SYNCHRONIZATION BETWEEN CHUA AND MODIFIED CHUA OSCILLATORS WITH PASSIVE CONTROL

Transcription

1 Acadmic Jornal o Scinc CD-ROM. ISSN: :: 040: SNCHRONIATION BETWEEN CHUA AND MODIFIED CHUA OSCILLATORS WITH PASSIVE CONTROL ılma Uarol Sakara Univrsit Trk Ur Erkin Kocama Ulda Univrsit Trk In this std th snchroniation btwn Cha and modiid Cha oscillators is invstigatd b mans o th passiv control mthod. First th Cha and cbic Cha oscillators ar dscribd and dind as a st o dirntial qations. Thir chaotic tim sris and phas portraits ar dmonstratd. Thn th passiv controllrs ar constrctd or th snchroniation. Th global asmptotical stabilit o rrors btwn th Cha oscillators is nsrd with th Lapnov nction. Atrwards nmrical rslts ar dmonstratd to conirm th thortical analsis. Th hav also shown that th passiv controllrs ar ctivl sd or snchroniing two dirnt Cha oscillators. Kwords: Cha s circit Modiid Cha oscillator Dobl scroll Chaos snchroniation Passiv control. Introdction Sinc Lorn introdcd th irst chaotic attractor in 96 [] th invstigations on chaos hav bcom an important sbjct in th nginring ara and man chaotic attractors hav bn discovrd. A nw thr-dimnsional chaotic Rösslr sstm was proposd in 96 []. Cha s dobl-scroll attractor was prsntd in 984 []. Sprott ocsd on simplr chaotic sstms in 994 and ncovrd 9 distinct chaotic lows which hav ithr iv trms and two nonlinaritis or si trms and on nonlinarit [4]. Chn attractor was proposd b Chn and Uta in 999 [5]. Lü t al. discovrd a nw chaotic sstm which rprsnts th transition btwn th Lorn and Chn sstms in 00 [6]. Thn Lü t al. sggstd a gnralid orm o th Lorn Chn and Lü sstms calld niid chaotic sstm []. In rcnt ars svral nw chaotic attractors hav also bn proposd [8 0]. Bsids nw chaotic attractors th snchroniation o chaotic sstms bcoms an important task and has rcivd incrasing attntion rom rsarchrs d to thir potntial applications spciall in scr commnication [ ]. Th ida o snchroniing two idntical chaotic sstms was introdcd b Pcora and Carroll in 990 []. Thn svral mthods hav bn sccssll applid to snchroni th chaotic sstms sch as activ control [4] passiv control [5 9] sliding mod control [0] implsiv control [] backstpping dsign [] and so on. Among thm passiv control is a signiicant chaos control mthod. It has bn sd or th snchroniation o Chn [5] niid [6] Rikitak [] btwn Rösslr and Gnsio-Tsi [8] and man othr chaotic sstms [9]. 5

2 6 Snchroniation Btwn Cha and Modiid Cha Oscillators With Passiv Control Introdcd b Lon O. Cha Cha s circit is a simpl lctronic circit that consists o on linar rsistor two capacitors on indctor and on nonlinar rsistor. Its simplicit and chaotic phnomna mak th Cha oscillator a wll-known circit. Its dnamical bhaviors and proprtis hav bn tnsivl invstigatd in man paprs [ 5]. Som modiid vrsions o Cha s circits ar also proposd [6 ]. Th control and snchroniation o Cha sstms hav rcivd lots o attntions rom rsarchrs. Rcntl th snchroniation o Cha oscillators is implmntd with activ control [8] adaptiv control [9] sliding mod control [0] control [] implsiv control [] and backstpping dsign [] mthods. According to th litratr rviw thr is no passiv control approach or th snchroniation o Cha oscillators. In this papr th snchroniation btwn two dirnt chaotic Cha oscillators is applid with th passiv control mthod. Sstm Dscriptions As sn in Fig. th Cha s circit contains 5 circit lmnts; indctanc L rsistanc R two capacitancs C and C and thr-sgmnt nonlinar rsistor g []. Fig.. Th Cha s circit. Th Cha s sstm is dscribd b th ollowing dimnsionlss orm whr > 0 and > 0 ar th sstm paramtrs dtrmind b th particlar vals o th circit componnts; is th nction dscribs th lctrical rspons o th nonlinar rsistor; and ar th stat variabls rprsnt th voltags across th capacitors C and C and th intnsit o th lctrical crrnt in th indctor L rspctivl. Th nction dpnds on th particlar conigration o componnts. It is gnrall considrd as b a b. Cha s circit hibits chaotic phnomna whn th paramtrs ar slctd as 9 00/ a 8/ and b 5/ with th initial condition []. Th tim sris o Cha oscillator ar dmonstratd in Fig. th D phas portraits ar dmonstratd in Fig. and th D phas plan is dmonstratd in Fig. 4. Cha oscillator is also known as dobl scroll bcas o its shap in th spac.

3 ılma Uaroǧl and Uǧr Erkin Kocama a b Fig.. Tim sris o Cha oscillator or a signals b signals c signals. c a b

4 8 Snchroniation Btwn Cha and Modiid Cha Oscillators With Passiv Control c Fig.. D phas portraits o Cha oscillator or a phas plot b phas plot c phas plot. Fig. 4. D phas plan o Cha oscillator. Hartl proposd a modiid Cha oscillator in 989 [6]. A cbic nonlinarit is sd in plac o th picwis-linar nonlinarit o Cha s circit. It changs th dnamics o sstm and th bircation strctr a littl. Th cbic Cha sstm is dscribd b whr > 0 and > 0 ar th sstm paramtrs; and ar th voltags across th two capacitors; and is th crrnt throgh th indctor. Th cbic Cha s circit sstm displas chaotic bhavior whn th paramtrs ar considrd as 0 and 00/ with th initial condition [4]. Th tim sris th D phas portraits and th D phas plan o cbic Cha oscillator ar dmonstratd in Fig. 5 Fig. 6 and Fig. rspctivl.

5 ılma Uaroǧl and Uǧr Erkin Kocama 9 a b Fig. 5. Tim sris o cbic Cha oscillator or a signals b signals c signals. c a b

6 0 Snchroniation Btwn Cha and Modiid Cha Oscillators With Passiv Control c Fig. 6. D phas portraits o cbic Cha oscillator or a phas plot b phas plot c phas plot. Fig.. D phas plan o cbic Cha oscillator. Snchroniation In this sction th snchroniation o two dirnt Cha oscillators with nknown paramtrs is applid b sing a passivit-basd control mthod. Two chaotic Cha sstms ar takn whr th driv Cha oscillator dnotd b th sbscript controls th rspons cbic Cha oscillator which is dnotd b sbscript. Th driv and rspons Cha sstms ar dind as ollows: 4 and 5

7 ılma Uaroǧl and Uǧr Erkin Kocama whr and in Eqation 5 ar th control nctions to b dtrmind. In ordr to obtain th control nctions or snchroniation th driv sstm is sbtractd rom th rspons sstm. Th and stat rrors btwn cbic Cha chaotic sstm 5 that is to b controlld and th controlling Cha chaotic sstm 4 is dind as. 6 It lads to. Th sstm is calld th rror sstm. Th snchroniation problm is to nsr th rror sstm asmptoticall stabl at th ro qilibrim point. B assming that th stat variabl is th otpt o th sstm and spposing [ ] T thn sstm can b dnotd b normal orm:. 8 Th passiv control thor has th ollowing gnralid orm: 0 a b p X 9 and according to sstm 8:. 0 0 a b p 0 Th storag nction is chosn as W V whr W is th Lapnov nction o 0 with W0 0. In ordr to obtain drivativ o W as ngativ th control nction is considrd b:

8 Snchroniation Btwn Cha and Modiid Cha Oscillators With Passiv Control. According to Eqation 0 b taking th drivativ o W d W W W 0 [ ] dt.. Sinc W 0 and W 0 it can b concldd that W is th Lapnov nction o 0 and that 0 is globall asmptoticall stabl []. So th ro dnamics o th rror sstm 8 is stabl basd on th Lapnov stabilit. Th snchronid sstm can b qivalnt to a passiv sstm and globall asmptoticall stabilid at its ro qilibrim b th ollowing stat dback controllr [6]: T W a b p v. Accordingl it is dtrmind or th rror sstm 8 as v whr is a positiv constant and v is an trnal inpt signal. B taking back convrsions th control nctions bcom v. Hnc th snchroniation btwn Cha and cbic Cha oscillators with nknown paramtrs is achivd b mans o passiv control mthod. Nmrical Simlations In this sction comptr simlations ar prormd to show th snchroniation btwn Cha oscillator and cbic Cha oscillator with th passiv control mthod. Th nmrical analsis is carrid ot sing Rng-Ktta mthod o ordr 4 with variabl tim stp. Th paramtrs o Cha s circit ar takn as 9 00/ a 8/ and b 5/; th initial condition is Th paramtrs o cbic Cha oscillator ar takn as 0 and 00/; th initial condition is Th passiv controllr paramtrs ar chosn as 0 and v 0. In ordr to prsnt th chaotic trajctoris bor and atr th control law is applid th controllrs ar activatd at t 5. Th simlation rslts o snchroniation ar dmonstratd in Fig. 8 and th rror signals ar dmonstratd in Fig. 9.

9 ılma Uaroǧl and Uǧr Erkin Kocama Fig. 8. Th tim rspons o stats or th snchroniation btwn chaotic Cha and cbic Cha oscillators whn th passiv controllrs ar activatd at t 5 or a signals b signals c signals. Fig. 9. Th tim rspons o rror signals or th snchroniation btwn chaotic Cha and cbic Cha oscillators whn th passiv controllrs ar activatd at t 5.

10 4 Snchroniation Btwn Cha and Modiid Cha Oscillators With Passiv Control As pctd Fig. 8 rom Matlab Simlink otpts shows that th passiv controllrs hav achivd th non-idntical snchroniation o chaotic Cha oscillators. Th rror signals that ar shown in Fig. 9 convrg asmptoticall to ro. Thror all th analtical rslts hav bn conirmd b th comptr simlations. Whn th controllrs ar activatd at t 5 th snchroniation is obtaind at t with th passiv controllrs. Hnc th passiv controllrs ar appropriat or th snchroniation o two dirnt Cha oscillators. Conclsion In this papr th passiv control mthod is proposd or th snchroniation btwn chaotic Cha and cbic Cha oscillators with nknown paramtrs. Basd on th proprtis o passivit th passiv controllrs hav bn constrctd to rali th snchroniation. Simlation rslts show that th passiv controllrs ar abl to snchroni th chaotic motion o cbic Cha oscillator to th chaotic motion o Cha s circit. So th thortical analss ar conirmd. Nmrical simlations also show that th proposd passiv control mthod is ctiv or th non-idntical chaos snchroniation o Cha oscillators. Rrncs. Lorn E. N. 96. Dtrministic nonpriodic low. Jornal o th Atmosphric Scincs Rösslr O. E. 96. An qation or continos chaos. Phsics Lttrs A Cha L. O. Komro M. & Matsmoto T Th dobl scroll amil. IEEE Transactions Circits and Sstms CAS Sprott J. C Som simpl chaotic lows. Phsical Rviw E Chn G. & Uta T t anothr chaotic attractor. Intrnational Jornal o Bircation and Chaos Lü J. Chn G. & hang S. 00. Th compond strctr o a nw chaotic attractor. Chaos Solitons & Fractals Lü J. Chn G. Chng D. & Clikovsk S. 00. Bridg th gap btwn th Lorn sstm and th Chn sstm. Intrnational Jornal o Bircation and Chaos Sahab A. R. iabari M. T. & Modabbrnia M.R. 0. A novl ractional-ordr hprchaotic sstm with a qadratic ponntial nonlinar trm and its snchroniation. Advancs in Dirnc Eqations hang B. J. & Li H. X. 0. A nw or-dimnsional atonomos hprchaotic sstm and th snchroniation o dirnt chaotic sstms b sing ast trminal sliding mod control. Mathmatical Problms in Enginring Pham V. T. Volos C. Jaari S. Wi. C. & Wang X. 04. Constrcting a novl no-qilibrim chaotic sstm. Intrnational Jornal o Bircation and Chaos Kinl W. Englrt A. & Kantr I. 00. On chaos snchroniation and scr commnication. Philosophical Transactions o th Roal Socit A a H. T. P. C. & Li S. C. 0. Application o a chaotic snchroniation sstm to scr commnication. Inormation Tchnolog and Control Pcora L. M. & Carroll T. L Snchroniation in chaotic sstms. Phsical Rviw Lttrs Bai E. W. & Lonngrn K. E. 99. Snchroniation o two Lorn sstms sing activ control. Chaos Solitons & Fractals

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