Generalized Projective Synchronization for Four Scroll attractor. M. M. El-Dessoky 1, 2 and E. M. Elabbasy 2.

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1 Lif Scinc Journal ;9() Gnralid Projctiv Snchroniation for Four Scroll attractor M M El-Dssok, and E M Elabbas Dpartmnt of Mathmatics, Facult of Scinc, King Abdulai Univrsit, P O Bo 8, Jddah 589, Saudi Arabia Dpartmnt of Mathmatics, Facult of scinc, Mansoura Univrsit, Mansoura, 556, Egpt dssokm@mansdug Abstract: This papr invstigats an activ control mthod is proposd to gnrali projctiv snchroni two idntical chaotic dnamical sstms b constructing th rspons sstm no mattr whthr th ar idntical Th proposd tchniqu is applid to achiv gnralid projctiv snchroniation for th Four - scroll attractor, whr all stat variabls ar in a proportional wa A stratg for practical implmntation of a scur communication stratg is also discussd Finall computr simulations ar don to vrif th proposd mthods, and th rsults show that th obtaind thortic rsults ar fasibl and fficint [El-Dssok MM, Elabbas EM Gnralid Projctiv Snchroniation for Four Scroll attractor Lif Sci J ;9():477-48] (ISSN:97-85) 58 Kwords: Projctiv snchroniation; Chaotic sstm; Activ control; Four-scroll attractor, Scur communication Introduction Rsarchrs from diffrnt aras, such as mathmaticians, phsicists, chmist, as wll as control nginrs hav dvotd thmslvs to amin th issu of snchroniation ovr th past dcad (Pacora and Carroll, 99; Carroll and Pacora, 99; Kocarv t al,99) Chaotic sstms, in particular, hav bn applid to th dvlopmnt of scur communications, chmical ractions, biological sstms and so on (Kocarv t al,99; Yuan and Jun, 9; Zhu, 9; Lu t al, ; Hua t al, 5; Juan t al, ; Rafal and Yu, 8 ) Th sstm which rcivd th most attntion among chaotic communication sstms prhaps is th Chua oscillator (Kocarv t al,99) This sstm blongs to gnral class of Lur sstms (Khalil, 996) Chaos control and snchroniation hav attractd a grat dal of attntion from various filds sinc Hubr publishd th first papr on chaos control in 989 (Hublr, 989) Ovr th last dcads, man mthods and tchniqus hav bn dvlopd, such as OGY mthod (Ott t al, 99), PC mthod (Pacora and Carroll, 99; Carroll and Pacora, 99), fdback approach, nonfdback control mthods, adaptiv mthod, nonlinar control, activ control, and backstpping dsign tchniqu, tc (Wang t al,, Elabbas t al, 4; Wang and G, ; Jiang t al, ; Sun and Zhang 4) Thr ar man applications to chaotic communication (Fallahi and Lung, ; Elabbas and El-Dssok, ) and chaotic ntwork snchroniation (Chow t al, ) Th tchniqus of chaotic communication can b dividd into thr catgoris, (i) chaos masking (Kocarv t al, 99), th information signal is addd dirctl to th transmittr; (iii) chaos modulation (Boutab t al, ), it is basd on th driv rspons (mastrslav) snchroniation, whr th information signal is injctd into th transmittr as a nonlinar filtr; (ii) chaos shift king (Parlit t al, 99), th information signal is supposd to b binar, and it is mappd into th transmittr and th rcivr In ths thr cass, th information signal can b rcovrd b a rcivr if th transmittr and th rcivr ar snchronid In 96, Lorn found th first classical chaotic attractor In 999, Chn found anothr similar but not topological quivalnt chaotic attractor th Chn attractor (Chn and Uta, 999) In, Lü and Chn found a nw critical chaotic sstm (Lü and Chn, ), baring th nam of Lü sstm It is noticd that ths sstms can b classifid into thr diffrnt tps b th dfinition of Vanĕŏk and Člikovsk (Vanĕŏk and Člikovsk, 996): th Lorn sstm (Lorn, 96) satisfis th condition a a, th Chn sstm (Chn and Uta, 999) satisfis a a and th Lü sstm (Lü and Chn, ) satisfis a a, whr a and a ar th corrsponding lmnts in th linar part matri A [ aij] of th dnamical sstm Th arl projctiv snchroniation (PS) is usuall obsrvabl onl in a class of sstms with partial-linarit (Xu t al, ; Xu and Ch, ; Xu and Li,, Wn and Xu 5), but rcntl som rsarchrs (Zhigang and Daolin, 477

2 Lif Scinc Journal ;9() ; El-Dssok, ) hav achivd control of th projctiv snchroniation in a gnral class of chaotic sstms including non-partiall-linar sstms, and trmd this projctiv snchroniation as gnralid projctiv snchroniation (GPS) (Yan and Li, 5; Changpin and Jianping, 6; El-Dssok and Salah, ) In this papr, w gnrali activ control to GPS, and dmonstrat this tchniqu b som tpical chaotic sstms, for ampl, th chaotic Lorn sstm and th chaotic Chn sstm such that GPS is achivd Th rsults from numrical simulations show that th mthod works wll Th papr is organid as follows In Sction, th gnralid projctiv snchroniation with activ control is applid to snchroni two idntical Four-scroll attractor and numrical simulations ar prsntd to show th ffctivnss of th proposd mthod In Sction, a schm of scur communication basd on th activ control of Four-scroll chaotic sstm is prsntd Conclusions ar finall givn in Sction 4 Gnralid projctiv snchroniation (GPS) idntical Four-scroll chaotic attractor Th projctiv snchroniation mans that th driv and rspons vctors snchroni up to a scaling factor a, that is, th vctors bcom proportional First, w dfin th GPS blow Considr th following chaotic sstm: f (, () g(, u(,, () n whr, R ar th stat vctor of th sstms () and (), rspctivl ; n n f, g R R R ar two continuous nonlinar vctor functions, u(,, is th vctor control input If thr ists a constant α (α ), such that lim, thn th GPS of th t sstms () and () is achivd, and w call α is a scaling factor Now, w appl th adaptiv fdback control mthod for gnralid projctiv snchroniation idntical Four-scroll attractor (Lü t al, & 4; Liu and Chn 4; Elabbas t al, 6; El-Dssok, ) which can b dscribd b: a b c () whr a, b and c ar positiv control paramtrs This sstm hibits a strang attractor at th paramtr valus a=4, b= and c=5 This sstm bridgs th gap btwn th Lorn (Lorn, 96) and Chn attractors (Chn and Uta, 999), i a a Th divrgnc of th flow () is givn b F F F F a b c, Whr F ( F, F, F ) ( a, b, c ) Hnc th sstm is dissipativ whn: a b c Scinc th sstm of Four-scroll attractor () is a dissipativ sstm thus th solutions of th sstm of quations () ar boundd as t for a b c If a b, a c and b c thn th sstm has fiv quilibrium points: E (,,), E ( bc, ab), E E 4 ( ( bc, bc, ab), E ab) ( bc, ab), Diffring from othr known similar sstms, sstm () has fiv quilibrium, and dos not hav Hopf and pitch bifurcations [8, 9] Of most intrsting is th obsrvation that this chaotic sstm not onl can displa a two-scroll chaotic attractor whn a=45, b= and c=5 (Figur ), but also can displa a Four-scroll chaotic attractor whn a=4, b= and c=5 ( Figur ) Figur : Th chaotic attractor - scroll attractor at a=45, b= and c=5 in -dimnsional 478

3 Lif Scinc Journal ;9() Figur : Th chaotic attractor of Four- scroll attractor at a=4, b= and c=5 in -dimnsional In this sction w appl th gnralid projctiv snchroniation idntical Fourscroll chaotic attractor In ordr to obsrv th gnralid projctiv snchroniation bhavior in th Four-scroll sstm, w hav two Four-scroll sstms whr th driv sstm with thr stat variabls dnotd b th subscript drivs th rspons sstm having idntical quations dnotd b th subscript Howvr, th initial condition on th driv sstm is diffrnt from that of th rspons sstm Th two Four-scroll sstms ar dscribd, rspctivl, b th following a c and a b b c u u u (4) (5) Thr ar thr control functions u i, (i=,, ) to b dtrmind latr Dfin th rror vctor as, and whr α is a dsird scaling factor Thn on obtains th rror dnamical sstm b subtracting (4) from (5) d d d a b c u u u (6) Rfrring to th original mthods of activ control, so w choos th thr control functions u, ( i=,, ) as follows: u u u v v v (7) thn th rror dnamical sstm (6) is dscribd b d a v d b v (8) d c v Th rror sstm (8) to b controlld is a linar sstm with a control input v, v and v as function of th rror, and As long as ths fdbacks stabili th sstm (8),, and convrg to ro as tim tnds to infinit, which implis that GPS idntical Four-scroll sstms is achivd with a scaling factor α Thr ar man possibl choics for th control v, v and v In ordr to mak th closd loop sstm (8) b stabl, th propr choic of th control should guarants that th fdback sstm must hav all ignvalus with ngativ ral parts For simplif, w choos v v v A a i (9) In this particular choic, th thr ignvalus of th closd loop sstm (8) ar -a, -b and -c Sinc th closd loop sstm has all ignvalus that ar found to hav ngativ ral parts, th sstm will b convrgnc In othr words, this choic will rsult in a stabl sstm and th GPS of two idntical Four scroll sstms What dsrvs to b mntiond is that th valus of th ignvalus pla an important rol in th stabilit of th rror sstm In ordr to quickn th rat of convrgnc, w should mak thm gt smallr Numrical Rsults B using Mapl, w slct th paramtrs of th Four-scroll attractor as a=4, b= and c=5 Th initial valus of th driv sstm and rspons 479

4 Lif Scinc Journal ;9() sstm ar takn as (), (), (), () 77, () 9 and () 44 rspctivl If w tak th scaling factor hnc th rror sstm has th initial valus () 9, () 7 and () thn th gnralid projctiv snchroniation btwn two idntical Four-scroll attractor ar shown in Figur If w tak th scaling factor hnc th rror sstm has th initial valus () 88, () 94 and () 54 thn th GPS btwn two idntical Four-scroll attractor ar shown in Figur 4 If w tak th scaling factor hnc th rror sstm has th initial valus () 54, () 8 and () thn th complt snchroniation btwn two idntical Four-scroll attractor ar shown in Figur 5 If w tak th scaling factor hnc th rror sstm has th initial valus (), () and () 76 thn th anti snchroniation btwn two idntical Four-scroll attractor ar shown in Figur 6 Figur 4 : Th trajctoris of, and idntical Four- scroll attractor with scaling factor for gnralid projctiv snchroniation Figur 5: Th trajctoris of, and idntical Four- scroll attractor with scaling factor for complt snchroniation Figur : Shows th trajctoris of, and idntical Four- scroll attractor with scaling factor for gnralid projctiv snchroniation Figur 6: Th trajctoris of, and idntical four- scroll attractor with scaling factor for anti snchroniation 48

5 Lif Scinc Journal ;9() Th application in scur communication In this sction, w will appl th adaptiv schm drivd abov to scur communication using th Four-scroll attractor Assum that m( is th mssag signal, adding it to th right hand sid of th scond quation for th transmittr (driv sstm), thn w hav a b m () t c (9) Slct th output ( of th sstm (9) as th transmittd signal, thn construct th rcivr as follows: a b p() t u c u p k ( ) Lt whr k is a positiv paramtr (), and p m Thn th rror sstm can b dscribd b b u c u dm k () Rfrring to th original mthods of activ control, so w choos th thr control functions u,( i,) as follows: i u and u () thn th rror dnamical sstm () is dscribd b b c dm k () Thn w tak th Lapunov function : V ( ) ( k ) (4) It is clar that th Lapunov function V () is a positiv dfinit function Now, taking th tim drivativ of quation (4), thn w gt dv ( ) k dm k k c k dm k c Sinc th ign-frqunc of th mssag signal m ( is much lss than th oscillating frqunc of dm th chaotic sstm in practic It is as to hav dv () This can driv p m as t, that is p( can rcovr th mssag signal m ( Taking m( t ) sin( t ), k, initial valus ( ), (), (), ( ) 9, () 4and p() 6 Figurs 7(a, b) show shows that th trajctor of of th rror sstm with scaling factor and, rspctivl 4 Fig 7 (a): displa th trajctor of of th rror sstm with scaling factor 48

6 Lif Scinc Journal ;9() Fig 7 (b): displa th trajctor of of th rror sstm with scaling factor 4 4 Conclusion In this papr, an activ control mthod is proposd for manipulating gnralid projctiv snchroniation in a gnral class of chaotic sstms This mthod is ffctiv and convnint to gnralid projctiv snchroni two idntical sstms and two diffrnt chaotic sstms Also w hav proposd a schm for a practical implmntation of scur communication basd on a activ control mthod Numrical simulations ar also givn to validat th proposd snchroniation approach Acknowldgmnts: This projct was fundd b th Danship of Scintific Rsarch (DSR), King Abdulai Univrsit, Jddah, undr grant no 4//4 Th authors, thrfor, acknowldg with thanks DSR tchnical and financial support Th authors would lik to thank th ditor and th anonmous rviwrs for thir constructiv commnts and suggstions to improv th qualit of th papr Corrsponding Author: Dr M M El-Dssok Dpartmnt of Mathmatics Facult of Scinc, King Abdulai Univrsit, P O Bo 8, Jddah 589, Saudi Arabia dssokm@mansdug Rfrncs Pcora LM, Carroll TM Snchroniation of chaotic sstms Phs Rv Ltt, 99, 64(8): 8- Carroll TM, Pcora LM Snchroniing a chaotic sstms IEEE Trans Circuits Sstms,99 8: Wu CW, Chua LO A unifid framwork for snchroniation and control of dnamical sstms Int J Bifurcat Chaos, 994, 4(4): Kocarv L, Hall KS, Eckrt K, Chua LO, Parlit U Eprimntal dmonstration of scur communications via chaotic snchroniation Int J Bifurcat Chaos, 99, (): 79-5 Wang Xing-Yuan, Wang Ming-Jun A chaotic scur communication schm basd on obsrvr Commu Nonlinar Sci Numr Simulat, 9, 4: Fanglai Zhu, Obsrvr-basd snchroniation of uncrtain chaotic sstm and its application to scur communications Chaos, Solitons and Fractals,9, 4: Lu JA, Wu XQ, Lü JH Snchroniation of unifid chaotic sstm and th application in scur communication Phs Ltt A,, 5: Changchun Hua, Bo Yang, Gaoiang Ouang, Xinping Guan A nw chaotic scur communication schm Phs Ltt A, 5, 4: Juan L Mata-Machuca, Rafal Martin-Gurra, Ricardo Aguilar-Lop, Carlos Aguilar-Iban, A chaotic sstm in snchroniation and scur communications, Commun Nonlinar Sci Numr Simul,, 7: 76- Rafal Martin-Gurra, Yu W Chaotic snchroniation and scur communication via sliding-mod obsrvr Int J Bifur Chaos 8;8:5-4 Khalil H K Nonlinar sstms nd d Englwood Cliffs, NJ: Prntic-Hall; 996 Hublr AW Adaptiv control of chaotic sstm [J] Hlv Phs Acta 989; 6: 4-6 Ott E, Grbogi C, York JA Controlling chaos [J] Phs Rv Ltt 99; 64: Wang G, Yu X, Chn S Chaos control, snchroniation and its application China: National Dfns Industr Publishing Hous; 5 Elabbas EM, Agia HN and El-Dssok MM Snchroniation of Modifid Chn Sstm Int J Bifur Chaos, 4, 4(): - 6 Wang C, G SS Adaptiv snchroniation of uncrtian chaotic sstm via backstpping dsign Chaos, Solitons & Fractals,, :9-6 7 Jiang G-P, Chn G, Tang K S A nw critrion for chaos snchroniation using linar stat fdback control Int J of Bifur Chaos,, (8): Sun J, Zhang Y Som simpl global snchroniation critrion for coupld tim- 48

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