Generalized Projective Synchronization Using Nonlinear Control Method

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1 ISSN (prin), (online) Inernionl Journl of Nonliner Siene Vol.8(9) No.,pp Generlized Projeive Synhronizion Using Nonliner Conrol Mehod Xin Li Deprmen of Mhemis, Chngshu Insiue of Tehnology Chngshu 55, Jingsu, P.R.Chin (Reeived My 9, eped July 9) Asr:Bsed on symoli ompuion sysem M ple nd Lypunov siliy heory, he generlized projeive synhronizion prolem of drive-response sysems is invesiged. The generlized projeive synhronizion of wo idenil hoi sysems is hieved vi he nonliner onrol, nd he synhronizion of wo differen hoi sysems is lso hieved vi he orresponding nonliner onrol. To illusre our resuls, numeril simulions re used o perform he proess of he synhronizion. The oris of drive sysems nd oris of he response sysems re pu in he sme plo for undersnding inuiively. Keywords:generlized projeive synhronizion; Lypunov heory; hoi sysems Inroduion Sine Peor nd Crroll [] disovered he synhronizion of he hoi sysem, he synhronizion prolem in hoi sysems hs een inensively nd exensively sudied in reen dedes. I hs een exensively exploied in vrious res, suh s seure ommuniions, life siene, nd so on. Up o now, here exis mny ypes of hos synhronizion in dynmil sysems suh s omplee synhronizion, pril synhronizion, phse synhronizion, lg synhronizion, niiped synhronizion, generlized synhronizion, e[-8]. In priulr, mongs ll kinds of hos synhronizion, projeive synhronizion repored y Minieri nd Rehek [9] is one of he mos noiele ones h he drive nd response veors evolve in proporionl sle he veors eome proporionl. The erly projeive synhronizion is usully oservle only in lss of sysems wih pril-lineriy [], reenly some reserhers [-] hve hieved onrol of he projeive synhronizion in more generl lss of hoi sysems inluding non-prilly-liner sysems. In he re of generlized synhronizion, Yng nd Chu [5] relized generlized synhronizion sed on he liner rnsform mehod. Reenly, Meng nd Wng [] proposed new generlized synhronizion lgorihm sed on nonliner onrol, whih expnded he ppliion rnge of generlized synhronizion. In his pper, he generlized projeive synhronizion of wo idenil hoi sysems nd wo differen hoi sysems is onsidered respeively. Using he nonliner onrol heory, novel generlized projeive synhronizion lgorihm is proposed. Two idenil Rössler sysems re invesiged o hieve he generlized projeive synhronizion y he lgorihm presened y us. For illusre he effeiveness of he lgorihm o wo differen hoi sysem, he hu irle nd he disk dynmo model re hosen o mke hem reh he generlized projeive synhronizion. Numeril simulions re used o perform he proess of he synhronizion nd we suessfully pu he oris of drive sysems nd oris of he response sysems in he sme plo for undersnding inuiively. The res of his pper is orgnized s follows: Generlized projeive synhronizion sheme vi nonliner onrol is given in seion ; In seion 3, generlized projeive synhronizion of wo idenil E-mil ddress: lovelixin@3.om Copyrigh World Ademi Press, World Ademi Union IJNS.9.8.5/8

2 8 Inernionl Journl of Nonliner Siene,Vol.8(9),No.,pp Rössler sysems is relized vi nonliner onrol; In seion, generlized projeive synhronizion eween he hu irle nd he disk dynmo model is oined; Finlly, some summry nd onlusions re given in seion 5. Generlized projeive synhronizion sheme vi nonliner onrol Firsly, we quoe some noions whih re used hroughou his pper: For veor x, x = (x T x) /, where x T denoes he rnspose of he veor x. For mrix A, le A indie he norm of A indued y he Euliden veor norm, i.e., A = (λ mx (A T A)) /, where λ( ) represens he mximum eigenvlue of mrix. Consider he following drive hoi sysem: ẋ = f(x), (.) where x = (x, x,..., x n ) T R n is he se veor of drive sysem, f( ) is oninuous veor funion. The onrolled response sysem is given y he following equion: ẏ = Ay + Bg(y) + u, (.) where y = (y, y,..., y m ) T R m is he se veor of response sysem, A nd B re sysem mries wih proper dimensions, g( ) is oninuous veor funion, nd u is he onroller. Usully, funion g( ) is glolly Lipshiz oninuous. Definiion Define he synhronizion errors of sysems (.) nd (.) s if he errors sisfy he following propery, e() = x my, (.3) lim e = lim x my = where m is nonzero onsn, hen we sy h here exis e he generlized projeive synhronizion eween sysems (.) nd (.), nd ll m sling for. Lemm (Brl lemm [7]) If f() is uniformly oninuous, nd lim hen f() when. f(τ) dτ is ounded, Theorem For g(y) of he response hoi sysem (.) sisfy he Lipshiz oninuous ondiion, if he onroller u is designed s u = m [f(x) Ax + ɛ(x my) Bg(x)] + B[g(my) mg(y)], (.) m where ɛ = dig(ɛ, ɛ,..., ɛ m ), nd sisfies min(ɛ i ) > (L B + A ); (.5) hen he generlized projeive synhronizion of sysems (.) nd (.) will e oined. Proof. The synhronizion errors re defined s Eq.(.3) e() = x my, hen wih Eq.(.), he error dynmil sysem n e desried s ė = ẋ mẏ = Ae ɛe + B[g(x) g(my)]. (.) IJNS emil for onriuion: edior@nonlinersiene.org.uk

3 X. Li: Generlized Projeive Synhronizion Using Nonliner Conrol Mehod 8 Le Lypunov error funion V () = et e = e. (.7) i is esy o know h V () us non-negive funion. Evluing he ime derivive of V () long he rjeory of Eq.(.7) gives V () = e T Ae e T ɛe + e T B[g(x) g(my)] A e min(ɛ i ) e + e B g(x) g(my) A e min(ɛ i ) e +L B e = ( A +L B min(ɛ i )) e. (.8) I is ovious h V (). Therefore, V () is uniformly oninuous. Le K = min(ɛ i ) (L B + A ), hen V () K e, where K >. Thus, he following n e drwn: V () V ()e K. (.9) From Eq.(.9), we n know h lim V ()d is ounded. Moreover, V () is uniformly oninuous. Aording o he Brl lemm, we n ge lim V () =. Nmely, lim e() =. Therefore, he error sysem (.) is sympoilly sle. Nmely, he drive sysem (.) nd he response sysem (.) n sympoilly hieve he generlized projeive synhronizion. This omplees he proof of he heorem. Remrk 3 In pper [], he generlized synhronizion vi nonliner onrol is invesiged. Here we pply he nonliner onrol mehod o he generlized projeive synhronizion. To relized he generlized projeive synhronizion of hoi sysem, nonliner onrol lgorism sheme is proposed in his pper. For he hrer of generlized projeive synhronizion, we suessfully simule he oris of oh drive sysem nd response sysem in he sme plo o oserve inuiively. In he following, o demonsre he vlidiy of he proposed sheme, he sheme is used o hieve he synhronizion of wo idenil nd differen hoi sysems respeively. 3 Generlized projeive synhronizion of wo idenil Rössler sysems vi nonliner onrol Consider he Rössler sysem ẋ () = x () x 3 (), ẋ () = x () + αx (), ẋ 3 () = β + x 3 ()(x () γ), nd he onrolled response sysem is defined s following ẏ () = y () y 3 () + u (), ẏ () = y () + αy () + u (), ẏ 3 () = β + y 3 ()(y () γ) + u 3 (), (3.) (3.) where u = (u, u, u 3 ) T is he onroller, α = β =., nd γ = 5.7. Rewrie sysem (3.) in he form of Eq. (.), where A =., B =, g(y) =. 5.7 y y 3 +. We n esily ge h A = nd B =. Here we hoose L = nd sele ɛ s ɛ =. 8 IJNS homepge:hp://

4 8 Inernionl Journl of Nonliner Siene,Vol.8(9),No.,pp I is esy o verify h Eq.(.5) is sisfied. Th is o sy Eq.(.9) is sisfied for he error funion e(). Bsed on he Brl lemm, we n ge h he wo idenil Rössler sysems oin generlized projeive synhronizion. The onroller u n e go from Eq.(.). Here we le m = nd hoose he iniil vlues of he drive sysem nd he response sysem s (x (), x (), x 3 ()) = (.5,., 3.) nd (y (), y (), y 3 ()) = (.,.5,.), respeively. Fig. shows he numeril simulion of he error e of he wo idenil Rössler sysems. Oviously, e, e nd e 3 onverge o zero finlly fer he onroller is ived. Fig. revels he numeril glol synhronizion eween hem wih differen iniil vlues s menioned ove. Fig.3 gives ou he simulion ori of he vriles of he drive sysem, nd he simulion oris of he response sysem fer synhronizion. From he Fig.3, we n esily oserve he rio of he mpliudes of he wo sysems ends o onsn sling for m...8 e e e Fig. : () denoes he ori of error funion e ; () denoes he ori of error funion e ; () denoes he ori of error funion e 3. 5 x3(y3) x(y) 5 5 x(y) Fig. : he drk one denoes for he response sysem, nd he oher one denoes for he drive sysem. x(y) 8 8 x(y) x3(y3) Fig. 3: () he oris of x nd y : he rel line denoes x nd he roken line denoes y ; () he oris of x nd y : he rel line denoes x nd he roken line denoes y ; () he oris of x 3 nd y 3 : he rel line denoes x 3 nd he roken line denoes y 3. IJNS emil for onriuion: edior@nonlinersiene.org.uk

5 X. Li: Generlized Projeive Synhronizion Using Nonliner Conrol Mehod 83 x3... x x y3 y y Fig. : () he ror of he hu s irle; () he ror of disk dynmo model wihou he onroller. Generlized projeive synhronizion eween he hu irle nd he disk dynmo model In he following, we use nonliner onrol mehod o synhronize he hu s irle nd he disk dynmo model o he fixed sling for m. The drive sysem (hu s irle) is desried s follows: x = α(x x f(x )), x = x x + x 3, (.) x 3 = βx, where x +, if x, f(x ) = x +, if x, x, oherwise, And he response sysem (disk dynmo model) is inrodued s elow y = y + y y 3 + u, y = y + y (y 3 ) + u, y 3 = y y + u 3, The wo ove rors re shown in Fig.. Rewrie sysem (.) in he form of Eq. (.), where y y 3 A =, B =, g(y) = y y 3. y y We n esily ge h A =.3 nd B =. Here we hoose L = nd sele ɛ s 8 ɛ = 7. I is esy o verify h Eq.(.5) is sisfied. The onroller u n e go from Eq.(.). Se α =, β = 5.8, =.78, nd =.888. We le m = / nd hoose he iniil vlues of he drive sysem nd he response sysem s (x (), x (), x 3 ()) = (.,.5,.) nd (y (), y (), y 3 ()) = (.,.3,.5), respeively. Fig.5 revels he numeril glol synhronizion eween he wo differen hoi sysems wih differen iniil vlues s menioned ove. Fig. gives ou he simulion ori of he vriles of he drive sysem, nd he simulion oris of he response sysem fer synhronizion. From he Fig., we n lso esily oserve he rio of he mpliudes of he wo sysems ends o onsn sling for m = /. A he end, Fig.7 shows he numeril simulion of he error e of he wo differen hoi sysems-he hu s irle nd he disk dynmo model. Oviously, e, e nd e 3 onverge o zero finlly fer he onroller is ived. (.) IJNS homepge:hp://

6 8 Inernionl Journl of Nonliner Siene,Vol.8(9),No.,pp x3(y3) 5.5 x(y).5 x(y) Fig. 5: he drk one denoes for he response sysem, nd he oher one denoes for he drive sysem. y 3 x(y) x3(y3) 8 3 Fig. : () he oris of x nd y : he rel line denoes x nd he roken line denoes y ; () he oris of x nd y : he rel line denoes x nd he roken line denoes y ; () he oris of x 3 nd y 3 : he rel line denoes x 3 nd he roken line denoes y 3. 3 e 8 e 8 8 e3 8 8 Fig. 7: () denoes he ori of error funion e ; () denoes he ori of error funion e ; () denoes he ori of error funion e 3. 5 Summry nd onlusions Bsed on symoli ompuion sysem M ple nd Lypunov siliy heory, we propose sheme o relize he generlized projeive synhronizion vi nonliner onrol eween wo idenil hoi sysems nd wo differen hoi sysems, respeively. Numeril simulions re used o perform he proess of he synhronizion nd suessfully pu he oris of drive sysems nd oris of he response sysems in he sme plo for undersnding inuiively. Wih he id of symoli-numeri ompuion, he sheme n e used for oher hoi sysems nd hyperhoi sysems. Referenes [] L M Peor, T L Crroll: Synhronizion in Choi sysems. Phys. Rev. Le. :8-8(99) IJNS emil for onriuion: edior@nonlinersiene.org.uk

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