chaos in dynamical systems by the poincaré-melnikov-arnold method

8/14/2019 Chaos in dynamical systems by the Poincar-Melnikov-Arnold method

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C h ~ o s in ~ n a m l c a l S y s t e ~ by thol 'oin"oco-Helnikov-Arnold Method

Je r ro ld E. Marsden

. ~ . Hethods for proving the exintence o f alaon in tho Senseof P o i n c a r ~ - B i r k h o f f - 5 m a l e horseahoes are presonted. We shall COn-. centra te on expl ic i t ly ver l f Jable resul to that apply to Ilpeci f ic; e ~ A c p l e s such as the ordinary di fferen t ia l equations (or a fQrced,pendu" ." , and for 5uperfluid ~ I e and the par t ia l dif fc rc l IUa l t' ~ P I l y t to eq ... t ion (1 . 1 ) oneproceeds as follo. ,s. Let x ~ , ~ I so Cl.l. becomes

2.


2/7

(1.4 '

fbc ho.... . ,Unlc orb i t . lor C. 0 aro colilput .d to bo g1v .... by

ill [ ~ I t~ ( t '-1 )an ( s hh t '

12 secht U.5,

nd Ona has :j1 2o l ~ , 2 > - COD 11.6)1I,,"ce 11.] ' 91 vr.

T [ t COil Idt dt-T [ [1 u., , ,htt- to ' coo ...t] dt -

Ch,,,,,,ln9 v.uln ' . D,ld uuln9 the fAct thAt oN:h i. oven "lid DJn is odd,w" 9ul

Tho integral i, eveluated by residuGs.

( l . l ,which cleady hel .1.1'10 zaro

1

I.

l

.. 2. Proof o f tho Polncarii-Holnlkov ThClOre.. There are two. convenient ways of V 1 5 u ~ l l z i n 9 tho dyn .. 1CS of (1.2" One C.nIntroduce the ~ l n c a r c ....p Pl,R2 J112,whlch is the tille T ....pfor (2.1, ater t lng e t t i ~ G. For C . 0, tho point Xo and tho

~ : " " , , ~ ~ n ; ; : : : ~ 1 : r : a : : r l : : t ; : ~ : ~ ~ ~ ; : " : : r ~ : : i r , , ~ n t tseddle. Xt for t > 0 , 500.\11, /lnd wo /lro interested In "hetheror not the stable and unstable ....n folds of tho point Xc for the

~ ~ p pO in tersect t rAn9verually l i f th is holds for one " i tI holds ~ o r a l l a) . It 50, we soy 11.2' a O.

to atudr 11.2' iD ty look di rect ly a t tho susJlI2 .. 5, where 5 Iltands for the circ le ,Th.. secolld waypended system onele"",nts of which11.2' becomes the are re ' l" rded aD tho T-perlodlc v ... l .bl .. e. Thenautonomous U S I ~ n d Q d systua

x o x, (fllx,O }o 12.1)Fro'" thlo point of vi ... tho curve

is periQdlc Qrblt fQr 12.11, whose IItable and unstAble "",nUQlds. s uWoIYO' Dnd WOIYO' /lro colncicJnnt. 'Or C > 0 tho hyperbolicclQsod orb i t Yo perturb ) to e Mllrhy hYPllrboltc closod orb i twhich hn . "tllhl" nnd ulIst.,blo .... "1101"" W;'CYrl "",I W ~ ' Y r I f

I w'\." all


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d.J,t"nc:e froll thl hOlloOCllnlc orbi t . IfOn W'(y I that 18 ap in tegra l curVe ofe t

IIIxel t , to ' , t , Is tho curvethe G u ~ p o d c d SY5t" . (2.11

and has an in i t la l condition x8 CtOto ' which 1s the perturbationIIof WoflO' (the ,1_ t 'ell 1n the DOnia direcUon to thohomocl n c orbit . then "0IX8tlto.to" ~ 9 u r l ' th i s n o ~ l dlGtanc:e.But

F ~ o ~ (2.2), we getT- J II II +' 0to ' 12.31

Since IT. t is c-clolle to I t - to ' Iw,lrormly as T .. _ . andI 0dCIIO + ell I IxcC t , o" U 0 uponontli lU)' as t .. - , And11 0 .110' D O , 12.1, becomes

SIIllUarly.

T , ...... 11. 11,, "f" .' nnlIolo"C-:l.1 "L 0 0JI'UIIt'l 1',$ .. _, wa

tr1I0 1cC-S,lO))tt /1I0.1I111;ll-lo"t, tit f 01 . 2 ,

-9

1I1I..,II)nll T nndT,t; .... 11IWI,

gilt

l AU' hA(ldJuIj

IhAtC2.4,

1I111'"c'r,. )) , II,,".1 (2 . > ' , lInd

..1. . ~ : .- t -- (UO,1I1 ) liCt-to" t ' d t + oce 2,12.6'

t followlI that if I to '. u IIxe l to . to ' has Xc I to ' to 'point iI(O) a t U.... to.

ha . a sUiple . e ~ o 1n tJlIe to ' thenmust lnteraect tranavereall) , near the

~ Since


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,1 :> Ii Il/2 C O h ( : ~ ) U.SIlIa,{i

and [ is SIOaH.A n o t h e ~ in teres t ing e.D.ple. due to MOntgomery [1984] concernsthe equations for superfluld llle. These are the Leggott Clquatlonu

and 0 shall confine ouull i l les to t h e phase far lIi1l1pllcity (seaMontgomery'S paper for additional resu l ts l . The equAtions are

4 - - JRill 206 - r Y i ) ~ - r.lyn 0111 wt. + r2 13.61IIln 201

Here II to the sr i n. 8 the ongle describ ing the o r d e ~ pacoJIDetell'And 1. X. Dce phYGlc . 1 cOII"tanto. The holllDcllnic orbUa for 0 Dre given by

iii - 2 tan 1 (e l i l t ) fI 2- 12

lle t 1 ltl i t I + e Ull t

On" calcul. , tcs u"i"g I l.61 "nd (J.71 In 13.1' that

no tha t ( l .6 ) hall ChAOIi In tho .ense o f hor" . ho" . 1 f

And If 1" " ...11.

ll . :> (:09hr .II w III

13.71

( l .BI

Cl.91

4. ~ : ~ . o n to 1'01:'/1.. TlII-,e 18 II YC'lIion o f th" 1 ~ . l l I c l l r c i -Helnlkov l h ~ o o c e m applicablo to I'UC's thllt Is due to 1101""5 lindM . . .d,," (1901). 0"" basically III II 1 u . . the {ormula (1.11 "hell'''

1 l

Xo K Xl now 1s replaced by the symplectic pa ir ing betveen Xoa"d XI' 1I ...ever. there are two new di ff ieul t i . in Addition tost n, ' . r'" tecl tn ical Dnalytic prob1e",s that Oll'lao .. th PDE's. Thef l r 9 l 111 tlt.at thel'o til II se r ious probl" . with uBOn"nces. Thesec ..n be deal t wHh using the aid of d""'plng - - tho und_red caSewould nccd on I n f i ~ l t e d i " ' e n ~ i o n a l version of " ~ n o l d dif fus ion - scc 16 below. Secondly, the problem io not reducible to t"O dimen(o.ion:;, tho hort lhoe involves u11 the modus. Indcclll, t.he highccIl1Od"s do SOL'll to bo I nyol ved I n the phYIl ica 1 buck Ung proc"" se9 forthe bo ... 1 Ildel discussed next.

,. rDC m a d ~ l for a bUCkled Corced beAM io;; + . . . . + r ... - .e I: [W.i1d&)W. - 11 COli wt -6 . (4.11

"horo wl&.tl 0 Z I describes the def lec t ion o f the be . . ., - 3/at , - 3/3& and r e... Are phVoleal constant . . FOrth i D cas". the thcory oba . t ha t if

(DI a2 < r < 4pJ ( f t r s t ROde 10 buckledl(bl ,2 a2 (j 202 _ r l , I w2 1 - 2.1 C r ~ o o n a n c e condition,Icl t n(r_1I2 , [w)i > ---;::& cooh lJr -w 2 (tcanovecsal zero" fOr K(to"Id) is > 0

and t 10 " . U . then 14.1, h"" horseslto" ExperhAolito o r F. Moon a t COrnell which ohow ' haas In a focced: buckled beam provided the lOtlvation which led to tho study of

(4 .11.TIlls kind of resu l t has r ..centlv been ugcd by Sh"orod and l landen

(t9011 for A study o f chaos in v . d,-r ", .U's f lu id Is ... Sl" . , rod ' .Ittclu,'c I thl' IU rroC'c:cdtllfJ .) and hy '.IO I, RAIIII .. lind Horrl ;un forSf'. i lon '(IULlllono. For c ttl,.lc.", In the d Alp,.d, lon.:cd i i nr.-( .onlonCfluiltlon our .... .. ch ..o t i c t r o & l w ~ i t . i o n ~ ... h .... n IJrr .. th.c,; 4lnd kink-nut .k' p.lJ r:. tlnd In tht:: U.njdnlau-Ollu Cflu.tt'on One C.ll1 h vc: Cholotlc

t ~ n " d t l o l ' " h.,tween uolutions I th dl ffcrcn t n ....honl or , .. le .AutOllf lN lWI lIi, .11 tonllU_" ~ l . t f " I D " . t'cJr U"",11 t I lnt .\U "yr.t('m-,. withlwo d ' C J ~ ~ , ~ ; o l f ; ; ; ; ; ~ l u m . i i ~ ; ~ : - ; - .,.,,'lir.rncl,r. [1'' '1.' '] "how how tho

""lnll" 'l1 .. 11,,,0.1 .....V be u.ccl to I,m"., thn cxl"lr,u:L- u( hor:lN.hOP-1ion '''9V fluclacoo in two degree o f fr.,eclo . n""rly Inte'JrilblcGyUl'ms. 'n . c la ss t system" studied "lOve II II.lmll tonl"n o f the fOml

a


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j '.1 : ,

21J('l.P. 0.1) F(q.p) .. GU) + Ill ( ' l .p.D.lI ... Olt ) (5 .1)whel'o (I I .U al'o acUon " '918 coordinAte" for the asci Hotor GG(OI - O. G' > O. I t ie .sDuaod t h ~ t F h A . a hOmocllnlc orb i tl f l t Iq( t ) . Pl t l l and th4t

( t , - [ (F,I I ) d to I- (5.2'Ithe integral taken along 1&lt.to', nt , I , ) has . ~ p l e zeros.Ulan IS.lI ha. hor5eshoe9 on energy surfaces neal" the surfacecorcc"pondlng to the hoaoocUnic orb i t and 5 ..... 11 I , tho IlOrlicshoClsoro taken ro la t ive to a Poincare s t robe . to the osci l la to r G.1101 .... " and Marsden 198241 a lso studlcll the effect o f posit ivoan" ne9ative da...,,11I9. " ' O ~ B r o s u l l ~ arc re la ted to t h ~ t in 2$lnce ono c.an olten re .uce a two " .. ,r"o o r (reedoDl lIa",Utolll"" "yst" .to Gone dU9ceo of freedom forcod a v u t c ~ .

For ;;Ome :y ;t .. 9 In "hic:h tht' v.ltt . l b l ~ I I do not SI_.lt "9 In 5 . 1,such .19 l nc.,rly synnc:tr ie heavy to"., one n("cds to explo i t It. f8ym-l Iu.."lryof the system and th in C'OIft,l'c.llc:J the sltU,.'lt.ion to SOlD.c . tcn t . The gc:ncrlll theory (or th is J5 ' lIven In lIoll11e and M.lesd"n{a90l] .lin" .. ls apl>1it'd to S'IOW t lw exiGtence o ( hol'''''5IlO''5 in then , , ~ r l y s y ~ e t r l c heavy tOPI see also so=o c lose ly related resu l t st Zigl in [1980.11].

This theory has been used. for example by Koil le r and coworkersIn a number of recent repr ints on vortex dynamics (Koiller andPinto de Carvalho [1983] seems to 1m the f i rs t to 9iva ,) correctproof of the non-inLa. ,rabll i ty of the res t r i c ted IoUI' vortex probIe . _. see S7 billow). There have ..Iso b ."n recent ap"licolUon9 tol e1rn,,,,,lcII of 9"lIer . l re lAt ivi ty fihowl"9 the ed ,a , : c nf horoe"haws In 1I1anchi IX IIDdell:. See 0150 Krlshnolprasltd [1901] furInto, .."tl"" ",1.llcoltlunR to I l - n l ' l " n p ~ c e c r " C t .(, 1I .. d nIH""I"n. IIrnold [' Jr.,,) "xt. .. I ... thn I'oln.,.u{..""I . ikoYlh.-ory"i'; "ysteJOs wilh 5C'v"ral .... 1'(,..0 oC [r,,,,.I0,,,. In

th is car-a the t l 'an" iYCf >C hotooclinic IJI,olllifuJd:. lire b a a ~ d On kAM to r iaud al ia . the poss ib i l i ty of chlloUc dr.i t t (roo> One tor ... to i>nolhcr.This dri f t . nov known .liS Arnold d l t u ~ i o n is a b a s i ~ Inqrc&llent Inthe stuely of cho>05 in lIamU tonlan rHo: . . . ( ..ee for Instance.Chlrikov [1979) and Lichlenberq and Lieber"",n [198l] "nd referencestherein). Inste"d of a aln91" Helnlkov fWlction. one JlI)w has atlelnlkov vector given 8che ...Ucal ly by

. ~II .I

9

.It f.l

(110 .111 ) dt jII - hk ' l ) d t 16.1,"hl'o Ik al'o l n t e 9 r a h (01 ' tho uupecturbc>d (co .pletoly in te9rable 'UystCA and where II now depends on to and on anglos conjugate toI I ' In ' One now requiros II to have transversal zeros In thevector 8enSe. This resu l t was 91von by Arnold for forced systems" .... vos exte ..... d to the autollonuus case by 1101....5 and H.nsden[1982b)". [1903].

1'hcse r,,,,ui ts apply to "rut"m such as a I"lnd"lwa coupl . d to11""".,,1 oscl l l a tors And t i l ... llY vnrtclC problem. I t has al so bt-onu'icd in I)()wcr' ~ v u t ( ~ a U l by Sol.'AI, H rndon Bud YllrdiY41 (J?O.,). bulldin9on Lhe ho n,,,::hoe case t rca ted by KOpc 11 and W ~ l h b u m [I J O ~ J. Seoa lso tho work of SalaDl and Sautry ..eporto" In these p r o e ~ ~ J i n 9 ~ .

1""1'0 havo: born a n...wer of athol' dlrf 'ctions o f research on tht'sotcchnlques. For exampl.,. Grun"l"r [1981) developed a lIIultidi ."ndona lvcrsion appUcable to the spherical pendu1U11 lind Creenspan and lIol ...s[1901) showed how it can be used to study subharmonlc bifurca t ions .7. F ~ p o n e n t i a l l y 5 ...11 "elnlkov Functions. Thore 15 a se . ousdl tf cu l ty thi> t ar sell whe" ono u .es tho He Ini kov mo thod noar an

e l l ipt io fixad point in " lI""'iltonliln syot" . . The di ff i cu l ty 1acloGely re l t d to the &lit f lcul ty I 'olnc"I''; encountered in tryln. ,tn prove nonintegr. lbU Ly 411" lho dlvl rgolRce o f Gerl"s e.p,ulOlion"lIo.lt occur In Ihe reDtric:tc:" J ho,'y "roble. . . Ile"r cUII>Llc /y in t s ,O,lO C . t ~ J homocJ l n l e orh t tn in nottn., l ron'I:'1 "'nd a r t ~ r G l c ~ m p o r l lr. -urn"u 'J lco h l(l A Ju UI u t I lU.lly lclty 111..1 " '''r'ttl., onel l l . l t f l lY' ' ' ' r turhll t loll lhAt 1" modrlle" Ity thn (ollnwl"., v ..r l t lon of 11.1).

"ln40- r. ,-o,,{1 :.) (7.\1I t one just bl indly COlD{lUtcll H( to ' OnO finds rro


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; I ' :"ItO.EI l ' 211 a e h [ ~ ~ J cos ( ':01 11.21

while t h i s hAS . ~ l tho proof o f the P O l n c . r e - " e l n l k ~ ~ / 2 etheores i. no longer valid . i nee I t o , e l i s now o f ardor eand the error anolysi. in the proof only gives e r rora o f order (2.In f ac t no expansion in powors of C can detect exponentially G ~ a l lto ,_ l iko 0- 11/ 2. (This i s the sor t o f dl f f i cu l ty t ha t see .. tooccur in the paper of z ig l in [1980b) on the four vortex probleQlsee also Sanders [1982].)

Reeent work o f lIolmn. H"rsden and 5cheurle all.lI to show thlltIndeed 11.11 ha:l horseshoes lo r s .... l1 C. The 14"0 is to exp"nd.xpression5 t o r the stOlbl0 and unstable manifolds In a Perron typeser i cs whoao tOI'lll5 or" oC ordor Cko-It/2c. do so , the extenslonof the system to c o ~ p e l x t ime plays a c ruc ia l ro le .

One can hal'' ' tha t f ouch rt '"ult l for (7.1I can ' c l Iy bf prov"n,then 1t 'y ... , ,..,5"lbl11 lo relurn lo I 'ulncoln;'o lll')() rk and cnml , l"to thn "''JU.., . t8 he l e r t unUnlohed.

{ I ) I' 11. 1I/1111:1.-f./IIJlK, J .r.. HllIISIII;" .I01 rr ."lnn I n ~ s . ~ ~ o n of1 ' 1 l " ~ . ll:t;t; (La A,.p\."oIr).

r .1 . Varal y". II .no 1do l'Owor S), Jt"m:-tT;iii)bl.[2) V. IIIUlOLD, In5L.,b lllt of Dr"""""", 5rs t ... l th Sr.veralDcqr"clI of Fr .edo . IJokl. Akad. II .uk, SSSR. lS6(1964) I'P. 9-12.[ l ) B.V. cllrnuov. A Universal In9 t . Jb iHtr of Many-Pllllen9lonal

O l1lator 5yst"."" ( 'hysic . Ri ports. S2(19791. W 26 >-179.[4] 5.N. CHOW, J.K. I ALE, and J . I1III.LET-PARET, An ExalllPle o fBifurcation to l Iorocl lnlc Oc-bit9, J . Diff . E'fns., l7TI9liol,

pp. 1 ~ r : j ' 1 l .[S) D. ( ; t " : I : N ~ ; I ' A N .....1 P. 1I01 ... s. Suhl . ""onlc hUurClI tlons And

, , ' l n l ~ o v ' S Huthod 1198]), pre'print..CKIINIlI.tR, TI.elll., Universi ty o f IIonh Carolln;, (1901).6] J .

(7] J . CUCKEHIIElHR and P. HOLlIES. Nonlinear OscUla t ions , IlynamlcolSrstelllS, and oifurcation o f Vector Fields, Springer. AppUedHath. Sciences. Vol. 42, 1991.

(a] P. 1IOI.llF.S, IIwragln2.J'. ' ' ' Chaotic MOtions in Forced OscillationlOr"H J . on AI'P1. tulh. 19. 69-80 .. id 40(1900). PI'. 167-160.

11

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., 1111, , . , , :[9] P.J . II0U1ES and J .E . HIIRSDDf. A Pae t ia l D if f e ren t ia l Equoltlonwith In f in l t e1r 11 ...y Periodic Orblta . Chaotic Osc i l la t ions ofa Forced B e ~ , IIrch. flat. Hech. lInal. 76(1981), pp. I lS-166.[10] P.ol. HOIJtS and P.E. KARSDN. lIorse9ho In Perturbations o flIamiltonillll Syste" ' ' ' i t h 'nora Oograes fa Fraedolll, COII\ftI. Hath.I 'hrs. 02(190Jal . Pl'. S2J-S44.[11] P .J . 1I0UU:S and J .E ARSDt:N, tlo lnikOV's ... ethod and ArnoldDiffusion for Perturbat ion t Int"grAble lIaolilton an 5r$t",,"5,

J . KaLIl. rhys. 231l9B2bl. pp. 669-615.{12] P. HOLMES and oJ. K A / t S D ~ I f , lIor5""lIo ~ ' . l c M t l In th 'norn-nrqr ... _~ " ~ ~ ~ ~ w t n ~ h ' " f l f 11:1: 'rr.lllu. Circuit l l and ~ ; y 9 t c m ; ; - -

2 ' J I 1 ' J l I ~ ' . I'l" 7)0-746.[II;) S. KOZIIlV, . 1 ~ 1 I _ n ~ ' ~ _ , > - ' . . t L > . 1 : " ' I 1 ~ I . ~ . ~ . Run. I1MhSurv,yu. ;lUII',IIIlI, 1'1'. 1-/(, .h7) I' .S. KIUf>II"lIrIIllSIID . ~ : l , .. . ' : ~ ~ ~ ~ . J l ' . . ~ J ~ ~

SI 'ac crar t, 190J Ipn l , r ln t ) .' I to] II .J. LICIITIIII1lC and .1\. L1t:UERfIllNN, 0"2ular and Stochastic~ ~ Sprill'.l:,,ow H U Suc. , 4i1l900Ol', ,,:181.---

12


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I; .: : : ' \ .

[26] S.L. ZIGLIN. Nonintegl:ablUty o f 4 "roh '", on tho IlOlIon ofFoul" l 'Oint VOl:ti Cl s. SOY. Holth. wk1 21 t190Ub). I'P. 296-299.[27] S.L. Zrr.r.ltl, A'-Anchil o f S o . t l o ~ . . ? : l ~ ~ ~ ~ . ' .

in llluaU tonilln Syntclln. Ilokl".IV IIk.,,1 .,uk. SSllll 2S"1I1\1Ulal. .I'P. 21i-2 1.

[2n] S 1 ZIGI.IN. ' . . : ~ . : : l 1 s ' . . ~ . ~ ' . . f " I ' ~ , , C _ ~ I I ' ' r I ~ E . : : _ ' .. : . . . I . . ~ l " . i ~ ' ' . : . . ~ . L E : - ~ : : . V E _ l L . ~ ~ . ' ~ i l : l . : ~ ~. .l ... :: l2 '1< . '; o f l 'r .. dom, ' r lkl. H. t l l . ~ I i > k . ~ . . i64- >66,

Lr.l l IG. lUI J 1I1'I,i. Holth. Hach. 4S(1901bl, 1'1'. 411-41 J

1)

~-

legal Ilot IceThis report w ~ s prepared as all account of work s ~ o n s o r e bythe Center for Pure and Applied M ~ t h e m ~ l l c s Nellher theCenter nor the Department of Hathematic5. makes any warranty

e ~ p r e s s e d or Implied, or assumes any legal l i ~ h l l i t y orresponsibility for the accuracy. completeness or usefulnessof any injormation or process disclosed.

chaos in dynamical systems by the poincaré-melnikov-arnold method

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