Bifurcations of fractional-order diffusionless Lorenz system

Transcription

1 Bifurcios of frciol-order diffusioless Lore ssem Keui Su * J. C. Sro Scool of Psics Sciece d Tecolog Cerl Sou Uiversi Cgs 483 Ci Derme of Psics Uiversi of Wiscosi-Mdiso Mdiso WI 5376 USA Asrc Usig e redicor-correcor sceme e frciol order diffusioless Lore ssem is ivesiged umericll. Te effecive coic rge of e frciol order diffusioless ssem for vriio of e sigle corol rmeer is deermied. Te roue o cos is eriod-doulig ifurcio i is frciol order ssem d some icl ifurcios re oserved suc s e fli ifurcio e ge ifurcio ierior crisis ifurcio d rsie cos. Te resuls sow e frciol-order diffusioless Lore ssem s comle dmics wi ieresig crcerisics. Kewords: Frciol-order ssem; Diffusioless Lore ssem; Cos; Bifurcio PACS: 545. Iroducio Frciol clculus s 3-er-old isor s old s clculus iself u is licios o sics d egieerig ve us egu []. M ssems re ow o disl frciol-order dmics suc s viscoelsic ssems [2] dielecric olriio elecrode-elecrole olriio d elecromgeic wves. M scieiss ve sudied e roeries of ese frciol-order ssems. More recel ere s ee growig ieres i ivesigig e coic evior d dmics of frciol order dmic ssems [3-2]. I s ee sow severl coic ssems c remi coic we eir models ecome frciol [8]. I [4] i ws sow e frciol-order Cu s circui wi order s low s 2.7 c roduce coic rcor. I [5] i ws sow ouoomous Duffig ssems wi order less 2 c sill eve i coic mer d i [6] cos i modified Duffig ssem eiss for ol ssem orders re d 2.. I [7] e frciol order Wie ridge oscillor ws sudied were i ws sow limi ccle c e geered for frciol order wi rorie vlue of e mlifier gi. I [8] coic evior of e frciol order Lore ssem ws sudied u uforuel e resuls reseed i er re o correc. I [9] e coic evior of frciol-order er model ws ivesiged i wic coic rcor ws oied wi e ssem order s low s 2. d cos corol of frciol order coic ssem ws reored i []. I [] Cos i e frciol-order Rössler ercoic ssem ws sudied i wic cos ws foud i e * Corresodig uor. E-mil ddress: eui@csu.edu.c K. H. Su sro@sics.wisc.edu J. C. Sro

2 frciol-order ssem wi order s low s 2.4 d ercos ws foud wi order s low s 3.8. I [2] ercoic evior of ieger-order olier ssem wi usle oscillors is reserved we e order ecomes frciol. I [3-5] e coic evior d is corol i e frciol order Ce ssem re ivesiged. M oer frciol-order olier ssems re coic suc s e frciol-order Areodo s ssem [6] e frciol-order Ce-Lee ssem [7] e frciol-order modified v der Pol ssem [9] d frciol-order roiol mecicl ssem wi cerifugl goveror [2]. Desie ese m emles e ifurcios of frciol-order olier ssems ve o e ee well sudied. I is er we reor e ifurcios occur i riculrl simle oe-rmeer versio of e Lore model clled e diffusioless Lore equios DLE descried i [2] d furer ivesiged i [22]. Te Kl-Yore dimesio of e diffusioless Lore ssem ws clculed d used s mesure is comlei i [23]. However e frciol-order vri of is ssem s o ee sudied d i is idel cdide for emiig ifurcios sice i s sigle ifurcio rmeer. Te er is orgied s follows. I Secio 2 e umericl lgorim for e frciol-order diffusioless Lore ssem is reseed. I Secio 3 e coic evior d ifurcios of e ssem re sudied. Fill we summrie e resuls d idice fuure direcios. 2. Frciol-order derivive d is umericl lgorim Tere re severl defiiios of frciol derivives [24]. Oe oulr defiiio ivolves ime-domi comuio i wic o-omogeous iiil codiios re ermied d ose vlues re redil deermied [25]. Te Cuo derivive defiiio [26] is give d f d f J d or d d f d f J d d were is e firs ieger wic is o less d iegrl oeror give J θ is e θ -order Riem-Liouville J θ θ θ ϕ τ ϕ τ dτ were θ is e gmm fucio wi < θ. 2 Te Diffusioless Lore equios re give & & 3 & R 2

3 were R is osiive rmeer. We R 5 coic soluios occur. Equio 3 s * * * equilirium ois ± R m R wi eigevlues sisf e 3 2 crcerisic equio λ λ Rλ 2R. Figure sows e coic rcor for e diffusioless Lore ssem for e vlue of R wic e Kl-Yore dimesio s is mimum vlue of Now cosider e frciol-order diffusioless Lore ssem give d d d 4 d d R d were deermie e frciol order <. Figure sows e coic rcor for e frciol-order diffusioless Lore ssem wi. 95 d R Fig. Coic rcors for e diffusioless Lore ssem i Eq. 4 wi R Tere re wo ws o sud frciol-order ssems. Oe is roug lier roimios. B usig frequec-domi eciques sed o Bode digrms oe c oi lier roimio for e frciol-order iegror e order of wic deeds o e desired dwid d e discrec ewee e cul d roime Bode digrms. Te oer is e Adms-Bsfor-Moulo redicor-correcor sceme [27-29] wic is ime-domi roc d us is more effecive. Here we derive geerliio of e Adms-Bsfor-Moulo sceme rorie for Eq. 4. Te followig differeil equio d f d T 5 2 L is equivle o e Volerr iegrl equio [27] 3

4 4! d f τ τ τ. 6 Oviousl e sum o e rig-d side is comleel deermied e iiil vlues d ece is ow. I icl siuio oe s << d ece e Volerr equio 5 is wel sigulr. I [27-29] e redicor-correcor sceme for equio 5 is derived d is roc c e cosidered o e logue of e clssicl oe-se Adms-Moulo lgorim. Se N T / d 2 N L wi T eig e uer oud of e iervl o wic we re looig for e soluio. Te e correcor formul for Eq. 6 is give 2 2! f f 7 were B usig oe-se Adms-Bsfor rule ised of oe-se Adms-Moulo rule e redicor is give! f 9 were. Now e sic lgorim for e frciol Adms-Bsfor-Moulo meod is comleel descried Eqs. 7 d 9 wi e weigs d eig defied ccordig o 8 d resecivel. Te error esime of is meod is m N O e L were. 2 mi Usig is meod e frciol-order diffusioless Lore equios 4 c e wrie s R R 3 2 } ] {[ 2 ] [ 2 } ] {[ 2 2

5 5 were R Bifurcios of e frciol-order diffusioless Lore ssem 3.. Frciol-order diffusioless Lore coic ssem I our simulios we ve visull iseced e ifurcio digrms o ideif cos. We lso ve cofirmed ese clculig e lrges Luov eoe i some cses usig e Wolf lgorim [3]. Here we le q d e effecive coic rge of e frciol-order diffusioless ssem wi differe corol rmeer is foud s sow i Fig. 2. Clerl ere eis ree differe ses corresodig o limi ccles cos d covergece. If e ieger-order ssem is coic e effecive coic rge of e frciol-order ssem decreses sligl s corol rmeer icreses. If e ieger order ssem is limi ccle e frciol-order ssem c roduce cos u e effecive coic rge of e frciol-order ssem sris sigificl s e corol rmeer icreses. Te lrges Luov eoe of e frciol-order ssem wi R8 is sow i Fig. 3. I is cosise wi e coic rge is vlue.

6 e lrges LE Fig. 2 Coic rge of e frciol order DLE. Fig. 3 Lrges LE of e frciol order DLE wi R Bifurcios wi differe corol rmeer R Here e frciol orders re equl d fied.95 wile e corol rmeer R is vried from.5 o. Te iiil ses of e frciol-order diffusioless Lore ssem re d For se sie i R is. d e ruig ime is 4s e ifurcio digrm i Fig. 4 ws oied. I sows e frciol-order DLE is coic wi oe eriodic widow we e ol order is We e corol rmeer R is decresed from e frciol-order ssem eers io cos eriod-doulig ifurcio s sow i Fig. 5 wi ses of.5. To oserve e dmic evior e eriodic widow is eded wi ses of.5 s sow i Fig. 5. Tere eiss ierior crisis we R 5.57 fli ifurcio we R 6.2 d ge ifurcio we R I e ierior crisis e coic rcor collides wi usle eriodic ori or limi ccle wii is si of rcio. We e collisio occurs e rcor suddel eds i sie u remis ouded. For e ge ifurcio sddle oi d sle ode colesce d iile oe oer roducig ori s eriods of cos iersersed wi eriods of regulr oscillio [3]. Tis sme evior occurs i e eriodic widows of e logisic m icludig e miiure widows wii e lrger widows [32]. I is eriodic widow we lso oserve e roue o cos eriod-doulig ifurcio. Fig. 4 Bifurcio digrm of e frciol-order diffusioless Lore ssem wi R for q.95. 6

7 Zm 5 Fli ifurcio 6 Ierior ifurcio Tge ifurcio R R Fig. 5 Bifurcio digrm of e frciol-order diffusioless Lore ssem wi R for q.95. R [9] R [57] 3.3. Bifurcios wi differe frciol orders Now le q d cge e frciol order q from.9 o u fi e corol rmeer R8. Te iiil ses of e frciol-order diffusioless Lore ssem d ruig ime e e sme s ove u wi vriiol se of R se o.5 gives e ifurcio digrm sow i Fig. 6. Tis cse sows e roue o cos for e frciol-order DLE s e frciol order decreses. I is ieresig o oe coic rsie is oserved we q is less.92. Te se sce recor is sow i Fig. 7 for q.9 wic suddel swices o er of oscillio decs o e rig equilirium oi. Te ime isor of i Fig. 7 lso sows eveul covergece o e fied oi. O e verge coic evior swices o dmed evior fer ou 7 oscillios. For lrger q < q. 92 coic evior ersiss loger. Similr evior s ee reored for e sdrd Lore ssem [33]. Te frciol-order ssem gives w o cos eriod-doulig ifurcio s sow i Fig. 8 wi ses i R of.2. I is coic i e rge of.92 o.962. We e frciol order is less.92 e frciol-order ssem coverges o fied oi. Edig e eriodic widow sows e dmic evior eer s i Fig. 8 wi ses of.. Tere eis ree ids of ifurcio i.c. ge ifurcio fli ifurcio d ierior crisis. Te frciol order rmeer c e e s ifurcio rmeer us lie e corol rmeer. Zm Fig. 6 Bifurcio digrm of e frciol-order diffusioless Lore ssem wi q for R8. 7

8 Fig. 7 Trsie coic evior i ssem 4 wi R8 d q.9. 3D view o e -- sce Te ime isories of vrile Zm Fig. 8 Bifurcio digrm of e frciol-order diffusioless Lore ssem wi q for R8. q [.95 ] q [.93.95] 3.4. Bifurcios wi differe frciol order for e ree equios Fi R8 d le vr. Te ssem is clculed umericll for [.4] wi icreme of equl o.2. Te ifurcio digrm is sow i Fig. 9. I is foud we e frciol-order ssem is coic wi oe eriodic widow c s sow i Fig. 9 wi ses of.2. We icreses from.4 or decreses from eriod-doulig roue o cos is oserved. 2 Fi R8 d le vr. Te ssem is clculed umericll for [.75] wi icreme of equl o.. Te ifurcio digrm is sow i Fig.. We decreses from eriod-doulig roue o cos is oserved d i coverges o fied oi we is less.8. Tus e ol smlles order of e frciol-order diffusioless Lore coic ssem is 2.8 i is cse. 3 Fi R8 d le vr. Te ssem is clculed umericll for [.8 ] wi icreme of equl o.. Te ifurcio digrm is sow i Fig.. Similr eome re foud u e coic rge of e frciol-order ssem is muc smller of e revious wo cses d i eers io cos eriod-doulig roue d e i coverges o fied oi we.86. Te dmic eviors i e eriodic widow re similr wi e cse 8

9 of 2 s sow i Fig. wi ses of.2. Zm Zm Fig. 9 Bifurcio digrm of e frciol-order diffusioless Lore ssem wi for. [.4] [.48.52] Zm Fig. Bifurcio digrm of e frciol-order diffusioless Lore ssem wi for Zm Fig. Bifurcio digrm of e frciol-order diffusioless Lore ssem wi for. [.8 ] [.886] 4. Coclusios I is er we ve umericll sudied e ifurcios d dmics of e frciol-order diffusioless Lore ssem vrig e ssem rmeer d e ssem order. Ticl 9

10 ifurcios suc s eriod-doulig ifurcios fli ifurcios ge ifurcios d ierior crisis ifurcios were oserved we o e corol rmeer d e frciol-order re cged. Comle dmic eviors suc s fied oi eriodic moio rsie cos d cos occur i is frciol-order ssem. Fuure wor o e oic sould iclude eoreicl lsis of e dmics of e frciol-order ssem s well s i-de sudies of cos corol d scroiio for e ssem. Acowledgmes Tis wor ws suored e Ci Scolrsi Coucil No.26A39 e Niol Nure Sciece Foudio of Peole s Reulic of Ci Gr No d e Niol Sciece Foudio for Pos-docorl Scieiss of Peole s Reulic of Ci Gr No We re greful for discussios wi Prof. George Rowlds. Refereces [] R. Hilfer Alicios of frciol clculus i sics; New Jerse: World Scieific2 [2] R. C. Koeller J Al Mec [3] T. T. Hrle C. F. L oreo Nolier D [4] T. T. Hrle C. F. Loreo H. K. Qmmer IEEE Trs Circuis Ss І [5] P. Are R. Coeo L. Foru D. Poro I: Proc. ECCTD Budes [6] Z. M. Ge C. Y. Ou Cos Solios & Frcls [7] W. Amd R. El-Kli A. El-Wil Elecr Le [8] I. Grigoreo E. Grigoreo Ps. Rev. Le [9] W. M. Amd J. C. Sro Cos Solios & Frcls [] W. M. Amd W. M. Hr Cos Solios & Frcls [] C. G. Li G. R. Ce Ps. A [2] W. M. Amd Cos Solios & Frcls [3] C. G. Li G. R. Ce Cos Solios & Frcls [4] C. P. Li G. J. Peg Cos Solios & Frcls [5] J. G. Lu G. R. Ce Cos Solios & Frcls [6] J. G. Lu. Cos Solios & Frcls [7] L. M. Tm W. M. S. Tou Cos Solios & Frcls [8] L. J. Seu H. K. Ce J. H. Ce & L. M. Tm Cos Solios & Frcls [9] Z. M. Ge A. R. Zg Cos Solios & Frcls [2] Z. M. Ge W. R. Jug Cos Solios & Frcls [2] J. C. Sro Ps Rev E [22] G. V. D. Scrier L. R. M. Ms Ps. D [23] J. C. Sro Cos [24] I. Podlu Frciol Differeil Equios; New Yor: Acdemic Press 999 [25] S. G. Smo A. A. Klis O. I. Mricev Frciol iegrls d derivives: eor d licios. Amserdm; Gord d Brec 993 [26] M. Cuo Geos J R Asro Soc [27] K. Dieelm N. J Ford J M Al Al [28] K. Dieelm Elecr Trs Numer Al [29] K. Dieelm N. J. Ford A. D. Freed Nolier D [3] A. Wolf J. Swif H. Swie. & J. Vso Ps. D [3] Y. Pomeu & P. Meville Comm i M Ps [32] J. C. Sro Cos d ime-series lsis; New Yor: Oford Uiversi Press23 [33] J. A. Yore E. D. Yore J of Sis Ps