Localized Lyapunov exponents and the prediction of predictability

Transcription

1 $ Phyc ttr A locat pla a Chrtn ocalz yapunov ponnt an th prcton of prctalty Zhmann a onar A. Smth ac urgn Kurth a Un rtat Potam Inttut fur Phy Nchtlnar ynam Potfach Potam Grmany Mathmatcal InttutUn rty of Ofor Ofor OX 3B UK c onon School of Economc onon CA AE UK Rcv 9 cmr 999; rcv n rv form 9 Aprl 000; accpt 3 May 000 Communcat y C.R. orng Atract Evry forcat houl nclu an tmat of t lly accuracy a currnt maur of prctalty. Two tnct typ of localz yapunov ponnt a on nfntmal uncrtanty ynamc ar nvtgat to rflct th prctalty. Rgon of hgh prctalty wthn whch any ntal uncrtanty wll cra ar provn to t n two common chaotc ytm; potntal mplcaton of th rgon ar conr. Th rlvanc of th rult for fnt z uncrtant cu an llutrat numrcally. 000 Elvr Scnc B.V. All rght rrv. PACS: a; a; Tp Kywor: yapunov ponnt; Prctalty; Chao; athr forcatng; Enml prcton; Nonlnar ynamc. Introucton Prcton of prctalty rfr to th quanttatv attmpt to a th lly rror n a partcular forcat a pror. Thr ar at lat thr ourc of Corrponng author. Tl.: fa: E-mal ar: chr@agnl.un-potam. C. Zhmann. ffculty n quantfyng prctalty: th pnnc of maur " of# prctalty upon th partcular mtrc aopt th pnnc of uncrtanty ynamc upon th magntu % & of th uncrtanty n th ntal conton an 3 th fact that rror n th mol ar oftn unnown untl aftr prcton ' ( ar orv to fal. yapunov ponnt 3 quantfy * prctalty through gloally avrag ffctv + growth rat of uncrtanty n th lmt of larg tm an mall uncrtanty; thu y contructon thy ar of lmt u. To otan a quanttatv tmat of th accuracy of a partcular forcat th local y- of uncrtant aout that ntal -namc conton / $ - front mattr Elvr Scnc B.V. All rght rrv. PII: S

2 B G C E F A * c tu } ½ Ä Ž ¼ Í ž Ê y z U 4 5 ar mor rlvant 4 9. By allowng ttr r amnt a prcton of prctalty of valu n any fl from phyc to conomc; wathr forcatng prov a partcular ampl rlvant to oth fl. Effct 8 + growth rat fn ovr a f uraton ar alo u to quantfy prctalty; thy ar 9mploy aly n th opratonal wathr : forcat ; cntr of Europ an North Amrca 0. In Scton th tncton twn what wll call fnt tm ponnt an fnt ampl ponnt hown to l n th partcular ntal = orntaton of th prturaton 9ach conr for a gvn ntal conton; th can rult n ramatcally ffrnt 8ffct 8 + growth rat. Both typ of ponnt ar call ocal yapunov ponnt an rcognzng th tncton twn thm rolv om confuon n th ltratur. In trm of prctng th forcat accuracy th fnt tm ponnt ar hown to th mor rlvant quantt n Scton 3 whr t alo provn that th man of th largt fnt tm ponnt o not prov an una tmat of th corrponng gloal ponnt an mlarly for th man of th mallt fnt tm ponnt. In aton n rgon of tat pac whr th largt fnt tm ponnt l than zro all prturaton wll hrn npnnt of thr orntaton; th nvtgat n Scton 4 whr uch rgon ar * provn to t n two common chaotc map. Each cla of yapunov ponnt cu n th papr aum th orvatonal uncrtanty nfntmal; of cour a long a t rman nfntmal t cannot lmt prctalty an onc t fnt t growth no longr quantf y yapunov ponnt. Thrfor th rgorou rult r- * trct to nfntmal uncrtant ar contrat wth numrcal montraton for fnt uncrtant n Scton 4. Part of th popularty of gloal yapunov ponnt tm from th fact that thr valu o not pn upon th mtrc or coornat ytm u; th not th ca for th ponnt a upon a fnt lngth of trajctory yt n practc only th lattr ar avalal. rturn to th u n Scton 5. Fnally thr th quton of mol rror n nonlnar forcatng thr paramtrc or tructural. Argualy mol rror may mor rponl for poor prcton of ral nonln- ytm than chao. Mol mprfcton 9ar ar ( C. Zhmann t al.3 Phyc ttr A not conr n th papr a thr no ytmatc mannr to nclu ytmh mol mmatch thu t aum throughout th papr that th prfct mol nown. I. ocalz yapunov ponnt Th ynamc of nfntmal uncrtant aout a pont n an K m-mnonal tat M 0 pac N ar + govrn y th lnar propagator OP > 0 t whch 9volv any nfntmal ntal uncrtanty R S T m 0 R aout forwar for a tm V > t 0 along th ytm trajctory to : t X Y Z[ ^ \ > _ t ] ` a. t 0 In crt tm map th lnar propagator ovr traton mply th prouct off acoan g along h th p trajctory q r that j lm 0 n o... vw 0. For hgh mnonal ytm ntrt tn to focu on upac whch ar lly to contan th fatt growng prturaton 40 {. Two orntaton of partcular ntrt ar that whch wll hav grown th~ mot unr th lnarz ynamc n ƒ aftr tp an th local orntaton of th gloally fatt growng rcton l ˆwhch omtm call th yapunov vctor 3. Th frt of th orntaton Š fn y th ngular Bvalu n Œ compoton 4 of th propagator: th ar mply th rght ngular vctor of. Each aocat wth a ngular valu n ; y convnton š œ. Th fnt tm yapuno 8ponnt 5 ar ± n n ª «$ log º ² n log µ» ¹. Proprt ¾ of th ponntà hav Á n not  à y ornz 4 Grargr t al. Aaranl Å5 Æ an rfrnc throf. By Ç n È É Olc Thorm n th lmt th convrg to a unqu t of valu th yapunov ponnt Î whch ar th

3 ø F ` R ü 5 ù U A $ ã ä ù ü ^ B Ú _ þ å ù Û ü æ G U ë H 8 U I [ é ê à á µ } À ¾ Œ o Ž am for almot all maur. Õ Ö Ð Ñ Ò Ø wth n Ô ( C. Zhmann t al.ï Phyc ttr A rpct to an rgoc Ü lm Ý Þ log Ù...m K 3 If th um of th â ngatv a volum 9lmnt n tat pac wll hrn on avrag a t 9volv along a trajctory an mot ntal conton wll volv towar an attractor of mnon l than K m. At ach pont on uch an attractor th orntaton l not th orntaton corrponng to ç that th orntaton towar whch almot vry uncrtanty è n th uffcntly tant pat woul hav volv whn th trajctory rach. Smlarly fn lm a th í î ïorntaton corrponng ð ñ to òì mó for tal. Numrcally l an l 0 Um 0 can appromat ô y ö volvng th ngular vctor of ú û ü ý ù j j that th j tp propagator aout th th j * pr-mag of forwar ù 0 j tp untl th trajctory rach 0. Thu ÿ j l j ü j m K. 4 0 j j ù ü j A j w pct l 0 to approach th orn- taton of l for an K 0 m lang to th fnton of th fnt ampl yapuno ponnt n $ % $ "# & log ' ( - * + l 5. n / 0 3 an 4 m mlarly fn ung lm. Both th n 8 an th 9 : n ; ar = oftn call local yapunov ponnt To avo th confuon of th polymy w wll call th > fnt- tm nc thy ar compltly fn C n y a fnt gmnt of trajctory an th E fnt-ampl nc thy ampl th growth of an orntaton fn y th gloal ynamc. Both nvolv th am lnar forwar propagator ut ach rflct th + growth n K of a ffrnt M orntaton: N th l for th n O P n an th for th S.A T U V oth n c approach an a yt for th rlatvly mall ovr whch forcat ar typcally ma thr proprt ar qut ffrnt an nthr contran y th valu of. f 3. Proprt of localz yapunov ponnt g Gnral contrant on th rlatv magntu of th largt an th mallt fnt tm an fnt ampl ponnt ar now rv from th fnton aov an thn llutrat low. By contructon h n j mamum l n m n growth corrpon to thu q r for ach an thrfor th t n- 9 n u v qualty z n { alo hol } ~ for th man valu w whr not an arthmtc ¼ avrag tan wth rpct to th natural maur an ƒ N a numrcal appromaton wth ampl z N.E- ampl from two chaotc ytm ar gvn n Fg.. Th Hnon map 4 ¼ ˆ Š a Œ y Œ y whr a.4 an Ž 0.3; th acoan npnnt of y an ha contant trmnant qual to Ž. Th Ia map 5 ˆ š œ ˆ co ž > t Œ y n > t Œ y ˆ n > t Œ y co > t ª «wth > t 0. 4 ˆ Œ y ± an ² 0.9 prov a rathr mor compl acoan tll wth contant trmnant qual to n th ca. Not th non-gauan hap of all th truton n Fg. partcularly tho for mall. Contratng th hap of th truton for th Hnon an Ia ytm uggt that uch truton wll trongly ytm pnnt. Alo not how th truton harpn wth ncrang n ¹ an that» for gvn th truton of º an n ¼ ½ ffr. If a pcfc Z not of ntrt appromaton of l tx Y \ ] at Z t along a numrcal trajctory can mply appromat y a vry long ntgraton of th ytm an tangnt quaton for an artrary ntal uncrtanty. Th qualty of th appromaton may unnown howvr 8 an th cuon n Scton 5. aum throughout that thr t a unqu natural maur whch wll appromat y th numrcal traton of th ytm.

4 n ö ã â 8 K '( O 4 ( C. Zhmann t al.á Phyc ttr A Â Ã Ä Å Æ Ç È É Ê Ã Í Î Ï Ð Ñ Ò Ô Õ Ö ÜFg.. truton Ý à of á a an Ø c n th Hnon Þ ß an th Ia ytm for Ù thc Ú 4 thn an Û 4 ah ach wth N 409. Th arrow at th top a ncat. argr n wth hav n u n th lowr panl. N ä n å æ Th man valu è ç é ¼ n ê ë N o ì not í n î ï ncra wth ncrang. In fact ð ñ ò for any a can n y conrng th matr óô th prouct of acoan ø ù.... v úû nto two u-prouct üý an þÿ ach of lngth :... ¼ n... Th frt ngular valu " of #$ rflct th mamum pol growth ovr th frt tp; th frt ngular valu of matr %& mut l than or 9qual to th prouct of th frt ngular valu of th matrc an * thu / Th qualty hol only f th frt lft ngular vctor u5 of algn ; = wth> th rght ngular vctor 9: of AB E.. F u C a n th unform G H I Bar map an Bar Apprntc Map. From Eq. th largt fnt tm yapunov ponnt 3 Th follow mmatly from th ngular valu compo- M N T ton SV O of a quar matr P R ST U V whr th uprcrpt T not th tranpo of a matr. X a agonal matr who largt ntry Y an Z[ an \] ar orthonormal rotaton matrc. Notng that rotaton matrc cannot nhanc growth yl th r rult. A rf proof gvn n th Appn

5 K š Æ Ú í ÇÈ n ž ì U å ß ì a Ä Å ÿ & ì ì ø % ù / ú O / þ / fn y _` mut l than or qual to th n c a 8rag of tho fn y f an gh : w j r lm no t p n q u v n ƒ log y log z z { n ~ }. Smlarly th mallt fnt ˆtm po- - ¼ n Š nnt fn y Œ mut atfy Ž U m - n n œ Um n Um. A th tru n for ach nvual altrnatvly ª «m th man of th largt mallt fnt tm yapunov 9ponnt wll not ncra not cra a n- cra y a factor of two: ¾ n ± ² n µ À n ¹ º» n ¼ ½ an Á  à m m. 8 hn an ÉÊ ar of ffrnt lngth an thn á Í Î â Û Ï n Ð Ñ Ü Ò Ý n Ô Õ Þ Ö n ã Ø Ù à ä 9 æ ç è ç an a mlar rlaton otan for ë é ê m.. ö ç î ï ð ç ñ ò ó ô th ar u-atv upr-atv m ( C. Zhmann t al.^ Phyc ttr A qunc of functon. hl th û ç ü ý o not guarant a monotonc cra of or n- ç cra of wth ncrang t o mply m ç an m m ç for all 0 provng that th man of th ç truton not an una tmat of th gloal " yapunov po- # nnt $ for any fnt 58. Th man of th truton of fnt ampl ' ponnt qual to ( y * fnton + npnnt ç - of. hl 0 n Fg. ach. N cra 3 wth ç 4 5 ncrang a ncat y th arrow th N conc provng a contncy chc 8 9 ç a : ; to whthr N mght larg nough> o that A = N appro- mat th lmtng valu B. C E F G H I K M N Ia ytm for P lght gry 4 gry an R S T U 5 lac traton. Not that n th Ia ytm t. Fg.. Contratng fnt ampl yapunov ponnt aca wth th corrponng fnt tm yapunov ponnt ornat n th

6 ¼ ' Õ Ù ä ÿ ê X æ é [ Y ì Ú r è Ö å Å o à û ÿ û ' V [ œ ü U Z & ï ' ñ ù ò E ð ú ó ô ï all If th acoan trmnant contant thn for m ^ _ \ ` log ] a c f g g h ç j l ç m n whr p q w t. Rcallng that t u v ç y z { ç } ~ an m m Hnon ytm an th Ia ytm ƒ ç ç Š ç Œ Ž ç log an ç š œ «ma ž log œ mn ª ç Z ( C. Zhmann t al.v Phyc ttr A t follow that for oth th ç ˆ fnng ± a trangl whch oun th truton of º ç ² µ ç ¹» a llutrat n Fg. for th Ia map. th ncrang th truton of pont approach th ln ½ ¾ ç À Á  ç Ã. For Ä Æ Ç ç ÈÉ Ê ç Í 5 Î Ï th largt orv valu of Ð Ñ Ò wa Yt th wth of ach of th truton c 0.3 ncatng that th varaton tm from th ffrnt ntal conton on th attractor not th ntal orntaton. Ô 4. Rgon of hgh prctalty n chaotc map yapunov ponnt ar oftn a to rflct prctalty an a potv gloal yapunov ponnt oftn a to troy any hop of long-trm prctalty. But nc thy ar fn va th lnar propagator yapunov ponnt n only quantfy th growth of nfntmal uncrtant n th ntal conton th a hgh prc to pay for nvaranc unr a mooth chang Ö Ø of coornat. Both th Ia Ú Û Ü ytm Eq. an th Hnon ytm Eq. ar conr chaotc for th paramtr conr aov nc n ach caý ç Þ t ß lv that á â 0; yt th o not mply ã 0 for any fnt. In t clar from Fg. that thr ar many pont on th attractor aout.5 % of th Ia ytm for whch th lang ç fnt tm ponnt ngatv.. thr ar tat aout whch è ry nfntmal uncrtanty wll hrn rgarl of t orntaton. now proc to locat th corrponng rgon n tat pac wth ngatv largt fnt tm yapunov ponnt ë ì ç í î 0 whch w ntrprt a lly rgon of rlatvly hgh prctalty: all nfntmal ntal uncrtant wll cra n th rgon. Rcntly 99 mlar rgon hav n trmn ö analytcally ø n th ornz 30 y- tm alo 03 ; w now prnt nw rult for th Ia ytm an th Hnon ytm. Th rult thn montrat numrcally to hol n vral ca for fnt uncrtant ut th act rult low ar ujct to th cavat of nfntmal uncrtant a ar all gnral argumnt rgarng th prcton of trmntc chaotc ytm. 4.. Infntmal uncrtant Naturally act rult ar mot aly otan for mall. Thrfor w conr only ý an þ analytcally; numrcal ÿ rult ar gvn for largr valu. In a map 0 mpl that th largt ngular valu of th acoan l than on. For th Ia ytm wth n th rang 3 3 th on-tp fnt tm yapunov ponnt pa through zro at two crcl aout th orgn wth ra r o c c whr c ( * " # $ %.. In th ca / for all pont thr wthn th nnr crcl.. tho wth y r 8 or out th outr crcl y : r o. Fg. 3 how ; pont on th Ia attractor whr th gn of = > ncat y th gry cal. For A 0.9 th ra ar rc 0.35 an ro.404 thu th attractor l wll wthn F r og an G th outr crcl # not vl n Fg. 3. A H approach on th rau of th nnr crcl go to zro. In th Hnon ytm thr ar no rgon n whch vryi uncrtanty wll hrn aftr on traton that M K N 0 for all. A hown low th not th R ca for O P howvr. Th mallt valu of S T foun for pont on th 5 y-a. Hr X Y \ 0 an thu all uncrtant hrn cpt tho algn wth whch rman unchang n magntu. Not that for pont on th 5 y-a paralll to th 5 y-a. now prov that thr] t a fnt rgon wthn whch all pont hav a ^ _ ` 0; th rgon nclu a porton of th frt prmag of th 5 y-a.

7 w v Ž Å Ä é ¼ ê ½ ë Ô ( C. Zhmann t al.c Phyc ttr A Fg. 3. Rgon of crang uncrtanty n th Ia ytm. Pont on th attractor ar color gry f h th crcular rgon nar th orgn o l m n p 0 for all. f g j 0 lac othrw. thn Th prmag of th q y-a th paraola y ua a y t an th two tp propagator for pont z{ a } ~ Å 3 ƒ Š 0 3 ˆ Œ 0 a a whr w hav u th fact that n th Hnon ytm š only a functon of 3. To locat tho 3 wth œ ç ž 0 w trmn th ngular valu of a 0 q r ª notng that «whr ar th root of th charactrtc polynomal of T±². Thu a µ 4 ¹ º» 0. 3 can now tt whthr ¾ for any À ; altrnatvly w can olv for Á  to fn Í Å Æ Î Ã 0.9 Ï Ð Ç È É Ê a Ñ Ò 0.3. Not that th paraola an th for a Õ -a ntrct at th pont 0-. At th ntrcton Õ Ö Ø 0 an t follow from th quaton aov that Ù Ú ç Û Ü Ý Þ ß ç à á â ã or quvalntly ì ä ç å æ ç ç è í é î ï ð that log 0. Showng that a pont ngatv n th rang gvn y Eq. 4 prov

8 û ù é Å ÿ Å 8 that all pont on th paraola wthn th lmt hav ò ó ç ô ö 0. By contnuty thr t a fnt rgon n th nghorhoo of th paraola for ø ú Õ û a a wthn whch th largt two-tp fnt-tm yapunov ponnt ngatv. A numrcal tmat of th rgon hown n Fg. 5 a. hat aout largr valu of Trajctor pang through th rgon of ü ý þ ÿ 0 ar oftn foun to hav ç 0 for a wll; ngatv valu of 4 ar clarly vl n Fg. a. In th followng w wll llutrat th rlaton twn th trajctor of pont on th attractor wth ç 0 for Å an th qy -a; namly that uch pont tn to l nar ( C. Zhmann t al.ñ Phyc ttr A prmag of th q y-a. Th frt thr prmag of th Õ -a can otan analytcally. Th q y-a th prmag of th Õ -a whl th paraola not aov th frt prmag of th q y-a. Th frt prmag of th paraola a Õ a q y a. a Å a Th frt 4 prmag of th Õ -a ar hown togthr wth th attractor n ach panl of Fg. 4 whl pont on th attractor wth " # ç $ % 0 ar hown for th pcfc valu & 345 n th four panl. ' ( Fg. 4. All panl * how th Hnon attractor + th -a th y-a an t lat thr prmag: th paraola ol -. / ln t con 0 prmag 3 long ah 8 9 an t: thr ; prmag hort ah. Th ffrnt panl how pont on th attractor wth for a 3 4 c an = 5.

9 ¼ H û X Y Z c R S It clar that th rgon ç A B 0 ar rlat to th ntrcton of th attractor an prmag of th qy -a. nt how that th vn mor vnt for ntal conton n th gnral vcnty of th attractor. A n Fg. 4 Fg. 5 how oth th prmag of th Õ -a an th attractor; n aton tt pont for whch C ç E F 0 ar plott a wll whr th tt pont wr rawn at ranom from th rgon hown n th fgur. Pont wth crang uncrtant for G 5 ar foun n th vcnty of prmag of th q y-a.. thy oftn hav trajctor whch nclu a nar approach to th q y-a. Typcally th occur towar th n of that trajctory: pont wth I K 0 ar clo to th frt prmag of th qy -a pont wth M N 3O P 0 ar clo to t frt an con prmag an o forth. ( C. Zhmann t al.> Phyc ttr A Nt w nvtgat th havor for vn largr n th Hnon ytm. Grargr t al. how that on houl pct T U ç V 0 for artrarly larg aumng a havor ntally l avrag of ranom varal corrlat only ovr hort tm. Th gnral pctur corrct although th tal ar omtm mportant a w hav argu l- whr 5. Hr w conjctur that u to th trmntc natur of th Hnon ytm th fracton of [ \ ç ] cra mor qucly than th ranom varal argumnt woul uggt. A hown n Fg. th fracton of ntal conton on th attractor wth ^ _ ç ` a 0 orv to cra ponntally wth a ar th corrponng fracton whn lnar propagator of th map ar comn at ranom. Th ranom ca for 4 an 8 ar hown; for ach Fg. 5. All panl f how th Hnon attractor g th -a h th y-a an t lat thr prmag: th paraola ol ln t con. j prmag n long o ah p q an tr thr prmagt u hort ah. Th ffrnt panl how pont n th vcnty of th attractor wth l 0 for m av 3 w 4 c an 5.

10 œ ' ' ¼ ' ª «0 ( C. Zhmann t al.y Phyc ttr A z {. } ~ Fg.. For th Hnon ytm a th fracton of pont wth ƒ 0 a a functon of n th trmntc ca quar. Alo hown ar th ˆ rult Š for ranom matrc whr th matrc ar rawn from th truton of th lnar propagator of th Hnon for j 4 an 8. Th ol ln rflct th t ft to an ponntal cay ovr th rang 8 Œ 40. For larg j map that aout 30 traton of th map whr conr n th trmntc ca. Not that trmnm a trong contrant rucng th llhoo of fnng ngatv fnt tm ponnt. th fracton cra l qucly than n th trmntc ca. rgarng th trmnm n th r of acoan la to frqunc of ngatv Ž ç whch for th largr c tho of th trmntc ca y orr of magntu. 4.. Fnt uncrtant From th practcal pont of vw of tmatng prctalty th nowlg of uch pont woul of lmt utlty for larg nc th hrnng rgon aroun ach pont may vry mall. Furthr rcall that all tmat of prctalty a upon yapunov ponnt aum an nfntmal ntal rror. Thrfor w nt conr fnt uncrtant plctly frt plorng Gauan trut uncrtant n th Ia map an thn uncrtant of unform magntu n th Hnon map. In ach ca w allow th pct magntu of th rror to vary an cu th rlaton twn rgon of nhanc prctalty an th rgon whr ç 0. In th Ia map w conr normally trut uncrtant n ach coornat of th ntal conton wth zro man an th am tanar vaton. Tang a pont on th attractor at ranom 8 orvaton wr gnrat an prct forwar tp; f th tanc from truth at fnal tm wa l than th ntal prturaton appl n mor than 50% of th 8 ca thn th orgnal pont on th attractor wa conr to wthn š a rgon of hgh prctalty for that valu œ of ž. For thr a rgon not hown of hgh prctalty cntr on th crcl rv aov an hown n Fg. 3. Pont of hgh prctalty hav n orv for an pr- 9 t for 8. hl all ar nar th orgn many fall out th crcl ut th not urprng a th fnton of hgh prctalty n th numrcal prmnt much l rtrctv than rqurng a ngatv fnt tm yapunov ponnt whch guarant 00% of th uncrtant to hrn f thy hav nfntmal magntu. Th am tt for 4 ar hown n Fg.. Th rult a upon nfntmal uncrtant hown n th uppr lft panl ar n to rflct th rgon of hgh prctalty vry wll up untl 8 whch a far

11 ¼ ½ Á «Â Ã Â Æ É Ñ É Ú Ã Ã É X È ¼ Ñ Â Ï Ä Ð Å ( C. Zhmann t al. Phyc ttr A nfntmal; of cour tructur mallr than ¼ cannot tct. It com harr to ntfy rgon wth ffrng proprt n prctalty wth ncrang prcton tm; ntv pnnc on ntal conton wll lmt th prcton of prctalty a wll a prcton tlf. Thu far w hav only conr th valu of an «ponnt at a partcular valu of ¾ K; altrnatvly on mght conr rgon n whch th ponnt ngatv for all À K wth corrponng uncrtant crang monotoncally for th total uraton of K traton. hl uch ut ar of ntrt thy ar not nvtgat hr nc th tnc of mall potv valu at ntrmat ar contnt wth rgon of hgh prctalty; n th pont omtt from th t of pont for a partcular ar tho that ar a to play rturn of ll n mtorology 3. Although yon th cop of th papr t woul ntrtng to amn th patal truton an fracton of ntal conton n th ut oth a a functon of an th magntu of th ntal uncrtanty. Fg.. All panl how th Ia attractor; th ot rprnt th attractor. Intal conton on th attractor wth nhanc prctalty ar mar wth a ± whn mor than 50% of 8 ntal uncrtant how cra magntu at fnal tm ² 4. Th ffrnt panl rflctng ffrnt ntal magntu p ar contrat wth th lnar ynamc n th uppr lft panl. fracton of th amtr of th attractor. tr that rult of th n wll trmly ytm pcfc. 4 For th Hnon map th portrat of ngatv µ rval n Fg. 5 contrat wth rgon of hgh prctalty for fnt uncrtant n ntal conton wth a much harpr tt than for th Ia map. In th ca ach orvaton plac at ranom on a crcl of rau aout th tru tat; 000 uch orvaton wr conr for ach tru tat an only f th fnal tm tanc of ry on of thm wa l than ¹ wa th pont rcor a hgh prctalty. Th hown for 3 ffrnt ntal magntu n Fg. 8. For mall magntu º» 0.00 th cartography qut mlar to that of 4.3. Cotnc of chao an rgon of hgh prctalty Ç Anothr ntrtng apct concrnng th rgon wth Ê Í Î 0 th ntrplay twn th rgon an th locaton of untal proc ort whch ar lv to form th lton of th attractor n many chaotc ytm 33. Clarly an untal pro- ort cannot contan a pont wthn a rgon for whch Ò Ô Õ 0 nc that pont woul thn tal. In hort ach pont on ach untal pro ort mut avo all rgon of th tat pac n whch Ö Ø Ù 0: t not ay to how th com aout f th ort ar n on th attractor an th ara of th rgon o not vanh. If th rgon o not vanh thn th orvaton uggt a nw angl from whch to vw th trm ntvty of th tructur of th attractor to mall chang n paramtr valu. It alo ntrtng to conr th mplcaton potv Û Ü Ý mght hav on numrcal rult whn th tru attractor a tal attractng proc ort; no pont n a proc ort n l n a rgon for whch Þ ß à á 0. For a â ã 0.3 th

12 ª Ñ Ú ( C. Zhmann t al.ä Phyc ttr A å æ ç è é ê ë ì Fg. 8. All panl how th Hnon attractor; th ot rprnt ntal conton wth nhanc prctalty for fnt uncrtant. Th îffrnt ï panl long to ffrnt ntal magntu of ntal uncrtanty: a p í 0 thu concng wth Fg. 5c. In panl c an a ot at ncat that th tanc twn th mag of th tru tat an ach on of 000 nact orvaton cra at ð 4. Th orvaton wr ntally trut on a crcl of rau p cntr on. Hnon ytm ha a tal pro 4 ort. Th majorty of pont on th ort hav ñ ò 4ó ô ø ù 0 ut 4 for vral ö 0; th largt orv valu ú û ü ý mplyng a magnfcaton factor of mor than a hunr wthn on cycl. hn þÿ nonnormal th magntu of th lang ngular valu may qut larg rgarl of whthr or not th ort aymptotcally tal. th a lght ncra n a th ytm appar chaotc; th ynamc tll rml tho of th tal ort ut th attractor now cont of 4 mall clarly parat chaotc rgon ach vt n turn. It woul ntrtng to amn th truton of lang ngular valu aout tal proc pont on th am ort a a functon of paramtr; thr a potv lowr oun on th angl twn th gnvctor X Th orvaton uggt an ntrtng pol paralll twn th mpl two mnonal map an th ont of turulnc n lamnar flu flow. It ha long n nown that har flow can com turulnt at Rynol numr wll low th crtcal valu a fn y th clacal lnar talty thory a on gnvalu 34 an rfrnc throf; for a rcnt ovrvw 35. Prturaton n th rcton of th ngular vctor may grow ½raply for a fnt tm ctng nonlnar trm an thry omnatng th ont of turulnc; th long trm havor cr y th gnvalu com rrlvant. Non-normalty mght hol mlar conqunc for th numrcal traton of nonlnar ytm. Th mallt nonzro numrcal prturaton fnt ng fn y th numrcal

13 Â «Â \ G H 9 M $ ½ S O P $ Â Â u _ h g 9 f j gr; an t coul qut ffcult to ntfy a tal pro ort wth 0 y numrcally tratng th map. Th mallt nonzro numrcal prturaton mght wll grow uffcntly to rng th nonlnar trm nto play rultng n utan complcat ynamc up to th tm-cal at whch th numrcal ort clo actly upon tlf a all trajctor on gtal computr wll 3. o not clam that th th ca n th map conr aov ut mrly not th ynamc mght appar mlar an thu tr th valu of prformng a furcaton analy n aton to numrcal traton. 5. cuon an concluon X Th rult prnt n th artcl hol mplcaton for two quton of gnral ntrt: th appromaton of largt yapunov ponnt an th tmaton of lly forcat accuracy. Notng that th fnt tm yapunov ponnt can comput accuratly y tanar mtho a lowr oun on% th & ' ( rror n * aumng + " - #. / 0 N gvn y N 3 N 4 5 whl prov an uppr oun on 8. hn a goo appromaton of th yapunov vctor 9 l avalal on can alo rqur for th ffrnc : ; twn = > A B C ampl an fnt tm ponnt E F. Yt nc oth 9 l an I K ar multplcatv «rgoc tattc uncrtanty n numrcal t- Ñmat of l rman largly unquantf. Smlarly namuch a matr multplcaton o not commut attmpt to tmat th uncrtanty n N va th tanar oottrap approach mut trat wth car for trmntc ytm 5. Th goal of a uffcnt conton for th convrgnc of R tmat rman alluv. uanttatv ncary conton along wth plct tt for convrgnc n alatorc T ytm tochatc U ytm V wth potv ar cu lwhr 83. In trm of ntfyng th wort forcat ut X th ar mor mportant than th Y mply cau th Z ar largr. hl t omtm argu that th corrponng ngular vctor may pont off th attractor th [ rman rlvant a pol uncrtant aout th tru ntal tat wll alo l off th attractor almot crtanly. Infntmal un- ( C. Zhmann t al. Phyc ttr A ] ^ crtant along l hav th avantag to fr of trannt ut f of fnt magntu thy alo may l off th attractor. An vn for fnt uncrtant on th attractor fnt tm growth not oun y. Th fact mply that th ` a upr-yapunov growth foun y Ncol t al. 8 to pct: aftr tm t an uncrtanty may magnf y Ñmor than th largr of c t an t vn f th ntal uncrtanty nfntmal. Ovr what ura- ton can raltc.. opratonal uncrtant trat a nfntmal Or quvalntly what th «tnt of th lnar rgm Th an ntrtng an opn quton vn n numrcal wathr for- catng 38. Not that computng ponnt for fnt tm omwhat gratutou n that any ncra wll yl a potv ffct «ponnt; a potv ponnt mpl ffctvly ponntal growth thn only n th lmt of nfnt tm. For fnt tm a potv «ponnt mpl growth ut not ponntal growth; t only rflct th tm pnnc of th uncrtanty unr th atonal aumpton that th growth wa ponntal. Th wth of th truton n Fg. an o not ncat unform «ponntal growth on th tm cal. An altrnatv approach to quantfy prctalty y computng th tm rqur to rach an uncrtanty thrhol contrat wth th u of ffctv rat n 9 whr ampl wth oth larg l an larg uncrtanty oulng tm ar cu. A provn n Scton 4 thr ar ntal conton for whch no prturaton grow for two paragm attractor; t woul ntrtng to nvtgat th rlatv locaton of rgon wthn whch m n o p 0 an untal pro ort for larg a a functon of paramtr n a varty of low mnonal map; th numrc nar tal pro pont wth q r t 0 may alo prov of ntrt. In th papr ach ytm ha n conr n t natural tat pac t orgnal phycally rlvant co-ornat ytm. It houl not that Únthr th typcal maur of forcat rror 4 nor th fnt tm yapunov ponnt nor th fnt ampl yapunov ponnt ar nvarant unr co- 4 v w S 39 for an atypcal approach.

14 Ñ ½ \ ½ X Ñ z Œ µ ² y Ñ Â Ñ E I O Þß F P j G H ò ó l ² 4 ornat chang or vn chang n a Rmannan mtrc. In th tuy th forcat rror mply th Euclan tanc twn two pont n tat pac { { T pcfcally } ~ whr ƒ th ntty matr. Th ngular vctor corrponng to th largt fnt tm ponnt mamz th tanc at prcton tm. Thr may phycally mor rlvant fnton of tanc twn two forcat. Thn th ntty mght rplac y anothr Rmannan mtrc for ampl th nvr of th covaranc matr may rv a a natural choc whn th ffrnt rcton n tat pac play ffrnt varanc. Altrnatvly may u to account for ffrnt lvl of no on ffrnt tat pac varal to targt th varal who prcton of partcular concrn or vn to c whch varal toˆ orv Š n orr to mnmz th prc- ton rror 40 an rfrnc thrn. Th applcaton trmn th choc of mtrc. In concluon w agan tr that th rlvanc of all thr typ of ponnt rtrct to ca whr th uncrtant ar uffcntly mall that thr growth wll appromat y th lnar propagator 38 : Ž act only for nfntmal uncrtant. Bhavor of largr fnt uncrtant rqur th u of nml of ntal conton «ach contnt wth th orvaton; th rlatv prformanc of nml n th upac fn y ar contrat wth tho fn n th u- pac fn y 9 l for vral chaotc flow n 9. Th contructon of nml for forcat valuaton n mprfct mol rman an mportant u for all nonlnar ytm. Appn A. Hr w talh that for a prouct of matrc š w hav œ ž. Th pctral norm ª «of a matr 4 fn y ma 0 ± whr th mamum tan ovr all nonzro vctor ². It oun ¹ th º amplfyng powr of a matr..»¼ ² ½ ¾ ². Th pctral norm of a rotaton matr whl that of a agonal matr corrpon to th mamum lmnt. Th ngular valu compoton compo any quar matr nto th prouct of a rotaton ( C. Zhmann t al. Phyc ttr A matr a agonal matr an anothr rotaton matr hnc th pctral norm À À of aáâ matr ntcal to t frt ngular valu ÃÄ Å Æ. Gvn ÇÈ É Ê Í thn for any nonzro ² w hav Î Ï Ð ² Ñ Ò Ô Õ Ö Ö Ø Ù Ù Ú Û Ü Ý à áâ ² ã äå æç ² è éê ëì ² í îï ðñ. ô ô ö ý vng ý þ ÿ y ² uttuton yl ø ù úû ü a r. Rfrnc V.I. Olc Tranacton of th Mocow Mathmatcal Socty P. Ecmann. Rull Rv. Mo. Phy Arnol Ranom ynamcal Sytm Sprngr Brln E.N. ornz Tllu H.. I Aaranl " R. Brown M.B. Knnl Int.. Mo. # $ Phy. B R. ornr B. Hungr. Martnn % S. & Gromann S. ' ( Thoma Chao Solton an Fractal 99 * Smth Phl. Tran. R Soc. on. A C. Ncol -. S. Vanntm.-F. Royr..R. Mtorol. Soc. / Smth C. Zhmann K. Frarch..R. Mtorol. Soc Z. Toth E. Kalnay Bull. Am. Mtorol. Soc : T.N. t al. Palmr Phl. Tran. R. Soc. on A. Smth Nonlnar ynamc an Stattc chaptr ntanglng Uncrtanty an Error: On th Prctalty of ; Nonlnar Sytm Brhaur Boton 000. = > S. Vanntm C. Ncol. Atmo. Sc G. Strang nar algra an t applcaton Hartcourt Brac A B ovanovch San go 988. C 5 C. Zhmann.A. Smth. Kurth Phyca P. Grargr R. Ba A. Polt. Stattcal Phyc E. Ott Chao n ynamcal ytm Camrg Unvrty K Pr Camrg Nw Yor Mlourn 993. M N 8 G. Froylan K. u A. M Phy. Rv. E S. Ncol G. Mayr-Kr G. Hau Z. Naturforch. 38a R 983 p S T U 0 V.M. N Phyca X 3. Y Z A.S. Povy Chao [ \ ] ^ B. Echart. Yao Phyca _ ` 3 U. Ful. Kurth A.S. Povy Phyca a. c 4 f M. Hnon Commun. Math. Phy. g h K. Ia Opt. Commun Rull Pulcaton Mathmatqu l Inttut Haut Etu Scntfqu

15 n p t q u r Œ Ž ( C. Zhmann t al.m Phyc ttr A o A. Kato B. Hallatt Introucton to th Morn Thory of ynamcal Sytm Camrg Unvrty Pr S. Ellnr R. Gallant. McGaffry. Nycha Phy. tt. A C. Zhmann-Schlumohm Vorhragtun n chaotchn Sytmn un n r Pra Ph th Fr Unvrtat Brln. Mtorologch Ahanlungn. Nu Folg Sr A. v w Ban 8 Hft y z 30 { E.N. ornz. Atmo. Sc H. Muougawa M. } Kmoto S. Yon. th Atmophrc ~ Scnc ƒ.. Anron H.M. van n ool P. Cvtanovc Phy. Rv. ttr ˆ 34. Borg U. Broa Z. Naturforch. 43a N. Trfthn A.E. Trfthn S.C. Ry T.A. rcoll Š Scnc 993 p A. Smth acunarty an Chao n Natur Ph th Columa Unvrty Nw Yor NY. 98 S Appn. 3. Smth C. Zhmann. Kurth I. Glmour Nonlnar mol valuaton: -haowng proaltc forcat an wathr forcatng Ph th Ofor Unvrty P. McSharry.A. Smth Phy. Rv. tt A. Hann.A. Smth. Atmo. Sc. 999 n pr.