On The Fractional Euler Top System with Two Parameters

Transcription

1 Irol OPEN ACCESS Jourl Of Modr Egrg Rsrh IJMER O Th rol Eulr To Sys wh Two Prrs Mh Iv d Ghorgh Iv Ws Uvrsy of Tşor r of Mhs Srul d Gor ş Toolog 4 B-dul V Pârv Tşor Ro Corrsodg Auhor: Mh Iv W rs h dyl hvor of fly of frol dffrl syss ssod o Eulr o sys wh wo rrs Th frol sly of ulru ss for h rurd frol Eulr o sys wh wo lr orols s sudd lly h url gro d url sulo of h r dsussd Ky words: frol Eulr o sys wh wo lr orols frol sly Mhs Su Clssfo : 6A 57 65L of Susso: 6--8 of : I INTROUCTION Th lo-posso syss r urlly y rs of hysl s d grg roos sl dys d sur ouos 9 ] A sl lss of lo-posso syss s ford y fly of dffrl uos o R whh dds y rl of rl rrs d wo rrs d lld h grl Eulr o sys wh wo rrs Ths fly os vrous grl syss for s: rulr s of R sys sl s of Rovh sys d y ohrs Th R sys srvs s odl for h rvrsls of olry of h Erh's g fld 4] Th Rovh sys odls h dys of hr rsoly ould wvs rrlly d ] Th frol lulus hs foud o or ool vrous flds suh s hs hyss grg hsry ology oos ho dys d ohr ol dyl syss 5 7 ] A lss of frol dyl syss s ford y fly of frol dffrl uos o R ssod o grl Eulr o sys wh wo rrs lld h grl frol Eulr o sys wh wo rrs Ths r s sruurd s follows I So h grl Eulr o sys wh wo rrs s rlzd s lo-posso sys I So w rodu h frol Eulr o sys wh wo lr orols Ths so s dvod o sudyg of h frol sly of ulru ss for h frol sys I So 4 h url gro d url sulo for h rurd frol Eulr o sys ssod o dys s dsussd II GENERAL EULER TOP SYSTEM WIT TWO PARAMETERS or dls o lo dys s g 9] W osdr h followg dffrl sys of Eulr y o R : whr d / d R for suh h d s h IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8

2 O Th rol Eulr To Sys Wh Two Prrs If s lld h grl Eulr o sys wh wo lr orols If os h grl Eulr o sys 6] gv y: ; whr R suh h IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 ; W wll do h vor of rrs y Rr I 6] s showd h h grl Eulr o sys y rlzd s lo-posso sys f ur of dffr wys El L suh h I I I I I I If w R h: ; ; Th dys s lld h rgd ody uos wh wo lr orols whr rrss h gulr vloy vor of rgd ody I I I h oos of s r sor d R r rrs If w h w o h Eulr uos of h fr rgd ody o orhogol grou SO 9] or d h sys os: ; ; 4 whr R wh r rrs Ths sys s lld h R-lo sys wh o udr orol 4 8] or d R h sys os: ; ; 5 No h h sys 5 s h grl Rovh sudd ] h s I s lld h Rovh sys wh wo lr orols or h dys 5 s h Rovh sys 5] Rr Aog h sudd os rld o h syss gv El w rll h osruo of lo-posso rlzos d olr sly rol for h syss wh 9] 4 8] d 5 wh 5] Prooso L R A lo-posso rlzo of h Eulr o sys s R P wh h Csr C R P C R whr C

3 O Th rol Eulr To Sys Wh Two Prrs IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 Proof W hv C C C Th P P P X whr T X T d s lo-posso sys Also C s Csr s C P Corollry A lo-posso rlzo of h R-lo sys 4 s P R wh h Csr R R C C whr P C A lo-posso rlzo of h Rovh sys 4 s P R wh h Csr R R C C whr P C Proof Th ov ssros follows fro Prooso y rl of rrs d wh h orrsodg vlus

4 Prooso Th fuos C C R R oo frs grls for h dys Proof Idd d / d Slrly w hv / d Rr S dc d O Th rol Eulr To Sys Wh Two Prrs g v y 7 d 8 r oss of h C r frs grls follows h h rors of oo of h dys r rsos of h surfs: os d os III RACTIONAL EULER TOP SYSTEM WIT TWO LINEAR CONTROLS L f C R d R Th ordr Cuo dffrl oror 5] s dsrd y f I f whr f rrss h ordr drvv of h fuo f N s gr suh h d I s h ordr R-Louvll grl oror ] whh s rssd s follows: I f s f s ds whr s h Eulr G fuo If h f df / d I hs r w suos h ] Th frol Eulr o sys wh wo lr orols ssod o dys s dfd y h followg s of frol dffrl uos: whr R for wh d R El If w os h frol R-lo sys gv y: ; or d R h sys ; ; h h o os: ; Th sys s lld h frol Rovh sys wh wo lr orols or s h frol Rovh sys Prooso Th ulru ss of h frol Eulr o sys r gv s h uo of h followg hr fls: IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8

5 E : R R E : R R O Th rol Eulr To Sys Wh Two Prrs E : R Proof Th ulru ss r soluos of h uos f whr f f d f R L us w rs h sudy of frol sly of ulru ss for h sys lly w wll dsuss how o slz h usl ulru ss of sys v frol ordr drvv or hs sudy w ly h Mgo's s ] Th Jo r ssod o sys s: J Prooso ] L ulru s of sys d J h Jo r J vlud s lolly syolly sl ff ll gvlus of h r J ssfy: rg J s lolly sl ff hr s syolly sl or h rl gvlus of J whh ssfy rg J hv gor ully o Prooso Th ulru ss r usl Proof Th hrrs olyol of h r Th hrrs olyols of rs J J d J I J d J r h followg: d ] J J Th uos hv h roo J J J s S rg for ll y Prooso follows h h ulru ss r usl for ll I h s wh s usl ulru s of h frol sys w sso o w frol sys lld h rurd frol Eulr o sys wh wo lr orols for h ulru s If o sls h rrs whh h h gvlus of Jo r IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 4

6 O Th rol Eulr To Sys Wh Two Prrs of h rurd frol sys ssod o sys for h ulru s ssfy o of h odos fro Prooso h h rors of h syolly rohs h usl ulru s h ss h l whr s h Euld or Th rurd frol Eulr o sys wh wo lr orols for h ulru s s dfd y: whr R r rl oss IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 5 Th Jo r of h rurd frol Eulr o sys 4 for J Prooso 4 L h rurd frol Eulr o sys 4 L d R If h s syolly sl ; If h s syolly sl r d s sl for If L h s whr s syolly sl for ll 4 d Proof Th hrrs olyol of J s ] Th roos of h hrrs uo r Cs Th W suos h d I hs s w hv d R S rg for ll y Prooso ls h s lolly syolly sl for ll R or d w hv d R Alyg Prooso

7 O Th rol Eulr To Sys Wh Two Prrs s lolly syolly sl for whr r h ssro holds If h s sl or L d R S J gvlu d so Cs s usl for ll R hs ls osv Thrfor h ssros hold Th Suos d Th for s usl R d for S h gvlus r ll gv for follows h s syolly sl for ll If d h for d for J hs ls osv gvlu d so s usl L d R S J hs ls osv gvlu d so s usl R h ssro holds If 4 w wh d R h o os h rurd frol R-lo sys for Corollry L h rurd frol R-lo sys for If d h d Proof S h ssro follows fro Prooso 4 s syolly sl R Th rurd frol Eulr o sys wh wo lr orols for h ulru s s dfd y: whr r rl oss R Th Jo r of h rurd frol Eulr o sys 5 for s 5 IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 6

8 O Th rol Eulr To Sys Wh Two Prrs J Prooso 5 L h rurd frol Eulr o sys 5 L d R If h If h s syolly sl ; s syolly sl r d s sl for If L h whr s syolly sl for ll d Proof L h hrrs olyol of r J Th roos of h hrrs roos of ] r Alyg h s rsog s h dosro of Prooso 4 s sy o rov h h h ssros d hold If 5 w d R Rovh sys for h o os h rurd frol Corollry L h rurd frol Rovh sys for s syolly sl If d h d Proof S h ssro follows fro Prooso 5 Th rurd frol Eulr o sys wh wo lr orols for h ulru s s dfd y: whr R r rl oss Th Jo r of h rurd frol Eulr o sys 6 for J Prooso 6 L h rurd frol Eulr o sys 6 L d R If h s syolly sl ; s 6 IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 7

9 If h r O Th rol Eulr To Sys Wh Two Prrs s syolly sl L whr d s sl for If d syolly sl for ll Proof L h h hrrs olyol of r J whr ] Th roos of uo r whr d d Cs d Th d W suos h d Th d R S rg for ll y Prooso ls h s lolly syolly sl for ll R or d d w hv d R Alyg Prooso s lolly syolly sl for whr d r Thrfor h ssro holds If h s usl for ll R s sl or L d R h d J hs ls osv gvlu s usl for ll R Thrfor h ssros hold Cs d Th I hs s d oly f d I follows h ll If 6 w h o os h rurd frol rgd ody uos wh wo lr orols for s r ll gv f d s syolly sl for Corollry L h rurd frol rgd ody uos wh wo lr orols for If h s syolly sl for ll R ssfyg h odo d Proof S h ssro follows fro Prooso 6 El L h rurd frol R-lo sys for or 75 h odos of Corollry r hvd Th 75 s syolly sl L rurd frol Rovh sys for If w sl 5 d 75 Th 755 s syolly sl for h h odos of Corollry r hvd IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 8

10 O Th rol Eulr To Sys Wh Two Prrs IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8 9 IV NUMERICAL INTEGRATION O RACTIONAL MOEL 4 Th frol dffrl syss 6 4 wr h grl for: 4 whr R r rl oss Th frol dys 4 s lld h rurd frol Eulr o sys wh wo lr orols for h ulru s or l f h sys 4 w d o os h rurd frol Eulr o sys 4 for h ulru s Cosdr h frol dffrl uos 4 whr S h fuos r ouous h h l vlu rol 4 s uvl o olr Volrr grl uo of h sod ordr 5] whh s gv s follows: 4 ds s s s s hl l usd h rdor-orror sh 4] sd o h Ads- Moulo lgorh o gr h uos 4 W ly hs sh o frol sys 4 or hs l h N h for N Th rurd frol sys 4 grd s follows: h h ] ] ] ] ] 4 4 ] ] ] ] ] ] ] ] whr: ] ] ]

11 O Th rol Eulr To Sys Wh Two Prrs Th ov sh gv y h rlos 44 s lld h Moulo-Ads lgorh for rurd frol sys 4 4] Th rror s for h lgorh dsrd y 44 s N ] ] O h L us w ly h lgorh 44 d sofwr Ml o gr wo rurd frol Eulr o syss wh wo lr orols of y 4 or hs w h N 5 5 d h l odos whr Ths osdros r lfd h followg ss L h frol odl 4 for h ulru s ssod o R- lo sys 4 wh 75 d whh hs dsussd El I h rlos 44 w : 5 75 d I h oord sys O h ors of soluos of frol R- lo odl 4 for h ulru s 75 hv h rrsos gv h fgurs g for 75 d g for g Ors of R odl for 75 IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8

12 O Th rol Eulr To Sys Wh Two Prrs g Ors of R odl for L h frol odl 4 for h ulru s ssod o Rovh sys 5 wh 75 d 5 whh hs rsd El I h rlos 44 w : d 5 I h oord sys O h ors of soluos of frol Rovh odl 4 for h ulru s 755 hv h rrsos gv h fgurs g for 75 d g 4 for g Ors of Rovh odl for 75 IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8

13 O Th rol Eulr To Sys Wh Two Prrs g 4 Ors of Rovh odl for Th url sulos show h vldy of h horl lyss V CONCLUSIONS Th dys of h frol Eulr o sys wh wo orol rrs ws sudd Th lyss of h frol sly of for h rurd frol odl ssod o sys hs vsgd lly h url sulos for soluos of rurd frol R-lo sys d rurd frol Rovh sys r gv ACKNOWLEGMENTS Th uhor hs vry grful o rvwrs for hr os d suggsos REERENCES ] Z Chg y lyss of frol ordr Rösslr sys I J Nolr S 7-5 ] O T Chş d M Pu Gorl d dyl ss h hory of Rovh sys I J Go Mhods Modr Physs ] vd d ol Mull L-Posso sruurs rduo d gor hss for h Mwll-Bloh rvlg wv uos J Nolr S ] K hl N J ord d rd Prdor-orror roh for h url soluo of frol dffrl uos Nolr ys ] Khl rol ffrl Euos Thory d Nurl Tr TU Brushwg Brushwg 6] Gh Iv Gorl d dyl rors of grl Eulr o sys Id J Pur Al Mh ] M Iv Gh Iv d Orş rol uos of h rgd ody o h sudo-orhogol grou SO; I J Go Mhods Modr Physs ] C Lăzuru d T Bâzr A R y sys wh udr orol I J of Bfuro d Chos gs 9] JE Mrsd TS Rţu Iroduo o Mhs d Syry Ald Mhs 7 Srgr-Vrlg 999 ] Mgo Sly rsuls for frol dffrl uos wh los o orol rossg I: Prodgs of h IMACS-SMC 96 July 996 Lll r ] AE Mou Chos fd orol d syhrozo of frol-ordr od_d Auooous V dr Pol-uffg ru Cou Nolr S Nur Sul ] AS Povsy d MI Rovh Sohs hvor of dssv syss S Rv C: Mh Pys Rv ] I Podluy rol ffrl Euos Ad Prss S go 999 4] T R Osllos of sys of ds dyos Pro Cr Phlos So ] RM Tudor d A Gâr O h lo dys d gory of h Rovh sys sr Co y Sys Sr B Mh Iv "O Th rol Eulr To Sys wh Two Prrs Irol Jourl Of Modr Egrg Rsrh IJMER vol 8 o 4 8 IJMER ISSN: wwwro Vol 8 Iss 4 Arl 8