Synchronization of different 3D chaotic systems by generalized active control

 Everett Lindsey
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1 ISSN , Englnd, UK Journl of Informtion nd Computing Siene Vol. 7, No. 4, 0, pp. 78 Synhroniztion of different D hoti systems y generlized tive ontrol Mohmmd Ali Khn Deprtment of Mthemtis, Grhet Rmsundr Vidyhn, Grhet, Pshim Medinipur, West Bengl, Indi (Reeived My 9, 0, epted Septemer, 0) Astrt. This pper designs sheme for ontrolling hoti system to period system using tive ontrol tehnique. We disuss this tking hoti Genesio system s exmple. We hve lso disussed the synhroniztion sheme etween two different oupled hoti systems (Genesio nd Nuler spin genertor(nsg)system) s well s two identil oupled hoti systems (Foursroll ttrtor) vi tive ontrol. Numeril Simultion results re presented to show the effetiveness of the proposed sheme. Keywords: Choti system, Chos ontrol, Chos synhroniztion, Ative ontrol, Genesio system, NSG system, Foursroll ttrtor.. Introdution Chos synhroniztion is n importnt topi in the nonliner siene. In 990, Peorr nd Crroll [] introdued the ide of hos synhroniztion. Two or more oupled hoti systems re lled synhronized if their ehviors re losely relted. Chos synhroniztion hs reeived muh ttention due to its pplitions in mny re suh seure ommunition, informtion proessing, iologil systems nd hemil retions. Usully two dynmil systems re lled synhronized if the distne etween their orresponding sttes onverges to zero s time goes to infinity. This type of synhroniztion is lled identil synhroniztion. A generliztion, of this onept for unidiretionlly oupled dynmil systems ws proposed y Rulkov et.l.(995) [] where two oupled systems re lled synhronized if stti funtionl reltionship exists etween the sttes of the systems. This kind of synhroniztion is lled generlized synhroniztion(gs). In 009 synhroniztion in unidiretionlly oupled Rossler system ws proposed y Khn nd Mndl [] nd lso synhroniztion in idiretionlly oupled systems reported y Tri et.l. [4,5]. In 0 Khn et.l. [6] hve disussed vrious type of ontrol for ontrolling hos in unified hoti system nd reently in 0 generlized ntisynhroniztion of different hoti systems ws proposed y Khn et.l. [7]. In 996, Korev nd Prlitz [8] formulted ondition for the ourrene of GS for the mster nd slve system. Using tehnique from tive ontrol theory, ERWeii nd Krl.E.Lonngren(997) [9] demonstrte tht oupled Lorenz systems n synhronize. Further ERWeii nd Krl.E.Lonngren [0] hve investigted sequentil synhroniztion of two Lorenz system using tive ontrol. In 00, synhroniztion of Rossler nd Chen hotil dynmil sytems using tive ontrol ws studied y Agiz nd Yssen []. In 00 MingChung Ho, YoChen Hung [] generlized the tehnique of tive ontrol theory nd pplied them to synhronize two different systems. Sinh et.l. [] proposed generl pproh in the design of tive ontrollers for nonliner systems exhiiting hos in 00. Synhroniztion of two hoti fourdimensionl systems using tive ontrol tehnique ws proposed y Youming nd Wenxin in 007 [4]. In 009 Sudheer nd Sir [5] ws investigted hyrid synhroniztion of hyperhoti Lu system vi tive ontrol. Reently in 0 Shhzd [6] ws investigted hos synhroniztion of n ellipsoidl stellite vi tive ontrol. Synhroniztion sheme for mny oupled hoti systems re not explored till now. Synhroniztion strtegy for oupled hoti Genesio system is very importnt from the theoretil point of view nd this strtegy is reported y us in this Pulished y World Ademi Press, World Ademi Union
2 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp pper. The synhroniztion strtegy etween two identil oupled hoti systems s well s two different hoti systems re lso very importnt from the pplition point of view. This motivtes us to study the synhroniztion etween Genesio system nd Nuler spin genertor systems nd synhroniztion etween Foursroll ttrtor systems in this pper.. Genesio system The Genesio system, proposed y Genesio et.l [7], it ptures mny fetures of hoti systems. It inludes simple squre prt nd three simple ordinry differentil equtions tht depend on three positive rel prmeters. The dynmil equtions re desried y = y = z () = x y z+ x where x, y nd z re stte vriles nd, nd re positive onstnts, stisfying <. The Genesio system exhiits hoti ttrtor t the prmeter vlues =., =.9 nd =6 shown in Fig.... Control of hoti Genesio system to period system In order to oserve the ontrol ehvior we tke three dimensionl period systems s mster system with three stte vriles denoted y the susript nd the slve systems denoted y the susript. The initil onditions for the mster system re different from tht of the slve system. The mster nd slve systems re defined s follows Mster system = y = z () = y Slve system = y + u = z + u () = x y z + x + u where U = [ u,, ] T u u is the ontroller funtion introdued in the slve system. Using tive ontrol tehniques, we sustrt eqution () from eqution () nd get JIC emil for susription:
3 74 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol = y + u = z + u = x y z + x + y + u where x = x x; y = y y nd z = z z. We define the tive ontrol funtions u, u nd u s u = V u = V (5) u = x + y + z x y + V This leds to = y+ V = z+ V (6) = V Eqution (6) desries the error dynmis nd n e onsidered in terms of ontrol prolem where the system to e ontrolled is liner system with ontrol input V, Vnd V s funtions of x, y nd z. Our im is to stilize the systems x, y nd z t origin. This implies tht two systems re synhronized with feedk ontrol. There re mny possile hoies for the ontrol V, V nd V. We hoose V x V = A y (7) V z where A is onstnt mtrix. For proper hoie of elements of the mtrix A, the feedk system must hve ll of the eigenvlues with negtive rel prts. In this se, the losed loop system will e stle. Let us hoose prtiulr form of the mtrix A tht is given y 0 A = 0 (8) 0 0 For this prtiulr hoie, the losed loop system hs eigenvlues tht re found to e , nd . This hoie will led to stle systems nd s we will oserve in numeril investigtion, led to the synhroniztion of Genesio system to e period system. (4) Fig.. Solution of Genesio system to e period system with the tive ontrol detivted JIC emil for ontriution:
4 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Fig.. Solution of Genesio system to e period system with the tive ontrol tivted.. Numeril Simultion We selet the prmeters =., =.9 nd = 6 to ensure the existene of the hoti ehvior(fig.). The initil vlues re tken s x (0) = 0.6, y (0) =.6 nd z (0) =.5. Simultion results for unoupled system re presented in Fig.. nd tht of ontrolled system re shown in Fig.. The synhronizing of Genesio system to period system n lso e oserved y monitoring the differene of the two signls x, y nd z. For numeril simultion the initil vlues of differene of two signls re tken s x(0) = 4.0, y(0) = 0.0 nd z (0) =.0. Fig.4. displys the differene signls with the tive ontroller disonneted into the iruit. Fig.5. displys the sme signls with the tive ontroller onneted into the iruit. Fig.4. Disply the differene signls when tive ontroller detivted JIC emil for susription:
5 76 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol Fig.5. Disply the differene signls when tive ontroller tivted.. Nuler spin genertor (NSG) system The nuler spin genertor prolem ws studied y Shdev nd Srthy [8] nd Hegzi et.l. [9]. They showed tht the system displys rih nd typil ifurtion nd hoti phenomen for some vlues of the ontrol prmeters. The system onsists suitle simple of mtter ontining proper nulei in reltively strong mgneti field defining Zdiretion, n exiting oil with xis in the Xdiretion, perpendiulr to Z, pikup oil with xis in the Xdiretion, perpendiulr to oth X nd Z nd highgin mplifier feeding the voltge indued in the pikup oil k to the exiting oil. We first reformulte its eqution in the following form = β x+ y = x βy( κz) (9) = βα ( ( z)) κy ) suh tht α, β nd κ re prmeters, where βα 0 nd β 0 re liner dmping terms, the nonlinerity prmeters βκ is proportionl to the mplifier gin in the voltge feedk. Physil onsidertion limits the prmeters α to the rnge 0< α. Fig.6. Choti ttrtor of NSG system t β =.75, α =.5 nd κ =.0.. Synhroniztion of two different hoti systems JIC emil for ontriution:
6 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Now, we wnt to synhronize two different oupled hoti systems. Tke Genesio nd Nuler spin generting systems into onsidertion. Mster system(genesio system) = y = z (0) Slve system(nsg system) = x y z + x βα κ = β x + y + u = x βy ( κz ) + u = ( ( z ) y ) + u () Sustrt (0) from () we get = β x + y + u = x βy ( κz ) z + u = βα ( ( z ) κy ) + x + y + z x + u () We define the tive ontrol funtions u, und u s u = β x + V u = x + βy( κz) + z+ V u = βα ( ( z ) + κy ) x y z + x + V () This leds to = y + V = V = V (4) Eqution (4) desrie the error dynmil system whih is liner system with ontrol input V, V nd V s funtions of x, y nd z. Our gol is hieved when these feedks stilize the system x, y nd z t zero. There re mny possile hoies for the ontrol V, V nd V. We hoose V x V = A y (5) V z Where A is onstnt mtrix. Let us hoose prtiulr form of the mtrix A tht is given y β 0 A = β 0 (6) 0 0 αβ The eqution (6) n e written s X & = BX (7) where JIC emil for susription:
7 78 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol β 0 B = β αβ Finlly, we hve to produe mtrix B suh tht three eigen vlues hs negtive rel prts. (8) Fig.7. Solution of Genesio system nd NSG systems with the tive ontrol detivted. Fig.8. Solution of Genesio system nd NSG systems with the tive ontrol tivted... Numeril Simultion We selet the prmeters for NSG systems s β = 0.75, α = 0.5 nd κ = to ensure the hoti ehvior shown in Fig.6. The initil vlues of Genesio system re tken s x (0) = 0.0, y (0) =.0, z (0) =.0 nd for Nuler spin genertor system x (0) =.0, y (0) =.6, z (0) =.5. Simultion results for unontrolled system re presented in Fig.7. nd tht of ontrolled system re shown in Fig.8. Fig.9. displys the differene signls with the tive ontroller into the iruit. JIC emil for ontriution:
8 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Fig.9. Disply the differene signl of Genesio nd NSG system when tive ontrol tivted 4. Foursroll ttrtor Consider the following Foursroll ttrtor = x yz = y+ xz (9) = z+ xy where, nd re positive ontrol prmeters. This system hs hoti ttrtor s shown in Fig.0. t the prmeter vlues = 0.4, = nd = Synhroniztion of two identil hoti systems To oserve the synhroniztion ehvior for Foursroll ttrtor, we hve two Foursroll ttrtor systems where the mster system with three stte vriles denoted y the susript nd slve system hving identil equtions denoted y the susript. The mster nd slve systems re defined s follows JIC emil for susription:
9 80 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol nd = x y z = y + x z = z + x y (0) Sustrting (0) from () we get = x y z + u = y + x z + u = z + x y + u = x + y z y z + u = y + x z x z + u = z + x y x y + u By suitle hoie of u, u nd u the system () eomes = x + V where = y + V = z + V u = y z + y z + V u = x z x z + V u = x y x y + V There re mny possile hoies for ontroller V, V nd V. We hoose V x V = A y V z Where A is onstnt mtrix. Let us hoose prtiulr form of the mtrix A tht is given y ( + ) 0 0 A = Therefore differene signl system eomes = x = y = z () () () (4) (5) (6) JIC emil for ontriution:
10 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp Fig.. Disply the solution of the oupled Foursroll ttrtor system of equtions with the tive ontrol detivted. Fig.. Disply the solution of the oupled Foursroll ttrtor system of equtions with the tive ontrol tivted. 4.. Numeril Simultion We selet the prmeters for Foursroll ttrtor systems s = 0.4, = nd = 5 to ensure the hoti ehvior shown in Fig.0. The initil vlues of oupled Foursroll ttrtor system re tken s x(0) = 0.0, y(0) =.0, z(0) =.0 nd x(0) = 5.6, y(0) =.6, z(0) =.5. The results of the simultion of oupled Foursroll ttrtor system with the tive ontroller detivted re shown in Fig.. Fig.. displys the sme sequene of signls when tive ontrol tivted nd differene signls re shown in Fig.. with the tive ontroller onneted into the iruit. JIC emil for susription:
11 8 Mohmmd Ali Khn: Synhroniztion of different D hoti systems y generlized tive ontrol 5. Conlusion Fig.. Disply the differene signl of x, y nd z when tive ontrol tivted. We hve disussed synhroniztion of Genesio system to period system using tive ontrol tehnique. This strtegy n e used to synhronize ny hoti system to periodi system with desired period. This tehnique my e useful for ontrolling hoti osilltions in eletroni systems, iologil ells et. We hve lso disussed the synhroniztion etween two different hoti systems onsidering Genesio nd NSG systems s well s two identil Foursroll ttrtor systems vi tive ontrol. We elieve these synhroniztion sheme my e useful for sending serete messge. We show tht hoti system n e synhronized to periodi system or ny other system. Aknowledgement I would like to express my sinere thnks to Dr. Swrup pori of the Dept. Of Applied Mthemtis, Clutt University for motivting me y vlule suggestions for preprtion of this work. 6. Referenes [] [] [] [4] [5] [6] [7] Peorr nd Crroll, Synhroniztion in hoti systems, Phys. Rev. Lett. 64 (990), pp.8. Rulkov N.F, Sushik M.M nd T. Simring L.S, Generlized synhroniztion of hos diretionlly oupled hoti system, Phys. Rev. E. 5 (995), pp M.A.Khn nd A.K.Mndl, Generlized hos synhroniztion of oupled Rossler systems,bull. Cl. Mth. So. 0 (009), pp A.Tri, S.Pori nd P.Chtterjee, Synhroniztion of idiretionlly oupled hoti Chen s System with dely, Chos Solitons nd Frtls, doi: 0.06/j.hos A.Tri, S.Pori nd P.Chtterjee, Synhroniztion of generlized linerly idiretionlly oupled unified hoti system, Chos Solitons nd Frtls. 40 (009), pp M.A.Khn, A.K.Mondl nd S.Pori, Three ontrol strtegies for unified hoti system, Int. J. of Applied Mehnis nd Engineering. 6 (0), pp M.A.Khn, S.N.Pl nd S.Pori, Generlized ntisynhroniztion of different hoti systems, Int. J. of Applied Mehnis nd Engineering. 7 (0), pp JIC emil for ontriution:
12 Journl of Informtion nd Computing Siene, Vol. 7 (0) No. 4, pp [8] [9] Korev L nd Prlitz U, Generlized synhroniztion, preditility nd equivlene of unidiretionlly oupled dynmil systems, Phys. Rev. Lett. 76 (996), pp.86. ERWeii nd Krl E. Lonngren, Synhroniztion of two Lorenz systems using tive ontrol, Chos, Solitons nd Frtls. 8 (997), pp [0] [] [] [] [4] [5] [6] [7] ERWeii nd Krl E. Lonngren, Sequentil synhroniztion of two Lorenz systems using tive ontrol, Chos,Solitons nd Frtls. (997), pp.04. Agiz nd Yssen, Synhroniztion of Rossler nd Chen hotil dynmil systems using tive ontrol, Physis Letters A. 78 (00), pp.9 MingChung Ho nd YC Hung, Synhroniztion of different systems using generlized tive ontrol, Physis Letters A. 0 (00), pp Sinh SC, Henrihs JT nd Rvindr BA, A generl pproh in the design of tive ontrollers for nonliner systems exhiiting hos, Int. J. Bifurt. Chos (00) pp L.Youming, X. Wei nd X. Wenxin, Synhroniztion of two hoti fourdimensionl systems using tive ontrol, Chos, Solitons nd Frtls. (007), pp K. Sestin Sudheer nd M. Sir, Hyrid synhroniztion of hyperhoti Lu system, Prmnjournl of physis. 7 (009) pp Genesio R nd Tesi A, A hrmoni lne methods for the nlysis of hoti dynmis In nonliner systems, Automti, 8 (99), pp Mohmmd Shhzd, Chos synhroniztion of n ellipsoidl stellite vi tive ontrol, Progress in Applied Mthemtis. (0), pp.6. [8] Shdev nd Srthy, Periodi nd hoti solutions for nonliner system rising from nuler spin genertor, Chos,solitons nd Frtls. 4 (994), pp [9] Hegzi,Agiz nd Eledessoky, Adptive ontrol nd synhroniztion of modified Chu s iruit system, Chos,solitons nd Frtls. (00), pp JIC emil for susription:
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