Eigenvalue and Eigenvector Analysis of Dynamic Systems Paulo Goncalves *

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1 Evlu d Evco Alyss of Dymc Sysms Pulo Goclvs * Absc Whl svl mhods md udsd h cuss of modl bhvo hv b poposd c ys, foml modl lyss ms mpo d chll sysm dymcs. Ths pp dscbs mhmcl mhod o copo vcos o h mo dol vlu lyss of dymc modls. Th poposd mhod dvs bsc fomuls h chc how ch l (o loop) fluc s bhvo l dymc sysms. Bsd o h shs dvlopd fom l hoy, I xd h mhod o ol dymc sysms by l h sysm vy po m d vlu h mpc o h dvd fomuls. Th pp cocluds wh pplco of h mhod o l sysm.. Ioduco Foml modl lyss ms mpo d chll sysm dymcs. Svl mhods md udsd h cuss of modl bhvo hv b poposd c ys (Kmpm 996; Mohddh 997; Goçlvs, Lppo d Hs ; Slh d Dvds ; Slh ; Mohddh, Rchdso d Ads ; Olv ; Olv d Mohddh ; Gülp 5; Hs 5; Kmpm d Olv 5; Slh, Dvds d Byoum 5). Ths mhods c bc wo hds modl lyss: h loop domc wo of Rchdso (995) d vlu lscy wo of Fos (98). Mohddh (997) d Mohddh, Rchdso d Ads () xd h loop domc wo fs poposd by Rchdso (995). Th sch poposs phwy pcpo mcs (PPM) o fd h sucu h mos flucs h m ph of v vbl. Th mhod povds locl ssssm of how chs s vbl of s fluc h ch of h sm vbl ( d x& ). Whl h mhod hs h dx dv of b compuolly smpl s o wll sud fo sysms h oscll, sc h lyss s locl d co cpu lobl mods of bhvo. Mos of h m sch cs bc o vlu lscy hoy poposd by Fos (98). Th mhod clls fo h compuo of vlus d h xplos how h vlus ch s l s ch, h s, l lscs. Fos showd h compl dscpo of l lscs llows o pcpl o clcul loop lscs. Ths suso houh v mplmd sofw, pomsd o povd sw o how modl * Asss Pofsso, Mm Scc Dpm, School of Busss Admso, Uvsy of Mm, Col Gbls, FL, USA. Pho: (5) 8-86/Fx: (5) 8-. pulo@mm.du

2 sucu, h s s of fdbc loops, dms modl bhvo. Th pcul clculo h Fos susd s cully o fsbl. As h ld l, Fos s susd ppoch suls sysm of quos h s ov-dmd ffc of h fc h h umb of loops css much fs h h umb ls. Kmpm dscovd h smll subs of loops s suffc o uquly dscb vlus (.. h bhvo) of sysm dymcs modl (Kmpm 996). Us Idpd Loop S (ILS) poducs smll sysm of quos, sysm h c b solvd. Th Idpd loop s (ILS) mhod hs h mpo dv of llow us o clcul loop s fom l s, wh h umb of ls modl s of smll. Howv, hs h dsdv of ly o d hoc pocdu o slc h dpd loop s (ILS). Goçlvs, Lppo d Hs () us Mso s ul o xpss h chcsc quo d s soluos (vlus) ms of loop s (sd of l s), whch llows hm o ob loop lscs dcly. Whl h mhod sdsps h poblms ssocd wh by slco of loops hs h shocom of qu h compuo of ll loops h modl, umb h ss qucy v wh mod s modls. Olv () povds xso o h mhod slc fs h shos loops. Th shos dpd loop s (SILS) povds sysmc pso of h fdbc complxy s smpls compos d s h mos ul dscpo of h sucu cycl po. Olv d Mohddh () comp h suls obd wh h SILS ppoch o h of PPM d fd h h loops h m dymcs of cludd h SILS. Mo cly, Kmpm d Olv 6 xplo h pplco of loop vlu lscy o h modls o ssss h pol of h mhod d fd h h shs dpd o h chc d dymcs of h modl. Th wo of Slh, Dvds d Byoum (5) s mos o ous s s udsd h cobuo of boh vlus d vcos o modl bhvo. Whl w focus o h lycl compuo of h fluc of vlus d vcos o modl bhvo, Slh l. (5) povd compuol mhod (mplmd Mlb) o clcul such fluc. Th movo fo hs pp s o povd mhmcl fmwo fo fuu wo o vco d vlu lyss. Ths wo follows h sch do of Fos (98). Smlly o pvous sch, ou s bw sucu d bhvo s xpssd ms of udsd how chs ls o loops ffc h m ph bhvo of s vbl. Ou wo dps fom pvous ffos ms of s focus o

3 lycl suls d mphss o h mpc h fs m dvvs of vlus d vcos hv o modl bhvo, sd of vlu lscs.. Bhvo L Dymc Sysms Th foml sucu of l sysm dymcs modl wh vco of s vbls x(), wh x() (x, x,, x ), vco of fs m dvvs of h s vbls x& (), wh x& () ( x &, x &,, x & ), mx J cpu h pl dvvs of h ch of s vbl wh spc o oh (h mx J x x & x s commoly ow s h Jcob of h sysm), d cos vco b, c b psd compcly h follow wy: x & Jx b () Cosd ow h soluo o h homoous sysm. A sdd sul l sysms hoy s h h vlus () of h mx J dscb h bhvo mods h h modl d h soluos of h chcsc polyoml (P()), wh ( P() I J ). Assum fo smplcy h h sysm mx J x hs compl s of lly dpd vcos (,,, ) wh cospod vlus (,,, ), wh vlus my o my o b dsc. Sc h vcos lly dpd, hy sp h dmsol spc, hfo by vlu of h s x() c b xpssd by h l combo of h vcos: x ( ) ( ) ( ) ( ) () wh (),,,, scls. Us h fc h by dfo mulplco of h sysm mx by h vcos suls h poduc of h vcos by vlus (J ), w c w quo () by mulply by h sysm mx J x. Jx ( ) x& ( ) ( ) J ( ) J ( ) J ( ) ( ) ( ) ( ) x& ()

4 Sc quo () dfs h s vco x(), w c s fs m dvv. I ddo, us h fc h vlus d vcos cos l sysms, w c w () o : ( ) & ( ) & ( ) & ( ) x& () Comp h h hd sd of () d (), w ob: ( ) & ( ) ( ) ( ) ( ) ( ) & (5) & Ad sc h vcos lly dpd, h quly c oly hold f: ( ) ( ) & (6) Th sysm bov c b psd mx fom s: & & & ( ) ( ) ( ) ( ) ( ) ( ) (7) Th soluo of h homoous sysm of dcoupld quos psd bov s ow: ( ) ( ) ( ) o ( ) (8) Subsu h sul (8) ou ol quo () ylds: ( ) x (9) No h w w h suls bov mo compcly mx fom df V s h x mx whos colums h vcos of J d df h colum vco () wh compos ( (), (), ()). x V. W c p h w quo s ch Df V h wy llows us o w quo () s vbl d us o w h dymc sysm, whch ylds: V &( ) JV ( ) o smply: ( ) V JV( ) &, wh h compuo of h vs of h mx of vcos (V - ) dpds o h vlu of ll h sysm vcos. Th w sysm ( & ( ) ) s ld o h ol o ( x& ( )) by ch of vbl. Th w sysm & ) mx (V - JV) cospods o h sysm ov h () s quos, wh h ch ch s ( dpds oly o h poduc of h ssocd vlu ( ) d h ow s ( ). Accodly, w c w V - JVL, wh L s h dol mx wh h vlus of J h dol.

5 . How Ls Ifluc Sysm Bhvo W focus ou o o quo (9) o udsd how chs l s (.., h sh of modl pms) fluc sysm bhvo. Th bhvo of ch s h sysm x () c b dscbd by: x ( ) () wh s h -h compo of h fs vco. Th quo suss h h dom bhvo mod of h s vbl x () wll b dmd by h lv s of ch -h compo of ch vco, wh o. W c w quo () bov mx fom: x x x ( ) ( ) ( ) () Equo (9) hhlhs h h bhvo of ch s s flucd boh by vlus ( ) d vcos ( ). I ddo, boh vlus ( ) d vcos ( ) dpd o h vlus of l s (.., pms h modl), bcus vlus soluos o h chcsc polyoml (P()), wh P() I J d h s of h Jcob (J) h pl dvvs o h l s ( ) sysm dymcs modl. Thfo, ch h of by l ( ) suls w Jcob d dff vlus fo boh vlus ( ) d vcos ( ). To udsd h u of h mpc of chs l s o sysm bhvo, w h pl dvv of ch s h sysm x () wh spc o s l s. Fom quo (), w ob h ch bhvo of ch s x () du o chs l ( ) s: x ( ) [ ] () Th l vlus of () c b obd ms of x() fom h ch vbl dfo: V x. 5

6 6 d h dvv of dvdul compos, w ob: x () W c w quo () mo compc wy s: x () If w sd how chs o l ffc ll s vbls, w c w: x x (5) Bcus h vlus d vcos l sysms cos, h dvv of h xpol of h -h vlu ( ) wh spc o s vlu ( ) yld m h dpds o m ( ). Thfo, w c w quo (5) o yld: x x (6) Equo (6) suss h fo ch compo (wh o ) chc h bhvo of s x (), h cobuo o h ch bhvo of s x () du o h ch l ( ) s composd of wo ms cospod o:. Th cobuo of h dvv of, h -h compo of h -h vco, wh spc o l ( ); d No h h compuo of h pl dvv of ch m ssums h h l s dos o dpd o h l. S s w s vbl obd f h ch of vbls v by ( x V ) wh s h l poso vco of h w s vbls d x s h l poso vco of h ol s vbls. Th vs of h mx of vcos ( V ) dpds o h vlu of ll vcos d hus vs wh chs h l. Howv, w do o dff wh spc o h loop s bcus w c smply p s ch h l poso.

7 . Th cobuo of h poduc of h, h -h compo of h -h vco, h dvv of h -h vlu ( ) wh spc o l ( ), d m (). Th fs m cpus ch sy h mod of bhvo du o h cobuo of h pl dvv of h h -h compo of h -h vco wh spc o l ( ). Aloously, h scod m cpus ch sy h mod of bhvo, bu s mo complcd. H, h ch sy ows wh m, h -h compo of h - h vco d h pl dvv of h -h vlu ( ) wh spc o l ( ). No h, f vlus () d vcos () complx h dvvs wll lso b complx. I such css, h xpols wll b mulpld by complx vlus whch wll fluc o oly h mplud of h bhvo mod, bu wll lso ld o phs shf. Th quo bov suss h ly m ( ), h bhvo mod wll b mly flucd by h fs m,.., h dvv of h vco wh spc o h l ; d l o (s ), h bhvo mod wll b mo flucd by h scod m,.., h dvv of h vlu wh spc o h l. Thfo, h bhvo of l sysm wll b hhly dmd by h scod compo wh fo hh vlu of d h dom mod of bhvo wll b dmd by h lv s of ch. Sc h moy of h sch modl lyss hs dl wh vlu lscy closly ssocd wh h dvv of h vlu wh spc o l (o loop) w hv lood myopclly h lo m bhvo h my o ply sfc ol h sho m bhvo of l sysms... Ip h Impc o Bhvo Mods To udsd d p h mpc h ch l s hs o h ol bhvo of s vbl, s usful o cosd h o of h bhvo of h s obd f ch h l o h ol o. Sc ch s s composd by l combo of dff bhvo mods, w mus lso vs h mpc of h l ch ch bhvo mod compo. Th l p of h o (of chd s bhvo o ol o) dms fco h mulpls h ol bhvo mod, h mplfy o dmp. Th complx p dms phs o h ol bhvo mod. To 7

8 ob h bhvo mod mpc, w mus dvd ch compo quo () by h cospod compo quo (): x ( ) x ( ) (7) Equo (7) mphss h ol h h fs m dvvs of boh vco d vlu wh spc o h l hv o h w bhvo of s x (). Sc h ulm ol of hs foml modl lyss s fom polcy, s mpo o compu h ovll mpc of chs by l (o loop) o h ovll bhvo of s. Ths ovll mpc qus ddo of h dvdul mpcs of dff mods. Sc h bhvo mods composd by mx of osclloy mods, xpol owh d dcy d h coffcs ch wh m uomd mplmo of h mhod wll povd mchsm o sly vsul h sul d slc h ls o loops o ch o ob h dsd bhvo... Sysm Bhvo: L Evlu d L Evco Ssvs Ru o quo (6), w obsv h h pl dvvs of h vlu ( ) d vco ( ) wh spc o l ( ), spcvly d, c b udsood h cox of pvous wo o l vlu lscy (Fos 98, 98). I hs sch Nh Fos (98, 98) susd msu h ssvy of vlu wh spc o spcfc l ( ) by smply compu h pl dvv of h vlu wh spc o h l ( ). Ths would llow o o udsd how h sh of l could mpc spcfc mods of bhvo. S (8) I ddo, w could oml h ssvy msu o sol h ffc of h ch l fom h mud of h vlu d l. Ths omlo could b obd mulply h ssvy by h o of h mud of h l ( ) o h S dvo ppdx A. 8

9 mud of h vlu ( ). H dfd hs msu vlu lscy wh spc o l o l (vlu) lscy. E (9) wh s h bsolu vlu of h l d s h Eucld om of polly complx vlu ( ). No h h pl dvv of h vlu ( ) wh spc o h l ( ) s ps h scod m of quo (6) chc how ch l would ffc h ovll bhvo of s x (). Whl hs b susd h vco lscy would b qud o udsd how sucu ulmly flucs bhvo, o pvous sch oh h ous d Slh l. (5) hs mplmd. To do so, df h vco lscy ( ) wh spc o l ( ) sml wy s h l vlu lscy. Fs, w c msu h ssvy of vco compo ( ), h -h compo of h -h vco, wh spc o spcfc l ( ) by smply compu h pl dvv of h vco compo ( ) wh spc o h l ( ), llow o o udsd how h sh of l mpcs h sy of h vco compo. S () Scod, w could oml h vco ssvy msu o sol h ffc of h ch l fom h mud of h vco compo d l. Ths omlo could b obd by mulply h ssvy by h o of h mud of h l ( ) o h mud of h vco ( ). Thd, sd of cosd spcfc vco compo w c ccou fo h whol vco ( ) d df hs msu s h vco lscy wh spc o l o l vco lscy. E () wh s h bsolu vlu of h l d s h Eucld om of h vco ( ). No h h pl dvv of h -h compo of h -h vco ( ) 9

10 wh spc o h l ( ) s ps h fs m of quo () chc how ch l ffcs h sy of h mod of bhvo of vlu ( ). Whl h oo of l vlu d vco lscs usful, o h quo (6) povds d wy o ssss how vlu d vco ssvy (.., h pl dvvs wh spc o l ) wo oh o fluc sysm bhvo. Rw quo (6) us vlu d vco ssvs, w ob: x ( ) ( S S ) Evco ssvy bhvo ( Evlu ssvy () S cpus ch sy h mod of ) du o ch l ( ) S cpus h ch h bhvo mod (.., ) du o ch h l ( ). Th cobuo of h vlu ssvy chs wh m d bcoms h m dm of bhvo wh m.. Bhvo Nol Dymc Sysms I s mpo o mo h h mhod of lyss s dscbd so f ppls oly o l sysms, ps vy smll subs of ypcl sysm dymc modls. Tdol sysm dymcs modls ol, wh vlus d vcos h vy wh m. Hc, ssum h w could fd soluo o h s vco x(), h fs m dvv of ol sysm (psd h dscpo of h mhod by quo ) would lso clud h dvvs of h vcos, ld o: ( ) [ & ( ) ( ) ( ) & ( )] [ & ( ) ( ) ( ) & ( )] [ & ( ) ( ) ( )& ( )] x & () No h h quo bov s much mo complcd h quo (). Sc h vcos lly dpd hy sp h -dmsol spc d w could w ch dvv of vco ( ) & s h l combo of s pocos o dff vcos. Howv, hs pvs us fom h dsd spbl s sul of

11 quo (6). Thfo, wh w cosd ol sysm h lyss bcoms much mo complcd. Dsp hs complcos, possbl wy o sll us h mhodoloy s o l h ol sysm of quos. Sc h ld soluos ood ppoxmo of ol sysms soluos clos o h op po, h shs obd loclly (houh lo) co b ld o h s of h sysm. Nvhlss, w c ccumv hs shocom by l h sysm vy po m ( pcc, vy m sp h smulo) d compu s vlus d vcos. Apply h mhodoloy o h ld sysm vy po m llows us o compu how ch l s fluc ch h bhvo of s. Equo () povds compc wy o ps how chs l ffc s vbl fo l sysm, fo ld sysm w could w sml soluo: x ( ) ( ) S S ( ) () wh ch ( ) fs o h poso of h sysm h lo m ( ). No h ly m ( ), h ch bhvo mod wll b mly flucd by h vco ssvy (fs m); d l m ( ), h ch bhvo mod wll b mo flucd by h poduc of h vlu ssvy d h vco compo (scod m). Sc h ld sysm povds ood ppoxmo o h ol sysm oly clos o h op po, w oly c bou soluos o quo () h hpp ly m ( ). Th sul of quo () l ms ( ) dps oo f fom wh h sysm s clos ppoxmo o h ol sysm. Hc, fo ol sysms h ld vy po m, h mpc of ch l o sysm bhvo wll b mly dmd by h fs m. Equo (5) povds ood ppoxmo of h mpc of ch l s o h bhvo of s x. x ( ) ( S S ) (5) Dsp h ddol complxy of ol sysms, by l h sysm vy po m d h cosd h mpc of h l s, w v l soluo

12 h s s o compu h h of l sysm. Whl h moy of h sch modl lyss hs focusd o vlu lscy closly ssocd wh vlu ssvy wh spc o l (o loop) quo (5) suss h vco ssvy lso plys mpo ol dm h mpc h ch sucu hs o modl bhvo. W hop h follow up sch mplm hs mhod o ol sysms c shd mo lh o s usfulss o dol sysm dymcs modls. 5. Applco o L Sysm: Th Ivoy-Wofoc Oscllo W llus h cocps bov wh vso of h fml wofoc voy modl. Th modl cpus smpl poduco sysm. Th modl mps o m dsd voy by dus poduco v h d f wos. Mo pcsly: Ivoy s h dffc bw poduco d shpms. Shpms dmd by dmd ducd by soc-ous, should voy fll oo low. Poduco dpds o h wofoc. Ad h wofoc s chod o h lvl cssy o m xpcd dmd. Th wofoc s csd bov hs cho f voy s oo low d covsly wofoc s dcsd blow h cho f voy s oo hh. Expcd dmd s smooh of cul dmd. A soc d flow dm of h modl s show blow. Th modl s composd of h s vbls, fou flows, h uxly vbls, wo xoous vbls, d fv coss. Wofoc (W) H/FTm (HFT) Poduc (P) H/F R HFR) Coco Tm (CT) Poducvy () Ivoy (I) Dsd wofoc (DW) Sls (S) Ivoy Coco (IC) Dsd Poduc (DP) Mmu Sls Tm (MST) Dsd Ivoy (DI) Dmd (D) Expcd Dmd (ED) Tm o Ch Expcos (TCE) Ch Expcd Dmd (CED) Fu Dm of l sysm dymcs modl.

13 I P S W D W HFR (DW W)/ HFT ED CED (D ED) / TCE IC (DI I)/ CT DP IC ED DW DP/ Th Jcob (J) of h sysm bov lds o h follow lo: J HFT CT / HFT HFT / TCE Th suls bov ps h chcsc polyoml d h vlus ms of l s. Aloously, w could hv w h chcsc polyoml d vlus ms of loop s. Sc hs sysm hs oly h loops: Loop. A mo blc loop ssocd wh Wofoc (W) wh -/HFT. Loop. A mo blc loop ssocd wh Expcd Dmd (ED), -/TCE. Loop. A mo blc loop l Ivoy (I), Wofoc (W), -/(CT * HFT). s sh fowd o s h h chcsc polyoml ducs o: 5 J I - J P( ) ( )( ) ( ) P() ( ) ( ) Ad, h vlus, fo h xmpl, ms of h loop s : 5 Th sd d c lso vfy h dvo of h chcsc polyoml ms of h loop s Goçlvs, Hs, Lppo ()

14 W c sly compu h vcos of h sysm us h l o loop s, l us pocd wh loop s. Th vcos v by: (J ), ; ; ; ; W c h ps h sysm bhvo mx fom: ED W I Expd h quos bov, w ob h sysm blow: I W ED Th sysm of quos bov pms us o compu h dom bhvo mods by comp h vco compos fo ch bhvo mod h fluc s. Wh hs pupos, w llow h m coss fo voy coco m (CT), h-f m (HFT),

15 5 d ch dmd xpcos (TCE) o qul (.. mohs), w ob h -/HFT- /, -/TCE-/, -/(CT * HFT)-/, d, povd us wh h follow vcos: ; ;. Subsu hm h quos dscb h bhvo of s vbls 5 I. 5. W. 5 ED. O mx fom:. ED W I Th dom bhvo of s ED() s h xpol dcy wh (.5). Comp h muds of h coffcs of h xpol ms I() d W(), w obsv h h dom bhvo of hos ss s dcy xpol, dmd by h p of complx vlus. No lso h hs smpl sysm, oly loop s ( ) d ( ) fluc h dom bhvo of I() d W(); d oly loop ( ) flucs h bhvo of ED(). To udsd how h s vbls mpcd by chs loop (o l) s, w d o compu boh h dvvs of vlus d vcos wh spc o h loop (o l) s. I h dvo h follows w us loop s. Equo () povds fmwo o hs mpcs d bls d pss h cssy dvvs.

16 6 Tbl Dvvs of vlus w loop s fo voy-wofoc xmpl. Evlu Evlu Evlu Loop H ( ) Loop Dmd Ad. ( ) Loop Ivoy-wfoc ( ) Fs, o h h dvv of h vlu d o flucd by loop (h dvvs qul o o.) Scod, loop dos o ffc h dmp of h complx vlus. I ddo, o h cs dcss h fqucy (css h pod) of oscllo. Th complx p h dvv hs dff s h h s of h vlu s complx p (b). Thfo, ch dcss h complx p of h vlu d sc f πb (o T π/b) low vlu of b lds o slow fqucy (o, lo pod.) Aloously, cs css h fqucy of oscllo, sc h complx p of h dvv hs h sm s s h s of h vlu s complx p (b). Tbl Dvvs of vcos w loop s fo voy-wofoc xmpl. Evco Evco Evco Loop H ( ) Loop Dmd Ad. ( ) [ ] [ ] Loop Ivoywfoc ( ) d d Bfo w pocd, w should cosd h mpc of h chs of loop s h vcos. Focus mly o h osclloy vlus l us cosd h dvv of wh spc o. Fs, h l p suss h vy cml ch cuss mulplco of (-/ ). Th complx p of h dvv suss duco h

17 7 complx vlu b, duc h phs l h could hv o h sysm bhvo. Sc h l d complx ps hv h sm s h phs l s posv. Loop hs sml mpc o h phs l. Icopo h suls fom bls d quo () povds d wy o ssss how h pl dvvs of h ss wh spc o loop fluc sysm bhvo. ED W I ED W I ED W I Ech mod of bhvo ( ) s mulpld by (polly complx) fco, fluc h sy of h ol bhvo mod d polly h phs l. Th suls my b s o p f w subsu vlus fo ch of h loop s. Subsu h vlus fo ch loop (, d ) d poducvy () suss h h osclloy mods m dom. 8. ED W I ;. ED W I ;

18 8 8. ED W I W c m ss of h mpc oducd by chs h loop s by comp h clls of ch of h h mcs bov wh clls h ol soluo mx (poducd blow), ccod o h sul fom quo (7).. ED W I Fo sc, s possbl o s h ch ( ) dos o hv mpc o h osclloy mod of bhvo. Th sul ms uv ss bcus loop, mo blc loop ssocd wh Expcd Dmd (ED), dos o cobu o h o of h osclloy mod, s c b s fom vlus d. Nvhlss, ch mpcs ll ss h sysm, cs h mplud ssocd wh h xpol dcy. No lso h h s of h ch s dpd o m, sul fom h mplfco of h ch ov m du o h ch loop. Th quos bov lso sus h chs loop ( ) do o mpc h bhvo of xpcd dmd (ED), whch c b s by ow of os h spcv mcs. Fuhmo, h ch mplfs h ol xpol dcy ( ) by fco of fou whl lso ch s s. Phps mo dffcul o udsd s h mpc o h osclloy mod of bhvo, s h coffcs fo boh d. A, h l p of h o (of h chd s bhvo o h ol o) dms fco h mulpls h ol bhvo mod; d h complx p dms phs o h ol bhvo mod. Cosd fs h mpc of ch o voy (I) s bhvo mod, h o bw chd d ol s suls 6. Th sul suss h h mpc dpds o m. Th complx coffc cobus o h mplfco wh h squ oo of h sum of squs of h l d complx ps

19 ( 6 ) d o h phs shf by h vs of h o of h l by h complx ps ( ). Wh m s clos o o ( ), h mplfco o h osclloy mod s v by fco of 6 d h phs shf s of π. To compu h mpc o h voy (I) bhvo spcfc m, would b qud o subsu h dqu vlu of m. Fo sc, h ch cuss mplfco o h osclloy mod by fco of.5 (sc. 5) d phs shf of ppoxmly -9 o o (sc 9 8 ). I s cssy o pocd sml wy o compu h mpc o dff bhvo mods. To fom polcy s sll qud o compu h ovll mpc of chs loop o h ovll bhvo of s, by dd h dvdul mpcs of dff mods d slc h dsd bhvo mods. 5. Dscusso Th movo fo hs pp s o povd mhmcl fmwo o udsd h cobuo h chs l (o loop) s hv o h m ph bhvo of s vbls l dymc sysms. Ou sch focuss o h lycl compuo of h fluc of vlus d vcos o modl bhvo. Ths wo follows closly h sch do sblshd by Fos (98). Ou wo dps fom pvous ffos ms of s focus o lycl suls d mphss o h mpc h fs m dvvs of vlus d vcos hv o modl bhvo, sd of vlu lscs. Th mhod dscussd bov hs h dv of oduc lycl udsd of h ol of vcos o fluc bhvo l sysms; s pcs, s poducbl; d povds sdd wy o ly l dymc modls. Scod, h mhod povds dc msu of h mpc of dff loops o h bhvo spos of h sysm. Thd, h mhod chcs msus d ums how dff loops fluc dff mods of bhvo. Fouh, h mhod cobus o udsd of s lyss sd of smply sdy s lyss. Flly, by l ol sysm vy po m, 9

20 w v l soluo h povds ood ppoxmo of h mpc of ch l s o h bhvo of s x. Th mhod lso hs umb of shocoms. Fs, soluos o h bhvo of ss h sysm qud o ob h lycl suls. Also, h dvos md h mpc of ch sucu o h bhvo of l sysms. Whl l sysms usd o dv h m suls, coscuv sysm lo xds h pplco o ol sysms. Ths sul s sd bu o xmpl s povdd. Fuh sch h mplms h compuo of vlus, vcos d h fs dvvs wh spc o h l (d loop) s d s dff ol modls qud o ssss h usfulss of h poposd mhod. As h l xmpl suss, h mhod poss chlls ms of p d vlu h mpc of vco d vlu cobuo o bhvo mods. Dsp s cu chlls d lmos, w hopful h h mhod povds usful sp o h lyss of how sucu flucs bhvo s wll s w dco fo fuu sch o h lyss of ol dymc sysms. 6. Rfcs Ebl, R.L Smplfco d Udsd of Modls. Sysm Dymcs Rvw. 5(). Fos, N. 98. A Dymc Syhss of Bsc Mcocoomc Polcy: Implcos fo Sblo Polcy Alyss. Upublshd Ph.D. Thss, M.I.T., Cmbd, MA. Fos, N. 98. Evlu Alyss of Dom Fdbc Alyss. Pocd of h 98 Iol Sysm Dymcs Cofc, Ply Ssso Pps. Sysm Dymcs Socy: Alby, NY. pp78-. Goçlvs P, Lppo C, Hs J.. Implm foml modl lyss. Pocds of h Iol Sysm Dymcs Cofc, B, Nowy. Sysm Dymcs Socy, Alby, NY. Gülp, B. 5. Towds Coh Loop Domc Alyss: Poss Evlu Elscy Alyss. Pocds of h 5 Iol Sysm Dymcs Cofc. Boso. Hs, J. 5. How o Vs G Modl L Yous. Pocds of h 5 Iol Sysm Dymcs Cofc. Boso.

21 Kmpm, C.E Fdbc Loop Gs d Sysm Bhvo. Pocd of h 996 Iol Sysm Dymcs Cofc Boso. Sysm Dymcs Socy: Alby, NY. pp Kmpm, C.E. d R. Olv. 6. Loop vlu lscy lyss: Th cs suds. Sysm Dymcs Rvw (fohcom). Mohddh, M.T A Ph T: Compu-Asssd Huscs fo Udsd Dymc Sysms. Upublshd Ph.D. Dsso, Rocfll Coll of Publc Affs d Polcy, S Uvsy of Nw Yo Alby, 996, Alby NY. Mohddh, M., G. Rchdso d D. Ads.. Us Ds o mplm h phwy pcpo mhod fo dc flul sysm sucu. Sysm Dymcs Rvw. ():-. Olv, R.. Modl sucu lyss houh ph hoy: Po huscs d fdbc sucu dcomposo. Sysm Dymcs Rvw. ():-6. Olv, R. d M. Mohddh.. Kp Smpl: Domc Assssm of Sho Fdbc Loops. Pocds of h Iol Sysm Dymcs Cofc. Oxfod, UK. Rsch, KJ. Mulv Cool: A Gph Thocl Appoch. Lcu Nos Cool d Ifomo Sccs Bl: Sp-Vl. Rchdso GP Loop poly, loop domc, d h cocp of dom poly. Sysm Dymcs Rvw (): Rchdso, G.P. Dom Sucu. Sysm Dymcs Rvw, 986. (): Slh, M. d P. Dvds.. A Evlu Appoch o Fdbc Loop Domc Alyss No-l Dymc Modls. Pocds of h Iol Sysm Dymcs Cofc. B, Nowy. Slh, M. d P. Dvds.. Th Os of Busss Cycls. Pocds of h Iol Sysm Dymcs Cofc. Al. Slh, M.. Th Chco of Modl Bhvo d s Cusl Foudo. PhD Dsso, Uvsy of B, B, Nowy. Slh, M., P. Dvds d K. Byoum. 5. A Comphsv Evlu Alyss of Sysm Dymcs Modls. Pocds of h 5 Iol Sysm Dymcs Cofc. Boso.

22 Appdx A Th Poduc of Complx Numb by Complx Expols To udsd h mplco of mulply complx xpol by complx umb, cosd h follow xmpl: ( c ) b d w c w h xpol s: ( c d ) c d d by dfo d cos( d) s( d ) so w c w h quo bov s: c ( b )( cos( d) s( d )) Mulply by ( b ) d df ( φ ) b b s( φ) w c w h quo bov s: b c c ( )( cos( d) s( d) ) ( b ) cos( d) s( d ) [( cos( d ) b s( d )) ( b cos( d ) s( d ))] c b, w obsv h cos ( φ ) d b c b b ( b ) cos( d) s( d ) cos( d ) s( d) b b c ( b ) cos cos( d) s( φ ) s( d) b [( φ ) ( s( φ ) cos( d) cos( φ ) s( d) )] Sc cos( d φ ) ( cos( φ ) cos( d) s( φ ) s( d) ) d s( d φ ) ( s( φ ) cos( d) cos( φ ) s( d )), w ob: c ( b ) cos( d φ ) s( d φ) c ( d φ ) b b c d b [ ] Thfo, h complx umb mulply h xpol cobus o h mplfco wh h squ oo of h sum of squs of h l d complx ps, d o h phs shf by h vs of h o of h complx by h l ps. Th vs of (x) s dfd h vl π π < φ < b. Th vs s vlu of o wh x s o; d s posv (v) vlu wh x s posv (v).

23 Appdx B - How loops fluc sysm bhvo To udsd how chs loop s (.., h sh of fdbc loop) fluc sysm bhvo, w follow dvo loous o h o sco. Th bhvo of ch s h sysm x () s dscbd by quo (), whch dmoss h h bhvo of ch s s flucd boh by vlus ( ) d vcos ( ). x ( ) Whl s mo commo o w h chcsc polyoml (P()) d vlus ms of h l s ( ), s lso possbl o w hm ms of loop s ( ). Loops, d h s, my b mo comphsv (b) wy o dscb sucu, sc modls of dcd o clud (o xclud) loops bsd o h dymc hypohss h hy blv mpo sysm. Sc w ulmly sd how sucu dvs bhvo, udsd how chs loop s fluc sysm bhvo my b mo ppop h loo how chs ls fluc bhvo. To cpu how loops fluc sysm bhvo, w h pl dvv of ch s h sysm x () wh spc o s loop s. Thfo w pl dvv of quo (), chc h bhvo of s x (), wh spc o loop ( ). x ( ) [ ] Whch fo l sysms, w c w quo s: (B) x ( ) (B) Equo (B) suss h fo ch compo (wh o ) chc h bhvo of s x (), h cobuo o h ch bhvo of s x () du o h ch loop ( ) s composd of wo ms. Th fs m cpus ch sy h mod of bhvo du o h cobuo of h pl dvv of h h -h compo of h -h vco wh spc o loop ( ). Aloously, h scod m cpus ch sy h mod of bhvo du o () m, (b) h -h compo of h -h vco d (c) h pl dvv of h -h vlu ( ) wh spc o loop ( ).

24 Wh hs suso of fd h chcsc polyoml ms of h loop s, Fos (98) xdd h suls of l ssvy d l lscy o loop ssvy d loop lscy. S d E (B) I ddo, w c xd h cocp of l vco ssvy d lscy oducd h pvous sco o loop vco ssvy d vco lscy wh spc o loop o loop vco lscy. S d E (B) Equo (B) povds d wy o ssss how loop vlu d vco ssvy (.., h pl dvvs wh spc o loop ) wo oh o fluc sysm bhvo. I pcul, w c w quo (B) s: x ( ) ( S S ) Loop vco ssvy of bhvo ( (B5) S cpus ch sy h mod ) du o ch loop ( ); Loop vlu ssvy S cpus h ch h bhvo mod (.., ) du o ch h loop ( ); d Th cobuo of h vlu lscy chs wh m, bcom h m dm of bhvo ov m. Loop vlu lscy cpus chs h mod of bhvo, h s, msus whh h s wll hv fs o slow owh, dcy, o oscllos. I u, loop vco lscy cpu chs h sy of h bhvo mod, h s, msus h mpoc of h bhvo mod o h ovll bhvo of h s. To compu h vlus ms of loop s ds dcd o Fos (98), Kmpm (996), Goçlvs, Hs d Lppo () d Kmpm d Olv (6).