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Randell Reynolds
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1 hi i a pe-pblihed veio. U Dae: -6- ime: : A New Pebaive Appoah i Noliea Siglaiy Aalyi a-leg Yee Depame of Mahemai ad Ifomaio ehology he Hog og Iie of Edaio ai Po New eioie Hog og Aba: he dy i devoed o he mio mehod whih eable oe o dy he iegabiliy of oliea diffeeial eqaio. A pebaive exeio of he mio mehod i ioded. he mio yem ad i fi pebaio ae he ilized o gai iigh io eai oliea eqaio poeig egaive Fh idie whih wee pooly deood i he lieae. I paila fo a o-piipal b maximal Pailevé family he fi-ode pebed eie olio i aleady a loal epeeaio of he geeal olio whoe ovegee a alo be poved. ey wod: Mio afomaio Pailevé e Siglaiy aalyi Odiay Diffeeial Eqaio ODE Pailevé e Siglaiy aalyi Mio yem maximal family pebaio expaio egaive Fh idie INODUCION he eleva lieae dy dae bak o oe ey ago whe Pailevé made a i-deph dy of iglaiie ad iiiaed he ow amed Pailevé aalyi of iegabiliy. Pailevé e p he poblem of deemiig all diffeeial eqaio whoe geeal olio ae igle-valed. Followig he pioeeig wok of Pailevé 9 he mehod of Gambie 99 Bea 96 Ablowiz e al. 98 ad Wei e al. 98 have bee evolved ad hey wee efl o apply i may ae. Howeve he mai dawbak of he mehod i ha oe of hem a bild eeay odiio a all iege vale of eoae. o be peifi egaive Fh idie ao be hadled by hee mehod. he eao why he mehod ao hadle egaive idie lie i he fa ha hei Lae eie i amed o be boded fom below. he mio mehod e he ew ool i iglaiy aalyi: mio afomaio ad egla mio yem whih wa fi ioded by H ad Ya 999. By hi mehod hey wee efl i he followig eveal ape. Fily he e of oig mio afomaio eable o ea eah piipal balae i he Pailevé e iglaiy e ad ymplei e of Hamiloia yem fom a ommo poi of view. Seodly H e al. howed ha he mio afomaio ae aoial fo fiie-dimeioal Hamiloia yem. Moeove Yee demoaed ha he lieaizaio of mio yem ea movable pole give he poibiliy o o he aoiaed Bakld afomaio of ome paial diffeeial eqaio ad he Shleige afomaio of ome odiay diffeeial eqaio. I he e wok o pimay goal i o iode a impoveme of he mio mehod o ha egaive idie eoae a be eaed. he e of he dy a ow be explaied. he pebaive Pailevé mehod i fi ioded. We demad iglevalede o oly fo ay pole-like expaio a i he Pailevé e b alo fo evey olio loe o i epeeed a a pebaio eie i a mall paamee ε. he age of he idea of he pebaive mehod pove o be emedoly beefiial fo he mio mehod a a ew impoveme. Ode-zeo i he al mio yem. Ode-oe ede o a lieaizaio of mio yem ea a egla iglaiy ad allow he iodio of all miig abiay oeffiie. Highe ode lead o he aalyi of a liea Fhia ype ihomogeeo yem. I paila egaive idie give ie o dobly ifiie Lae eie. A illaive example of Bea eqaio i alo peeed ad fially he olio follow. MAEIALS AND MEHODS Pebaive Pailevé aalyi: Now we fi pee he pebaive mehod oigially developed by Coe e al. 99. he mehod allow o exa he ifomaio oaied i he egaive idie h bildig ifiiely may eeay odiio fo he abee of movable iial iglaiie of he logaihmi ype. he Hog og Iie of Edaio Libay Fo pivae dy o eeah oly. No fo pbliaio o fhe epodio.

2 hi i a pe-pblihed veio. Ed ime Le oide a oliea odiay diffeeial eqaio: E x whih i polyomial i ad i deivaive aalyi i x. he adad Pailevé expaio ake he fom i he expaio vaiable x : E E E p q i whih he egaive iege p ad q ae he epeive iglaiy ode of ad E. We eek a Lae expaio fo ay olio whih i ea o he olio obaied by he adad Pailevé mehod. We do hi by oideig a pebaio expaio. Fo a o-piipal b maximal Pailevé family he pebaio exed he paila olio io a epeeaio of he geeal olio. Le defie he Pailevé expaio E a he olio of pebed poblem ad look fo a eaby olio fomally epeeed by a ifiie pebaio eie i powe of mall paamee ε: aylo ε x E ε E ε. Le deoe fo he eqaio E k he e of idie fo k h family { - } wih he followig ampio: all idie ae dii iege ad o be he malle idex -; all k familie ae maximal familie wih a mbe of idie eqal o he ode of he eqaio; a lea oe of he k familie i piipal ay maximal family wih apa fom - all iege idie o-egaive. Now he odiio ha he pebaio expaio ill be a olio geeae a ifiie eqee of eive diffeeial eqaio: E x E whee i he Féhe opeao aig o. A eah level of pebaio we o a pole expaio b he ode of he pole ieae wih he ode of he pebaio. he elig ifiie pebaio expaio i a dobly ifiie Lae expaio: E ε ε E p q E p q. I geeal pebaio heoy paially alway yield divege eie. Howeve by oideig he pebaio eie olio ε ad expadig i wih oeffiie depede of ε we a pove he ovegee. hi a be doe by iodig a ew afomaio fo he mio yem. Eveally we eed o dede a egla exeded mio yem wih egla iiial daa. he Cahy-owalevki heoem i he applied ad ovegee follow aodigly. he impoae of he pebed olio ε i ha: fo a o-piipal b maximal Pailevé family i i aleady a loal epeeaio of he geeal olio. I he followig we iveigae i deail eah ode of ε: Wih E x whee a eihe paila o geeal olio of he oigial oliea eqaio whih i deemied by adad Pailevé aalyi p A B A. Wih E whee he geeal olio of homo eqaio a paila olio of ihomo eqaio p A B - 5 whee A B ae abiay oeffiie ioded a level oe. he Pailevé eie a Lae eie whih i boded below i bied io he lieaized eqaio he elig eqaio fo i of Fhia ype he movable iglaiy of he oigial ODE i a egla iglaiy fo he lieaized eqaio ad i Fh idie ae ip whee i ove he Pailevé eoae. A hi fi ode a abiay oeffiie i ioded a eah idex. No all of hee ae ew ie we aleady have a oeffiie i oepodig o eah poiive iege idex. he oeffiie ioded io a he oepodig idie ip i a poiive iege peb he aleady abiay oeffiie he Hog og Iie of Edaio Libay Fo pivae dy o eeah oly. No fo pbliaio o fhe epodio.

3 hi i a pe-pblihed veio. Ed ime o add ohig ew ad i i o hamfl o e hem zeo a hi level. Howeve all ohe idie give ie o ew abiay oeffiie. heefoe he expeio ε aleady oai a may abiay oeffiie a hee ae idie i he family. Wih E - whee p A B - p C D - 6 whee A B ae abiay idepede of A i B i i - -h level oeffiie ha a be abobed by. Wiho he lo of geealiy we e A B fo eah. heefoe fo we oly oe abo a paila olio of eah ihomogeeo eqaio. he oeffiie C D ae depede of he pevio efl oeffiie whih belog o a be of {A i B i i }. A hee -h ode eah fio aifie a ihomogeeo liea diffeeial eqaio. he idiial eqaio i he ame fo all b fo he leadig behavio of i deemied by he iglaiy ode of he h fio o by. ESULS ad 6 d 6 9 whee : - ad ae abiay. Now we ae applyig he pebaive Pailevé aalyi o he eod family F ad he el ead: A 6 [ A A 6 d h he elig ifiie pebaio expaio fo F i a dobly ifiie Lae expaio: ε ε ε he loal epeeaio of he geeal olio i give by. ]. We begi o illae hogh a imple b iive example ha he aalyi o mio yem migh be pefomed i a pebaive appoah h ha egaive ad poiive idie a be eaed a he ame ime. We illae he algoihm of pefomig he ew pebaive appoah o he mio yem hogh he followig ODE example of hid-ode kid amely he Bea eqaio. We alo aim a howig he poof of ovegee of he o piipal balae of mio yem. he Bea hid-ode ODE i E whee - - -d 7 whee ad d ae fio of. By he adad Pailevé e we obai wo familie of olio wih iglaiy ode ad Fh idie i he followig: F p - {- } F p - {-- }. he Pailevé eie of F-F ae epeively: 8 * ε εa εa 6. o demoae expliily he exeio of he mio mehod we iode he oepodig mio afomaio fo he oigial eqaio give by 7: v w. he egla mio yem i give by: d. he above mio yem a be expeed a whee. Baed o he domia balae we obai he wo familie of olio of he he Hog og Iie of Edaio Libay Fo pivae dy o eeah oly. No fo pbliaio o fhe epodio.

4 Ed ime mio yem wih he followig iglaiy ode ad Fh idie: F p -- {- } F p -- {-- }. he Pailevé eie of F-F ae epeively: [ ] [ ] ad. 5 6 he pebaive expaio fo he mio yem i: ε 7. he fi few em ae deemied by : : : whee. Fially he eive lieaizaio of mio yem a be deemied ow. We oide he opiipal balae F oly. Wih. d 8 whih deemie a give by 6. Wih. 9 whih give. whee ad ae idepede abiay oa ioded a he zeo ad he fi level whih oepod o idie - ad - epeively. So i he ew impoa paamee ha we ae lookig fo. A hi level we e aohe wo abiay oa o zeo wiho ay lo of geealiy ie he abiay oa a idie - ad ae aleady epeeed io. Sie he family F i maximal he he pebed olio ε i a loal epeeaio of he geeal olio. Oe ideed a oie o look fo highe level pebaio i ode o obai a dobly hi i a pe-pblihed veio. he Hog og Iie of Edaio Libay Fo pivae dy o eeah oly. No fo pbliaio o fhe epodio.

5 hi i a pe-pblihed veio. Ed ime ifiie expaio. We li he eod level lieaizaio of mio yem below ad he ifomaio p o i good eogh fo o ppoe. Wih. DISCUSSION he ew afomaio fo he mio yem a be deemied baed o he above el. I he followig le alo pove he ovegee of he pebaio eie olio ε of F. Agai we hall e a ew afomaio o ove he oigial mio yem io a ew egla yem of fi-ode diffeeial eqaio wih egla iiial daa. Wih he family F we dede he followig Lae eie baed o 6 ad : ε ε ε 6 5 ε 6 ε whee ad ae he Pailevé eoae a Fh idie - ad epeively. We eaily ee ha he olio blow p whe o. We obeve ha fa ha i he oly eoae vaiable blowig p i he ode of O. So we fi iode he ew vaiable δ by ad fomally ive - io a eie of δ. I hi example i i δ ε δ ε δ O δ. Nex we fomally expad ad io eie of δ : εδ ε ε δ. By aig he δ eie fo a he loaio of he fi eoae o iode a ew vaiable ξ ad imilaly aig he δ eie fo a he loaio of o iode ξ we he dede he ew afomaio δ ξ : ξ ξ δ δ. 5 ξ he exeded mio yem beome: δ ξ ξ δ δ ξ ξ ξ ξ δ ξ d. he Lae eie fo δ ξ ξ ae: δ ε ξ ε ε ξ whih give he iiial daa: 6 7 δ ξ ξ ε. 8 he ovegee of he geeal olio a ow be died. Fo a o-piipal b maximal family F he fi-ode pebed eie olio i aleady a loal epeeaio of he geeal olio. I ode o how he ovegee of we eed he afomaio: δ ξ ξ. 5 he Hog og Iie of Edaio Libay Fo pivae dy o eeah oly. No fo pbliaio o fhe epodio.

6 hi i a pe-pblihed veio. Ed ime By he Cahy-owalevki heoem he exeded mio yem 6 wih he iiial daa 8 ha a iqe aalyi olio δ ξ ξ ea. he ξ δ δ ξ i a olio of he oigial mio yem ea. Moeove fom he odiay powe eie mehod we a fid he expaio fo δ ξ ξ. he a eay allaio eveal ha he Lae eie of ξδ δ ξ ae exaly. he ovege powe eie olio of he exeded mio yem lead o ovege Lae eie olio of he oigial mio yem beae of he eqivalee bewee he yem. hi pove he ovegee of ε i. I paila he eie of Fom o ε i ovege. ε ε ε ε we a fid he expaio fo ad hey ae exaly p o he ode whee all he eoae appea. hi pove he ovegee of Lae eie olio ε whih i loally epeeig he geeal olio fo F. EFEENCES Ablowiz M. J. A. amai ad H. Seg 98. A oeio bewee oliea evolio eqaio ad odiay diffeeial eqaio of P-ype. J. Mah. Phy. :75-7. Bea F.J. 96. Diffeeial Eqaio wih fixed iial poi Aali di Ma. pa ed appliaa LVI -6. Coe. A. P. Fody ad A. Pikeig 99. A pebaive Pailevé appoah o oliea diffeeial eqaio. Phyia D 69: -58. Gambie B. 99. S le éqaio difféeielle d eod ode e d pemie degé do l iégale gééale a e poi iiqe fixe hèe Pai. Aa. Mah. :-55. H J. ad M. Ya 999. Siglaiy aalyi fo iegable yem by hei mio Nolieaiy : 5-5. H J. ad M. Ya. he mio yem of iegable eqaio. Sd. Appl. Mah. :67-9. H J. M. Ya ad. L. Yee. Mio afomaio of Hamiloia yem Phyia D. 5: -. Pailevé P. 9. S le éqaio difféeielle d eod ode e d ode péie do l iégale gééale e ifome. Aa Mah. 5: -85. Wei J. M. abo ad G. Caevale 98. he Pailevé popey fo paial diffeeial eqaio J. Mah. Phy. :5-56. Yee. L.. Lieaizaio of mio yem. J. Noliea. Mahe. Phy. 9: -. CONCLUSION I hi dy we ae yig o iode a pah o he mio mehod o ha he egaive Fh idie a be eaed. hi oideaio exed he e of mio afomaio o a lage la of diffeeial eqaio. Baed o he example de oideaio we ae efl i eaig he egaive Fh idie. Ode-zeo pebaio give he odiay mio yem. Ode-oe ede o a lieaizaio of mio yem ea a egla iglaiy ad allow he iodio of all miig abiay oeffiie. he mehod eveal ha ε i aleady a epeeaio of he geeal olio whoe ovegee a alo be poved. 6 he Hog og Iie of Edaio Libay Fo pivae dy o eeah oly. No fo pbliaio o fhe epodio.

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