Normal Form for Systems with Linear Part N 3(n)

Transcription

1 Applid Mathmatics Publishd Oli ovmbr ( ormal Form or Systms with Liar Part () Grac Gachigua * David Maloza Johaa Sigy Dpartmt o Mathmatics Collg o Tchology Kimathi Uivrsity yri Kya Dpartmt o Mathmatics Kyatta Uivrsity airobi Kya Pur ad Applid Mathmatics (P) Dpartmt Jomo Kyatta Uivrsity o Agricultur ad Tchology airobi Kya * wambuigachigua@gmailcom dmalo4@gmailcom jksigy@yahoocom Rcivd August ; rvisd Sptmbr 7 ; accptd Sptmbr 4 ABSTRACT Th cocpt o ormal orm is usd to study th dyamics o o-liar systms I this work w dscrib th ormal orm or vctor ilds o with liar ilpott part mad up o coupld Jorda blocks W us a algorithm basd o th otio o trasvctats rom classical ivariat thory kow as boostig to quivariats i dtrmiig th ormal orm wh th Staly dcompositio or th rig o ivariats is kow Kywords: Trasvctat; Equivariats; Box Product; Staly Dcompositio Itroductio Thr ar wll-kow procdurs or puttig a systm o dirtial quatios x Ax vx (whr v is a ormal powr sris startig with quadratic trms) ito ormal orm with rspct to its liar part A Our cocr i this papr is to dscrib th ormal orm o th systmm x Ax vx that is th st o all v such that Ax vx is i ormal orm whr A is th liar part rom th Staly dcompositio o th rig o ivariats Our mai rsult is a procdur that solvs th dscriptio problm whr is a ilpott matrix with coupld Jorda blocks providd that th dscriptio problm is alrady solvd or ach Jorda block o tak sparatly Our mthod is basd o addig o block at a tim This procdur will b illustratd with xampls ad th b gralizd Th ida o simpliicatio ar a quilibrium gos back at last to Poicar (88) who was amog th irst to brig orth th thory i a mor diit orm Poicar cosidrd th problm o rducig a systm o oliar dirtial quatios to a systm o liar os Th ormal solutio o this problm tails idig aridtity coordiat trasormatios which limiat th aalytic xprssios o th oliar trms Cushma t al [] usig a mthod calld covariat o spcial quivariat solvd th problm o idig Staly dcompositio o Thir mthod bgis by cratig a scalar problm that is largr tha th vctor problm ad thir procdurs ar drivd rom classical ivariat thory thus it was cssary to rpat calculatios o classical ivariats thory at th lvls o qui- * Corrspodig author variats Maloza [] solvd th sam problm by Grobr basis mthods oud i [] rathr tha borrowig rom classical thory Murdock ad Sadrs [4] dvlopd a algorithm basd o th otio o trasvtats to dtrmi th orm o ormal orm o a vctor ild with ilpott liar part wh th ormal orm is kow or ach Jorda block o th liar part tak sparatly Th algorithm is basd o th otio o trasvctats rom th classical ivariat thory kow as boostig to modul o quivariats wh th Staly dcompositio or th rig o ivariats is kow amachchivaya t al [5] studid a gralizd Hop biurcatio with o-smisimpl : Rsoac Th ormal orm or such a systm cotais oly trms that blog to both th smisimpl part o A ad th ormal orm o th ilpott which is a coupld Taks- i i Bogdaov systm with A i i This xampl illustrats th physical sigiicac o th study o ormal orms or systms with ilpott liar part Our rsults ar maily basd o th work oud i [4] that is applicatio o trasvctat s mthod or computig ormal orm or th modul o quivariats o ilpott systms I sctio two ad thr w put togthr backgroud kowldg or udrstadig th cott o this work Sctio our orms th ctral part o this papr whr w shall comput th modul o quivariats Copyright SciRs

2 64 G GACHIGUA ET AL Ivariats ad Staly Dcompositios m Lt j dot th vctor spac o homog ous polyomials o dgr j o with coicits m i whr dots th st o ral umbrs Lt m b th vctor spac o all such polyomials m o ay dgr ad lt b th vctor spac m o ormal powr sris I m bcoms th rig o ormal powr sris o whr dots th st o ral umbrs For such smooth vctors ilds it is suicit to work polyomials For ay ilpott matrix w di th Li oprator by L : j j L vx vxx vx ad th dirtial oprator by Th maig that I additio () : x j j x x x x x () x is a drivatio o th rig g g g () L v v L v (4) A uctio is calld a ivariat o Ax i At x t or quivaltly kr A Sic t g g g gg it ollows that i ad g ar ivariats so ar g amd g ; that is kr is both a vctor spac ovr ad also a subrig o kow as th rig o ivariats Similarly a vctor ild v is calld a qui- At At variats o Ax i v x that is t t v kr L A Thr ar two ormal orm styls i commo us or ilpott systms th ir product ormal orm ad th sl() ormal orm Th ir product ormal orm is did by iml krl whr is th cojugat traspos o To di th sl() ormal orm o irst sts X ad costructs matrics Y ad Z such that X Y Z Z X X Z Y Y (5) Copyright SciRs A xampl o such a () X YZ is X Y Z sl triad Havig obtaid th triad X Y Z w crat two additioal triads ad XYZ as ollows Y X X L Y L Z L Z (7) Y X Z (6) Th irst o ths is a triad o dirtial oprators ad th scod is a triad o Li oprators Both th oprators ad XYZ ihrit th triad proprtis (5) Obsrv that th oprators XYZ map ach ito itsl It ollows rom th rprstatio thory sl that imykrx imxkry (8) Clarly th kr ia s subrig o th rig o ivariats ad it ollows rom (4) that kr X is a modul ovr this subrig This is th sl() ormal orm modul Boostig Rigs o Ivariats to Modul o Equivariats I this sctio w dscrib th procdur or obtaiig a Staly dcompositio o th modul o quivariats (or ormal orm spac kr X ) wh th Staly dcompositio o th rig o ivariats is kow Th modul o all ormal powr sris vctor ilds o ca b viwd as th tsor product x x ad i act th tsor product ca b idtiid with th ordiary product (o a ild tims a costat vctor) sic th ordiary product satisis th sam algbraic ruls as a tsor product Spciically vry ormal powr sris vctor ild ca b writt as x x x x whr th i ar th stadard basis vctors o xt th Li drivativ X L ca b xprssd as * th tsor product o ad that is X I I Udr th idtiicatio o

3 G GACHIGUA ET AL 64 with ordiary product this mas X v v whr x x ad v i agrmt with th ollowig calculatio i which v bcaus v is costat X v L v v L v v v x v * v v This kid o calculatio also shows that sl rprstatio (o vctor ilds ) with triad XYZ is th tsor product o th rprstatio (o scalar ilds) with triad ad th rprstatio (o with triad M H that is kr X kr r It ollows that a basis rom th ormal orm spac kr X is giv by wll did trasvctats i v as rags ovr a basis or kr x x ad v rags ovr a basis or kr Th irst o ths bass is giv by th stadard moomials o a Staly dcompositio or kr Th scod is giv by th stadard basis vctors r such that r is th idx o th bottom row o a Jorda block i It is usul to ot that th wight o such a r is o lss tha th siz o th block Th w di th trasi vctat v as i i j i j j r W M j g i j i r j i j j g j W M g j r From hr th computatioal procdurs o box products ar th sam as thos usd i dscribig rigs o ivariats rom [4] xcpt that iiit itratios vr aris 4 ormal Form or Systms with Liar Part () Bor gralizig w shall cosidr th ormal orm or oliar systms with liar part havig two ad thr blocks that is ad as xampls 4 Systm with Liar Part Th Staly dcompositio or th rig o ivariats with liar part is giv by: kr (s [6]) Sic ad has wight zro it is covit to rmov thm sic w do ot xpad alog trms o wight zro by sttig ad writ kr I this cas th basis lmts ar ad Thr- 6 or w d to comput th box product o th rig kr with 6 which ar both o wight Thror kr X kr 6 Distributig th box product thr ar two cass to cosidr Cas : Thr ar our products amly: a) b) c) d) Rcombiig trms givs Cas : Similarly w hav Copyright SciRs

4 644 G GACHIGUA ET AL Addig trms i cas ad w obtai: kr X Fially to complt th calculatio it is cssary to comput th trasvctats that appar Ths ar o th orm i ad i 6 or i whr w w w w M w M W igor th ozro costats ad bcaus w ar cocrd with computig basis lmts For th basis w hav: Thror th ormal orm or systm with liar part is: kr X Copyright SciRs

5 G GACHIGUA ET AL Systm with Liar Part Th Staly dcompositio or rig o ivariats o a systm with liar part is giv by: kr (s [6]) Th basis lmts or kr ar ad 6 9 Thror w d to comput th box product o th ivariats rig kr with 6 9 Thus kr X kr 6 9 Lt th kr X = () () 6 9 Thr ar thr cass to cosidr Computig ad simpliyig th cass w obtai th ormal orm as: Copyright SciRs

6 646 G GACHIGUA ET AL kr X i i i i i i i i i i i i i i i i i i i i i i whr i ad such that 6 ad 9 I gral rom th abov xampls w coclud that th ormal orms ar obtaid by computig th box product kr X kr Th basis o th ormal orm o i kr X ar trasvc- tats o th orm: whr is th stadard moomials o Staly dcompositio o th rig o ivariats kr i ad As a xampl w id th ormal orm or a systm with liar part w irst id th rig o ivariats kr whr x y usig x yz By i- y z spctio x ad y xz ad this grats th tir rig; that is kr (4) To chck this w ot that th wight o is two ad is o wight zro so th tabl uctio o is Hc T dw d w T w d this implis () Th xt stp is to comput kr X as a modul ovr kr cotais o Jorda block o siz hc th dirtial oprators x y y z y z x y I this cas th basis lmts is wight thror th ormal orm is kr X kr X which is o Copyright SciRs

7 G GACHIGUA ET AL 647 x x y y z z W comput: x y x z y x 4 x x y y z z x x y y z Th dirtial quatios i yx sl ormal orm ar: x yh x y xz x y x x x y zg x y xz xh x y xz y zg x xh x y y xx x xx y z x xy z x y xz x g x y xz y h x y xz z x xg x yh x z x x z x x x x x y x xyxz Th ormal orm upto quadratic trm is: x x x y y x xy z z x xy xz Rmark: Th ormal orm o a dyamical systms is a powrul tool i th study o stability ad biurcatios aalysis From th practical poit o viw oly th o rmal orm with prturbatio (biurcatio) paramtrs is us ul i aalyzig physical or girig problms I this papr th computatio o th ormal orm has b maily rstrictd to systms which do ot cota i prturbatio paramtrs by sttig th paramtrs to zro to obtai th simpliid ormal orm Havig oud th ormal orm o th rducd systm w shall th add uoldig trms to gt a paramtric ormal orm or biurcatio aalysis REFERECES [] R Cushma J A Sadrs ad Whit ormal Form or th (;)-ilpott Vctor Fild Usig Ivariat Thory Physica D: oliar Phoma Vol o 988 pp 99-4 doi:6/67-789(88)98- [] D M Maloza ormal Forms or Coupld Taks-Bogdaov Systms Joural o oliar Mathmatical Physics Vol o 4 pp doi:99/jmp48 [] W W Adams ad P Loustauau A Itroductio to Gröbr Bass Amrica Mathmatical Socity Providc 994 [4] J Murdock ad J A Sadrs A w Trasvctat Algorithm or ilpott ormal Forms Joural o Dirtial Equatios Vol 8 o 7 pp 4-56 doi:6/jjd76 [5] Sri Amachchivaya M M Doyl W F Lagord ad W Evas ormal Form or Gralizd Hop Biurcatio with o-smisimpl : Rsoac Zitschrit ür Agwadt Mathmatik ud Physik (ZP) Vol 45 o 994 pp -5 doi:7/b F9458 [6] G Gachigua ad D Maloza Staly Dcompositio o Coupld Systm Procdigs o th st Kyatta Uivrsity Itratioal Mathmatics Corc airobi 6- Ju pp 9-5 Copyright SciRs