o γ=1/3, β=1, f 1 =.74, l =, m=.8, n =.7, a=1, b= 1, c = 5, f 2 =.3, A=1,1,1, B=1,1,1, and α =.98,so G +KC T 1 x 2 1 +k k 1 k 1 k 1 + k k 2 k 2 k 2 = k 3 k 3 k 3 1+k 3. k 4 k 4 1 3y 2 1 +k k 4 35 If 275 < k 1 < 117, k 2 =, k 3 = 1,andk 4 = 4, which saisfy 34, he eigenvalue equaion of he equilibrium poin is locally asympoically sable. We choose k 1 = 2, k 2 =, k 3 = 1,andk 4 = 4. Theiniialcondiions ofhemasersysem1andhemasersysem2areakenas x 1 =, x 2 =.2 and y 1 =.2, y 2 =.3, he iniialcondiionsofheslavesysem1andheslavesysem 2areakenasX 1 =.3, X 2 =.4 and Y 1 =.5, Y 2 =.6, so he iniial condiions of he error sysem are se o be =.2, e 2 =.2, e 3 =.3, and =.3. InFigures3 and 4, wecanseehaallerror variableshaveconvergedozero;hais,weachievehedual synchronizaion beween he Van der Pol and he Duffing 5. Conclusions In his work, we consruc a heory frame abou dual synchronizaion of wo differen fracional-order chaoic sysems and propose a mehod of dual synchronizaion. In addiion, his mehod is used for designing a synchronizaion conroller o

6 6 The Scienific World Journal achieve he dual synchronizaion of wo differen fracionalorder chaoic Finally, he proposed mehod is applied for dual synchronizaion of he Van der Pol-Willis sysems and he Van der Pol-Duffing The numerical simulaions proves he accuracy of he heory. Acknowledgmen ThisworkissupporedbyheFundamenalResearchFunds for he Cenral Universiies of China under Gran no. CQDXWL References 1 G. M. Mahmoud, E. E. Mahmoud, A. A. Farghaly, and S. A. Aly, Chaoic synchronizaion of wo complex nonlinear oscillaors, Chaos, Solions & Fracals,vol.42,no.5,pp ,29. 2 Z. Xu, C.-X. Liu, and T. Yao, Sudy on a new chaoic sysem wih analysis and circui experimen, Aca Physica Sinica, vol. 59,no.1,pp ,21. 3 D. Maignon, Sabiliy resuls for fracional differenial equaions wih applicaion o conrol processing, in Proceedings of he Inernaional IMACS IEEE-SMC Muliconference on Compuaional Engineering in Sysems Applicaions, vol.2,pp , Lille, France, T. Carroll and L. M. Perora, Synchronizaion of chaoic sysems, Physical Review Leers, vol. 64, pp , J.-B. Hu, Y. Han, and L.-D. Zhao, Synchronizing fracional chaoic sysems based on Lyapunov equaion, Aca Physica Sinica, vol. 57, no. 12, pp , J.-B.Hu,Y.Han,andL.-D.Zhao, Anovelsabliliyheoremfor fracional sysems and is applying in synchronizing fracional chaoic sysem based on back-sepping approach, Aca Physica Sinica,vol.58,no.4,pp ,29. 7 L. S. Tsimring and M. M. Sushchik, Muliplexing chaoic signals using synchronizaion, Physics Leers A, vol. 213, no. 3-4, pp , Y. Liu and P. Davids, Dual synchronizaion of chaos, Physical Review E,vol.61,pp ,2. 9 D. Ning, J.-A. Lu, and X. Han, Dual synchronizaion based on wo differen chaoic sysems: Lorenz sysems and Rössler sysems, Compuaional and Applied Mahemaics, vol. 26, no. 2, pp , H. Salarieh and M. Shahrokhi, Dual synchronizaion of chaoic sysems via ime-varying gain proporional feedback, Chaos, Solions & Fracals,vol.38,no.5,pp , D. Ghosh and A. R. Chowdhury, Dual-anicipaing, dual and dual-lag synchronizaion in modulaed ime-delayed sysems, Physics Leers A,vol.374,no.34,pp ,21.

7 Advances in Operaions Research hp:// Volume 214 Advances in Decision Sciences hp:// Volume 214 Applied Mahemaics Algebra hp:// hp:// Volume 214 Probabiliy and Saisics Volume 214 The Scienific World Journal hp:// hp:// Volume 214 Inernaional Differenial Equaions hp:// Volume 214 Volume 214 Submi your manuscrips a hp:// Inernaional Advances in Combinaorics hp:// Mahemaical Physics hp:// Volume 214 Complex Analysis hp:// Volume 214 Inernaional Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Mahemaics hp:// Volume 214 hp:// Volume 214 Volume 214 hp:// Volume 214 Discree Mahemaics Volume 214 hp:// Discree Dynamics in Naure and Sociey Funcion Spaces hp:// Absrac and Applied Analysis Volume 214 hp:// Volume 214 hp:// Volume 214 Inernaional Sochasic Analysis Opimizaion hp:// hp:// Volume 214 Volume 214